16 The First Reckoning: What Did You Make One?
Arithmetic with Nisaba
16.1 Before You Begin
This appendix is for people who have learned arithmetic before, people who were once told they had learned it, and people who have spent years arranging their lives so the subject would not notice them.
The story that follows attempts to teach rather than merely summarize. Calculations appear when the characters need them to count, allocate, or challenge an account. After the coda, four “reckonings” name the ideas the characters discover, work through the arithmetic, and provide checks for your own understanding. Confident readers may skim that review and return when they need it. Readers rebuilding their confidence may find it better to work through it in order. Nothing in this chapter rewards speed.
By the end, you should be able to:
- distinguish a thing, a quantity, a number, a numeral, and a unit;
- compare quantities with equality and inequality after checking their units and boundaries;
- explain place value in decimal and sexagesimal notation, including zero as a count and as a placeholder;
- interpret addition, subtraction, multiplication, and division in context;
- use signed quantities to describe change without confusing a change with a final amount;
- connect fractions, decimals, ratios, rates, proportions, and percentages;
- distinguish exact results from useful approximations;
- estimate before using a machine and check whether its result answers the intended question;
- interpret exponents as repeated multiplication; and
- factor a positive integer into primes and state what the Fundamental Theorem of Arithmetic guarantees.
16.2 Act I: Six Circles of Stone
16.2.1 The spring reckoning
At first light the sheep came through the western gate of Reed Gate in a river of dust.
The settlement stood on raised ground between a branch of the canal and a mud-brick storehouse larger than any of its houses. The gate that gave Reed Gate its name had been rebuilt from bundled reeds so many times that no one knew which gate had named the place. The storehouse was different. Its thick walls, plastered bins, sealed jars, and shaded loading yard belonged to a temple household in Nippur. Barley obligations from the surrounding fields entered there; seed, rations, and measured advances left under seal. Wool waited there for the road to the city. The people of Reed Gate did not own what stood behind its doors, but almost every household depended on something that passed through them.
Lamassatum already waited beneath the loading awning with an empty jar and a clay token for barley owed to her household. She counted the people ahead of her twice, although there were only three.
Re’u the shepherd entered the gate with more of the flock, then stood to one side with his sling looped at his belt and his waterskin hanging from the opposite shoulder. The morning was cold enough for the woolen cloak he had wrapped twice around himself before dawn, but the day would be hot before the last animal was through. Spring made promises in this country that summer collected with interest. This year the canal had already fallen from the dark line it usually held against the bank. Re’u had noticed. He did not yet know what the difference would mean.
The temple’s order named six groups of sixty for this flock. Re’u had helped select the animals from the district flocks before dawn. The gate count would establish whether all of them had arrived.
He moved one stone from his left hand to his right for every sheep that entered the fold. When his right hand filled, he dropped the stones into a shallow bowl and began again. Ten stones made a little mound. Six mounds made a circle. When a circle was complete, he set one larger stone beside it. The scribe Ili-iddinam called a group of six mounds šūši, sixty. Re’u usually just called it a circle.
The flock pressed forward. Re’u watched backs, ears, feet, and the dark seams between animals. The count could be lost in any of them. One ewe turned at the gate when her lamb cried behind her. Re’u caught the ewe with his knee, drew the lamb through with one hand, and did not move a stone until he had decided which animal he was counting.
Ili-iddinam, sat beneath a reed awning whittling a new stylus with a wet clay tablet across his knees. He had arrived from Nippur for the reckoning wearing a clean fringe and an expression that suggested the dust had breached an agreement. He marked each completed circle while Abum-ilum, overseer of the storehouse, watched the gate and the tablet in alternation.
Abum-ilum held the storehouse seal and answered for whatever entered or left its doors. Ili-iddinam was not his household servant. He wrote the account that would go back to Nippur, where a loss that could not be followed on clay might become Abum-ilum’s loss whether or not he had caused it. The overseer could order a jar opened or an animal transferred. He could not make the tablet say that the jar remained sealed.
Lamassatum lifted her empty jar toward Abum-ilum. “Before the day grows hot?”
“After the flock is witnessed,” he said.
Without looking up from the clay, Ili-iddinam added, “And after the storehouse seal is witnessed.”
Abum-ilum looked at the young writer. “Yes. After that.”
“Five,” Ili-iddinam called after the fifth large stone.
“Five circles,” Re’u said.
“Five sixties.”
“Then write five sixties.”
The young man looked wounded. “That is what I am writing.”
Re’u let the sixth group pass and set down the sixth large stone. The remaining bowl was empty.
“And six šūši make three hundred sixty exactly,” said Ili-iddinam.
Abum-ilum nodded. “Six sixties of sheep from the canal district entrusted to Abum-ilum at the spring reckoning.”
Re’u knew the number without needing the tablet. He knew more than the number. He knew which ewe favored her left hind leg, which ram would strike if approached from the front, which yearling had learned to nose open a reed hurdle, and which three animals would remain close to water even when better grass lay beyond the ridge. The tablet knew none of these things.
Erishtum came from the wool shelter carrying a basket of cords. She was the widow of Re’u’s older brother, Ilum-bani, and had remained in the household with her two young children. Her son, Ahu-waqar, approached every new task with grave determination and usually misplaced one necessary part of it. Her younger daughter, Amat-Sin, had not yet learned that an adult’s silence might be intended to end a conversation. The wool, the sealed jars, and the measures of barley passed through her hands often enough that Re’u no longer remembered when he had begun asking her before promising any of them. He occasionally remembered after promising them. Those were different conversations.
She counted the six large stones, then looked at the flock.
“All of them?”
“All that were entrusted,” Abum-ilum said.
Erishtum did not answer him. She asked Re’u, “All of them?”
“All.”
Only then did she carry the cords inside.
16.2.2 The bronze girl
The shouting began beyond the outer wall.
At first Re’u thought a dog had worried the flock. Then the sheep nearest the western gate moved away from it together, not in panic but with the grave unanimity animals sometimes displayed before a storm.
Something stood where the path met the wall.
It was shaped like a girl not yet grown to a woman’s height. Its garment fell in simple folds to its ankles, but the folds, the feet, the narrow hands, and the face were all one bronze body. There was no pin, seam, hinge, or opening that Re’u could see. No household he knew possessed so much bronze. No temple official would have allowed it to stand unattended in a field.
The figure stepped forward.
Bronze should not have moved that way. It should not have moved at all.
Abum-ilum dropped his staff. Ili-iddinam pressed both hands into his tablet and spoiled the last line. Someone behind Re’u said the name before the figure spoke.
“Nisaba.”
The name moved through the people at the gate. Lady of grain. Mistress of the reed stylus. Keeper of accounts. Some knelt. Some backed away. Erishtum remained standing, although she tightened her grip on the basket.
The bronze figure looked from the people to the six large stones and the emptied bowl.
“Nisaba,” she said.
Her mouth moved only a little. Her voice did not sound as though it had traveled through a bronze throat.
Abum-ilum recovered his staff and his office at approximately the same moment. “Lady,” he said, “this flock belongs under the protection of the storehouse.”
Nisaba looked at him.
“The flock,” he added.
She turned back to the stones.
Re’u was relieved to discover that a divine visitation could also leave an overseer uncertain where to put his hands.
Nisaba crouched beside the six circles. She touched one of the larger stones, then one of the small ones. She separated three small stones from a mound, returned them, and touched the ewe nearest the gate.
“One,” she said.
She touched another ewe.
“One.”
She touched one stone.
“One.”
Then she looked at Re’u.
He understood the question before she found words for it.
“One stone for one sheep,” he said.
Nisaba placed a stone before the ewe. The ewe stepped around it. Nisaba moved another stone before another sheep. Soon a line of animals had passed while a line of stones accumulated in the dust.
“The stones are not sheep,” Ili-iddinam said.
Nisaba looked at him, waiting.
“But there is one for each,” he finished.
She set three stones in a row and waited until three sheep stood opposite them. With one finger she drew a line from each stone to one animal.
“Do the lines mean to say that they are the same?” Ili-iddinam asked.
“No,” Nisaba said after studying the lines. “Only that there are as many stones as sheep.”
It was the first complete answer she gave them.
16.2.3 The man beside the canal
By midmorning the lowest watering cut had clogged with reeds. Re’u left the crowd at the fold and followed the canal path with his staff across his shoulders. Sheep could wait for gods and officials to finish speaking. They could not wait long for water.
An old man sat where the path divided beside the cut. He wore a patched garment of no cut Re’u recognized. He was older than anyone Re’u knew how to place. His scalp was bare except for a dirty white fringe above his ears, and a mangy beard hung in thin knots from his chin. Both eyes were filmed white and fixed on nothing. A stick lay across his knees.
Before Re’u spoke, the man turned his blind face toward him.
“Re’u.”
Re’u stopped. “Who told you my name?”
“You did.”
“I have not.”
The old man considered this. “Then you have not done it yet.”
Re’u cleared the reeds from the cut while the stranger listened to the water begin moving again. When Re’u finished, the man held out one hand without asking where the path led. Re’u put that hand on his forearm.
“There is a bronze girl at the fold,” Re’u said.
The stranger’s grip tightened. “Yes.”
“You know her?”
“I remember her.”
Re’u led him back to Reed Gate. The man’s stick found the ground before his feet did, but he knew when the canal path narrowed and where the storehouse wall would return the sound of their steps.
At the fold, Re’u gave the introductions the stranger had not asked for. “Abum-ilum oversees the storehouse. Ili-iddinam writes its tablets. Erishtum is of my household and knows what the wool and jars contain better than I do.”
With each name, the old man’s face changed before the named person spoke. It was not surprise.
“You look as though you know us,” Erishtum said.
“I do.”
“We do not know you.”
“You will,” he said, and his smile faded.
He offered no name.
Nisaba turned toward him. The old man’s smile began before she spoke, as though he recognized even the length of her silence. It changed his whole ruined face.
“Nisaba,” he said.
“You know my name.”
“I have known you for a very long time, nearly all my long life, in fact.”
He raised one hand toward her, stopped before touching her, and closed it again around his stick. She watched the halted hand.
He walked to the stones without bowing. Re’u caught his elbow before his foot entered one of the circles. The old man accepted the correction without thanks or embarrassment.
He held out one hand toward the fold. Re’u guided it to a ewe’s shoulder.
“What is that?”
“One,” Nisaba said.
“One what?” asked the stranger.
She touched the animal. “Sheep.”
He crouched and searched the dust until his fingers found the stone Nisaba had paired with the ewe. “And that?”
“One.”
“One what?”
Nisaba looked from the stone to the sheep. “Stone.”
“The stone is not the sheep. The word is not the sheep. The mark is not the sheep. What have you counted?”
Nisaba touched the stone, then the ewe. “One beside one.”
“You have named the arrangement.”
The ewe moved to follow her lamb. The stone remained where Nisaba had placed it.
“The sheep,” Re’u said.
“Then how did the stones help?”
She followed the three lines she had drawn in the dust. “One stone for each sheep. No sheep twice. None omitted.”
The old man traced one of the lines with his stick. “Move this stone so that it claims the same animal as that one.”
Re’u moved it. “Now?”
“The stones now say that one animal was counted twice.”
“Stones are poor liars. Who made the pairing?”
Nisaba restored the lines. “I did.”
At the sound of a lamb, he extended his stick toward the ewe and the animal pressed against her side. “How many?”
“Two sheep.”
“Could they be one?”
Nisaba paused.
Re’u answered before she could. “One ewe and lamb if I am assigning a nursing pair. Two animals if I am counting mouths.”
The old man nodded. “The animals did not change. The count did because he changed what would count as one.”
He tapped the edge of the nearest circle with his stick.
Nisaba counted the circles, then touched one of the large stones at their centers. “Six groups. Sixty sheep in each.”
“Then what did you make one?”
This time the question did not sound foolish to Re’u. One sheep. One ewe-and-lamb pair. One flock. One stone standing for one animal. One mound of ten stones. One circle of sixty. The world had not supplied a single answer before the counting began.
“What I choose to count as one,” Nisaba said.
“A unit,” the stranger said.
The choice was hers and the word was his. Re’u understood both through the things she touched.
“And after you choose it?” the stranger asked.
“Count without repeating or omitting it.”
“What have you counted?”
“What I made one.”
The old man seemed pleased and terribly tired. “That is enough for a beginning.”
Abum-ilum stepped closer. “Do you know what she is?”
The stranger turned his blind face toward Nisaba. “Before spring returns, the tablet writer will ask whether an account is correct. She will tell him another hand can follow it.”
Ili-iddinam looked offended to have his future question answered in public. “How do you know what she will say?”
“I remember her saying it,” the man said flatly.
“I did not say that,” Nisaba said.
For the first time, the stranger’s smile faltered.
“No,” he said, softly. “Not yet.”
The old stranger bowed low to Nisaba. It was not the frightened bow the others had given her. Re’u had seen friends part at the canal road with less sorrow.
“You smiled when you saw me,” Nisaba said.
“Now you bow as people bow when a friend is leaving. Why?” Re’u added.
“You will see.”
He found the western path with his stick and took his leave.
16.2.4 A count with a purpose
By midday the wool workers had begun. The sheep were gathered in smaller pens, and the loose spring wool came away beneath practiced hands. Nisaba watched the work openly. The first fear did not disappear, but work pressed around it. There were animals to hold, fiber to gather, cords to tie, water to carry, and children to keep from teaching themselves dangerous uses for a bronze visitor.
Abum-ilum wanted Nisaba taken to the storehouse. Two men tried to guide her there by standing on either side and explaining the importance of records. She walked instead to the shade beside Erishtum’s wool baskets.
“She has chosen the wool,” someone whispered.
“She has chosen shade,” Erishtum said.
Nisaba watched Erishtum tie ten small bundles into a larger one.
“One,” Nisaba said.
“Ten,” said Erishtum.
Nisaba touched the cord around the whole bundle.
Erishtum understood. “One bundle. Ten smaller bundles. More wool than either of us wants to carry.”
Nisaba looked at the work yard, where people had begun using her name with increasing confidence and no increase in agreement about what she was.
“Purpose,” she said.
Erishtum pulled the knot tight. “Usually.”
Many days later, at sunset, Ili-iddinam brought a fresh tablet. He had copied the flock total again. Nisaba set one stone on the tablet’s bench and made a new mark beside it. She added another stone and another mark. Ili-iddinam watched her continue until nine stones stood in a row:
\[ 1\quad2\quad3\quad4\quad5\quad6\quad7\quad8\quad9. \]
He could follow the counts, although the signs were hers.
At the tenth stone, she gathered all ten into a bowl and wrote
\[ 10. \]
Ili-iddinam tapped the first mark. “You have returned to one.”
“One group of ten,” Nisaba said.
He tapped the second.
She pointed to the empty bench beside the bowl. “No loose stones.”
Nisaba drew ten bowls in the dust, with ten small strokes inside each. Ili-iddinam followed her finger along them. “A hundred.”
She drew a larger enclosure around the ten bowls, then nine more enclosures like it. “Ten hundreds,” he said. “A thousand. We have names for those.”
“Then read these,” she said, writing \(100\) and \(1{,}000\) beside the drawings.
He pointed to the first numeral. “One hundred, no tens, no loose ones.” He followed the places in the second. “One thousand, no hundreds, no tens, no ones.”
Re’u brought the stones from the six circles. They regrouped them into three collections of a hundred and six mounds of ten. No stone was added or taken away. Nisaba wrote \(360\) beneath the arrangement.
“Three hundreds, six tens, no loose ones,” Ili-iddinam said.
“We change place after sixty,” Ili-iddinam said.
“These signs change after ten.”
He turned his tablet toward her and pressed the marks he used for six. “Six jars,” he said.
On a second scrap of clay he pressed the same marks. “Six sixties of sheep.”
Nisaba moved the scraps until they lay side by side. “The marks are the same.”
“The account tells us what place they hold. We leave an empty place when the account supplies what is missing.”
Re’u put the stones back into their six circles of sixty. Nisaba pointed first to the six large stones, then to the circles of sixty they named. Ili-iddinam’s marks gave the total through their place and through the account that held them. She wrote both reckonings together:
\[ (6,0)_{60}=360_{10}=3\times100+6\times10+0. \]
“Six groups of sixty and no loose ones,” she said, touching the two places on the left. The comma kept them apart. The small sixty below named their grouping rule; the small ten below the other numeral named hers. The two short lines between the numerals said that both gave the same count.
The young writer frowned at the zero after the six.
“We would leave that place empty.”
“How do you distinguish six from six sixties?” Nisaba asked.
“The account tells us.”
Nisaba touched the zero. “Now the numeral does.”
The old stranger laughed once from the darkness beyond the awning at the scribe’s apparent confusion.
“You asked her what she made one,” Ili-iddinam called to him.
The stranger turned toward the young man’s voice. “Did I?”
Ili-iddinam laughed, then stopped when no one joined him.
That evening Re’u found him copying Nisaba’s nine signs onto a scrap of clay small enough to close in his hand.
“Will you take them to Nippur?” Re’u asked.
Ili-iddinam covered the scrap with his palm, then appeared annoyed that he had done so. “When I can explain them without beginning with a bronze girl beneath a sheep awning.”
“Why leave out the best part?”
“Because I want them to listen after it.”
He uncovered the tablet and corrected the tail of the mark for seven.
16.3 Act II: The Count Becomes an Account
16.3.1 What changed
Nisaba remained after the spring reckoning. She remained through the wool gathering and the days when the cleaned fleeces were tied into bundles for storage. By then, people no longer stopped work whenever bronze feet crossed the yard.
Abum-ilum sent twice to Nippur for instructions and received two answers that did not agree. Ili-iddinam wrote both questions; Abum-ilum impressed the storehouse seal into the clay before they left Reed Gate. One ordered him to guard a manifestation belonging to the temple. The other ordered him not to make claims about a divine presence without witnesses competent to prove the claim. He solved the contradiction by declaring that Nisaba was under his protection whenever she happened to be near the storehouse.
Nisaba solved it by going where she pleased.
The temple sent three officials with a wooden carrying frame and six men to bear it before the wool gathering ended. They wrapped the crosspieces in cloth so the bronze would not be scratched, placed the frame beside Nisaba, and asked her to sit. She did. Six men took the poles. The frame did not rise.
They brought six more men and two ropes. The ropes tightened, the poles bent, and one bearer made a noise he later denied making. Nisaba watched them until Abum-ilum ordered everyone to stop before the temple lost either a manifestation or twelve useful men.
Then Amat-Sin called to Nisaba from the wool shelter. Nisaba stood, stepped off the frame, and crossed the yard. No one tried the ropes again.
One bent pole later supported the wool awning. For the rest of the season, anyone who told the story beneath it pointed to a different part of the bow. The temple bearers said the wood had been green. Ibiya said he had heard Nisaba’s feet take root in the yard. Ahu-waqar said only that his sister had called and Nisaba had stood up, which no adult found sufficiently explanatory.
She walked with Re’u when the flock moved beyond the fields. She never appeared tired and never drank from his waterskin, but she stopped when the animals stopped and learned the ground by crossing it. At first she knew every count and almost nothing in the count.
At an irrigation cut, Re’u laid down a narrow plank before remembering how much bronze had defeated the carrying frame. Nisaba crossed it. The plank gave one small creak, less than it gave beneath Re’u. He looked back at her and then at the water.
“Do not ask me,” she said.
“I was going to ask the plank.”
She could say that one ewe lagged twelve paces behind another. Re’u could say she would be lame by evening if they did not remove the thorn between her toes.
Nisaba could divide the flock into two equal collections. Re’u could say which animals must remain together if he wanted the two collections to stay where he put them.
She could preserve the exact number of mouthfuls an animal took from a patch of grass. She could not tell from that number whether the animal had eaten enough.
“What do you see?” she asked him after a thin ewe refused water.
“Her sides. Her mouth. The way she stands.”
“Which one?”
“All of it.”
Whenever Re’u answered with the whole of what he saw, Nisaba asked him to name each sign separately. Re’u disliked losing sheep while she waited for every sign to have a name.
They kept working.
The work changed the way people approached her. At first they had brought offerings: a handful of grain, a twist of clean wool, a cup of beer placed where she stood. Nisaba examined each gift and left it untouched. Soon people brought disputes instead. The potter Ibiya and his brother agreed about how many jars had left their kiln but disagreed about where two had broken and who must bear the loss. Nisaba arranged pebbles until their accounts contradicted one another, then stopped.
“Which number is true?” one brother asked.
“The jar counts are the same,” she said. “You disagree about the two that broke.”
She would not decide what no count had established. By sunset, half of Reed Gate had heard that the bronze Nisaba could expose a false account. The other half heard that she refused justice.
Re’u understood both reports. People did not come to a goddess to hear that their own story was the part she could not count.
Children were less disappointed. They brought her questions that adults had learned not to ask aloud. Could she bend? Could she sleep? Was she hollow? Would she turn green if left in the rain? Nisaba answered the first two by crouching and by remaining upright through the night. For the other questions, she said, “I do not know.”
That answer frightened adults more than children.
The old stranger came and went by the canal paths. During the wool gathering he thanked Erishtum for a loaf she had not yet brought him. He told her the scorched edge had been better than the middle. When she baked the loaf two days later, one edge blackened against the oven wall. She carried it to him without cutting it. He tore off the scorched piece first.
A week later he asked Erishtum her name. He had known where the storehouse wall ended before Re’u led him there; now he sometimes stopped at a familiar turning and waited for someone to name the road. He remembered what had not happened and forgot what had.
Ibiya’s daughter Simat-Adad called him moon-struck. Her mother told her not to say such a thing where an old man could hear, then used the same word beyond the wall that evening. After that, Reed Gate had a name for him.
People who called him moon-struck in daylight found him beside the wall after sunset and asked what he remembered.
“Will the rains come?” Lamassatum asked while Nisaba listened.
“I do not remember,” he said.
“You claim to remember the days ahead.”
“Some of them.”
“What use is that?”
“Very little.”
Lamassatum left him half a loaf anyway.
When people later credited Simat-Adad with naming the stranger, she denied it. She said everyone had already been thinking the word and she had merely been the first person careless enough to say it beside her mother.
When the stranger joined Nisaba beside the wall, he struck the ground once with his stick before sitting one forearm’s length to her left. He did it each time. Nisaba never invited him and never moved away.
“Why do you remember only some days?” Nisaba asked.
The stranger turned toward her voice. “Why do you?”
Nisaba had no answer.
Before the feast in the longest days, a final tablet arrived from Nippur. Abum-ilum read the ruling aloud before the storehouse: Nisaba would remain wherever she chose to remain and would not be moved except by her own going. The temple had taken most of a season to command exactly what had already happened. Abum-ilum placed the ruling among the storehouse tablets with care. It was the first instruction on the matter that he could obey.
During a feast in the longest days, Amat-Sin and Simat-Adad brought Nisaba a circlet of green rushes threaded with the few flowers they had found beside the canal.
“Lower your head,” the older girl told her.
Adults had knelt before Nisaba. Officials had tried to direct her with orders and claims of protection. Neither had moved her so readily as two children who could not reach.
Nisaba bent. The girls settled the wreath above her brow.
“What does it mark?” she asked.
“Summer,” said Amat-Sin.
“Nothing,” said Simat-Adad. “We made it for you.”
Nisaba straightened with the wreath still on her head. By midday, the small flowers had begun to curl against the hot bronze.
When the flowers dried, Amat-Sin tied the rush circle to the bent carrying pole above the wool awning. Simat-Adad maintained that this had been her idea. The wreath remained there after both girls had forgotten to argue about it.
Ahu-waqar, who had begun carrying wool bundles but was still young enough to forget where he set them, placed his palm against Nisaba’s forearm. He pulled it back at once.
“Hot,” he said.
Nisaba had stood in full sun since morning. Re’u poured a little water across the bronze. It darkened, ran over her wrist, and struck the dust. She watched the wet path disappear.
The two girls each took one of Nisaba’s hands and led her beneath the awning. She went with them.
“The same body changes with the day,” she said.
“Everything does.”
“A count does not.”
“A count changes when the things do.”
“The stone does not.”
Re’u picked up one of the counting stones. “This stone does not become another number either. But in the morning it was a sheep, at midday it was a bundle of wool, and now it is only in my hand.”
Nisaba took the stone. “The stone was never the sheep.”
“No. But if I forget what I made it stand for, its number will not help me.”
Amat-Sin had been listening from beneath the awning. While they spoke, she collected the smoothest stone from the last completed circle and carried it away to finish a game.
Nisaba counted the stones again. “There are fewer.”
Re’u looked through the gate at the flock. “There are not.”
Amat-Sin returned when Erishtum called her name, the missing stone clenched in one dusty hand. She had not stolen a sheep or altered a number. She had broken the arrangement that allowed someone else to recover the count.
Nisaba accepted the stone from her. “The number remained. The record failed.”
“The stone was being a house,” Amat-Sin said.
Nisaba considered this and restored the stone to its circle.
The next day she asked him to count a flock without using stones. Re’u made notches on a reed. Then she asked him to do it without the reed. He divided the animals among three pens and remembered a subtotal for each. Then she asked him to count them without stones, reeds, pens, or memory.
“I cannot.”
“Then number requires a material thing.”
“Counting requires me to keep my place. Today I use stone. Ili-iddinam uses clay. You use marks no one else has seen. The number may not require them. I do.”
Nisaba looked toward the tablet writer beneath his awning. He could show her a missed ewe, an erased mark, a broken reed, and an account restored because two records had been kept apart.
So he did.
Later, when she drew her signs in dust, she began to ask who needed to read them after the wind passed. Ili-iddinam taught her which clay held a clean wedge and how wet a tablet could be before a line collapsed. Erishtum taught her that a cord tied around wool could preserve a group where a written sign would be useless to the person carrying it. Re’u taught her that an animal marked yesterday might still have crossed the wrong gate today.
Nisaba gave them a notation that could travel farther than a mound of stones.
Not everyone who sat beside her brought a dispute or a lesson. An old basket maker named Shubultum came in the evenings and spoke while splitting reeds. She told Nisaba the names of her husband and two sisters, all dead, and which pair of hands had taught her to bend a rim without breaking it.
“What account should I make?” Nisaba asked.
“None. I know their names.”
“Then why tell me?”
“I wanted someone else to know who they were.”
The following evening, Shubultum found Nisaba waiting beside the wall.
By the hottest month, they had found a practice between them. Re’u named what he noticed. Nisaba made the distinctions visible. He corrected what she had put together or kept apart. She showed him when two explanations that sounded different came to the same end.
The old stranger sometimes walked with them. His stick searched the ground before him. On steep paths he took Re’u’s shoulder without apology; on level ground he refused it. His hands still trembled, though less than they had at the spring reckoning. He no longer knew the path around the deep irrigation cut or which household kept a water jar outside its wall. Re’u blamed age for the lost knowledge and warmer weather for the steadier hands, then avoided placing the two explanations beside each other. The stranger could sit beside Nisaba for a long time without asking what she was.
One day he lowered himself on her right, near enough that his sleeve touched her arm. Nisaba shifted away.
“You sit on my left,” she said.
“Do I?”
“Every time. One forearm away. You strike the ground once before you sit.”
He moved to her left and measured the distance with his hand. “Here?”
Nisaba moved his hand the width of two fingers farther. “Here.”
The stranger smiled, though not as he had at the first reckoning. “I will remember.”
Re’u came to suspect that the old man had once known exactly how to be near her and was now learning it from her.
16.3.2 Four kinds of absence
The number of animals present did not remain 360. The spring tablet still recorded what had been entrusted.
Abum-ilum ordered twenty-four animals transferred to another flock after a dispute over pasture. Ili-iddinam made a tablet for the transfer. Re’u watched the twenty-four go and set their stones in a separate bag. They were absent from his flock, but they were not lost.
Twelve adult animals died before the rains. Re’u preserved what evidence he could and reported each death. Those animals were absent in another way.
One hundred fifty lambs were born. Thirty died before the end of the dangerous first weeks. The surviving lambs were present, although none had existed when Abum-ilum made the spring tablet.
At the fold, Ili-iddinam tried to make the several tablets agree:
\[ 360-24-12+150-30. \]
Nisaba watched his stylus stop.
“It should be more than 360,” Re’u said.
“Why?” asked Nisaba.
“We added more than we took away.”
“How much more?”
Re’u moved stones while Ili-iddinam computed in clay.
The removals were
\[ 24+12+30=66. \]
The births added 150. The net change was therefore
\[ 150-66=84. \]
So the new total was
\[ 360+84=444. \]
Nisaba wrote the account a different way:
\[ 360+(-24)+(-12)+150+(-30)=444. \]
“You have made the dead less than nothing,” Ili-iddinam said.
“No,” said Nisaba. “I made the change negative.”
Re’u looked at the expression. The dead animals had not become negative sheep. The transfer had not made negative sheep either. The signs showed which way the count had moved: into the flock or out of it.
“The same sign for death and transfer,” he said.
Nisaba waited.
“The total needs that,” Re’u continued. “The account does not.”
He laid the twelve death stones apart from the twenty-four transfer stones.
Nisaba studied the two groups. Both reduced the total by the same operation. They did not mean the same thing.
“The total is right,” she said. “It does not say why they are gone.”
The old stranger, sitting against the wall, lifted his head.
“What have you counted?” he asked.
“Changes to the flock.”
“And what did the total leave out?”
Nisaba looked at the death stones and transfer stones. “Why the animals were absent.”
“You are learning to be dangerous,” he said. “Someone may prefer the total without those two piles.”
Nisaba appeared to consider this praise.
Shubultum, splitting reeds beside them, tied the death stones into one corner of a worn cloth and the transfer stones into another.
“The dead have names,” she said. “The transferred have another shepherd.”
Nisaba watched how she tied the two corners.
16.3.3 The tablet and the household
Erishtum used the count of 444 for a different purpose.
More animals meant more mouths to feed before they meant more wool to sell. The surviving lambs would not contribute equally to the next wool gathering, and the ewes nursing them would not endure the lean months on the same ration as dry adults. She asked Ili-iddinam to place the adult animals and lambs on separate lines.
He objected that the official total was already correct.
“Correct for what?” she asked.
He indicated the tablet. “For the account.”
“The account of what?”
“The flock.”
Erishtum turned to Re’u. “Did you teach him that answer?”
“He arrived with it.”
They divided the present animals into two broad kinds:
\[ 324\text{ adults}+120\text{ lambs}=444\text{ animals}. \]
The adult count followed from the original flock:
\[ 360-24-12=324. \]
The lamb count followed from births and early deaths:
\[ 150-30=120. \]
The two subtotals recombined to the same total. Nothing in the arithmetic forced the separation. Erishtum’s purpose did.
She then separated the adults again: 180 breeding and nursing ewes, and 144 other adults. Re’u corrected three classifications after examining teeth and udders. Nisaba revised the marks without protest.
“Your first number was wrong,” Ili-iddinam said to her.
“I put three animals on the wrong line,” Nisaba answered. “The calculation followed the line.”
The young man glanced at his tablet. He did not seem comforted.
“That is worse,” he said.
“Usually,” said Erishtum.
16.3.4 The first shortage
The rains were poor.
Water came, but not enough and not in the expected rhythm. Pasture that usually held through the cool months thinned early. The canal fields received what the local officials could preserve for barley, and the flock ranged farther for less.
Re’u noticed the change before the storehouse did. The animals returned sooner from grazing. The ewes crowded the trough. The lambs nosed the ground after the adults had moved on.
He estimated how long the pasture would hold.
“Thirty days short,” he told Abum-ilum.
“Twenty,” said the overseer.
“Thirty.”
“The tablet from last year says twenty-two.”
“The sheep have not read it.”
Abum-ilum did not smile. He was responsible for the barley in the storehouse, the claims already sealed against it, and the consequences of opening jars too early. If he released too much, no calculation would put the grain back.
“Bring the account tomorrow,” he told Ili-iddinam. “Bring all of it.”
The young writer looked at the several tablets now required to describe one flock.
“Which one is all of it?”
No one answered before the rain began again, lightly and without conviction.
16.4 Act III: Enough for Whom?
16.4.1 Forty-one measures
By the middle of the lean season, the failed rains had become visible in every trough and storage jar. At sunrise Abum-ilum showed the witnesses that the clay sealing the storehouse door was unbroken. He opened it; Ili-iddinam recorded that he had done so.
The inner rooms were cooler than the yard. Plastered bins and sealed jars stood above the damp that could creep through mud brick. Reed tags and clay sealings connected each group of containers to a purpose that Ili-iddinam could find on a tablet. Abum-ilum knew the stores by handling them. Ili-iddinam knew what Reed Gate would later have to answer for in Nippur.
The storehouse contained forty-one gur of barley when they opened the account.
The local measures nested inside one another:
\[ 1\text{ gur}=5\text{ barig}=30\text{ ban}=300\text{ sila}. \]
The relationships were familiar to everyone present except Nisaba. Ili-iddinam explained them with the pride of a man briefly permitted to teach a goddess something.
“One ban is ten sila,” he said. “Six ban make one barig. Five barig make one gur.”
Nisaba wrote:
\[ 1\text{ gur}=5\times6\times10\text{ sila}=300\text{ sila}. \]
“Yes,” he said. “Except no one would write it like that.”
“You understood it.”
“Yes.”
“Then one person would.”
Ili-iddinam began to object, reconsidered the available evidence, and stopped.
The forty-one gur did not form one unclaimed heap. Thirty had been assigned to the flock. Six stood against household claims already witnessed. Five remained under the overseer’s control for a failure elsewhere: a broken delivery, a sick work team, a field whose yield did not arrive.
For each amount, Abum-ilum indicated the containers and Ili-iddinam read the corresponding line. Neither man’s memory was sufficient for the other man’s responsibility.
Nisaba converted each amount to the smallest measure they were using:
\[ 30\text{ gur}=9{,}000\text{ sila}, \]
\[ 6\text{ gur}=1{,}800\text{ sila}, \]
\[ 5\text{ gur}=1{,}500\text{ sila}. \]
Together:
\[ 9{,}000+1{,}800+1{,}500=12{,}300\text{ sila}=41\text{ gur}. \]
The total balanced. No barley appeared.
16.4.2 The overseer’s plan
Abum-ilum began with the spring tablet.
“Three hundred sixty entrusted animals,” he said. “Thirty gur for thirty days.”
Ili-iddinam divided 9,000 sila by 360 animals and then by 30 days. Nisaba wrote the same relation in her notation:
\[ \frac{9{,}000\text{ sila}} {360\text{ animals}\times30\text{ days}} =\frac56\frac{\text{sila}}{\text{animal}\cdot\text{day}}. \]
“Five-sixths of a sila for each animal each day,” the young writer said.
Abum-ilum nodded. The plan used the barley already set aside, the count on the spring tablet, and the thirty days they had chosen. Another overseer could follow the marks and obtain the same result.
Re’u looked at the lines dividing one kind of animal from another.
“Which 360 animals?”
“The entrusted count.”
“Twenty-four of those are with another flock. Twelve are dead.”
“Their changes are recorded.”
“And 120 lambs are standing in the fold.”
Abum-ilum pressed his lips together. “The reserve was assigned against the entrusted flock.”
“The barley will be eaten by mouths, not by the tablet.”
The overseer’s irritation was not stupidity. If a new count could silently enlarge every obligation, then no reserve was a reserve. A shepherd could arrive with borrowed animals, point to their mouths, and consume grain owed elsewhere. If Abum-ilum ignored sealed claims whenever suffering appeared at his door, the storehouse would soon reward whoever could assemble the most visible suffering.
“I opened a jar once for a man who brought hungry animals to this gate,” Abum-ilum said. “Half belonged to his wife’s brother. When Nippur asked where the barley went, it did not ask how the animals had looked. It asked whose seal had been broken.”
“Give me a rule I can defend when you are gone,” he said.
Re’u had no answer yet.
16.4.4 Feeding units
Re’u did not know the exact amount of barley each animal would require. No one did. The animals would still graze what they could find, weather would change their need, and sickness would make any fixed rate less trustworthy.
He did know enough to reject equality by headcount. After several days of observation, he proposed a weight for each kind of animal when they divided this barley:
- each of 180 breeding or nursing ewes counted as 1 feeding unit;
- each of 144 other adults counted as \(\tfrac34\) of a feeding unit; and
- each of 120 lambs counted as \(\tfrac12\) of a feeding unit.
The weights did not claim that a ewe ate exactly one sila of total food every day. They showed how much of this barley each kind should receive compared with the others, for this plan only.
Erishtum divided a small measure of barley into four equal-looking heaps. All four stood for the ewe’s share, three for another adult’s, and two for a lamb’s. The comparison held whether she used a small measure or a large one.
Re’u asked her to fill a one-sila vessel. “This much for each breeding ewe each day,” he said. “With whatever grazing remains. The others receive their smaller shares.”
Erishtum looked through the doorway toward the pens. “We will have to change it if the pasture worsens.”
“Or if they leave it uneaten.”
She set the vessel beside the four heaps. The heaps showed the relative shares. The filled vessel showed how large Re’u proposed to make the ewe’s daily share. It was a supplement chosen from what they had observed, not a measurement of everything the animal needed.
Nisaba calculated the daily total:
\[ 180\times1=180, \]
\[ 144\times\frac34=108, \]
\[ 120\times\frac12=60. \]
Therefore,
\[ 180+108+60=348\text{ feeding units}. \]
For thirty days:
\[ 348\text{ feeding units}\times30\text{ days} =10{,}440\text{ feeding-unit-days}. \]
At Re’u’s proposed rate of one sila per feeding unit per day, the plan required 10,440 sila:
\[ 10{,}440\text{ sila}=34\text{ gur}+4\text{ barig}. \]
Only 30 gur, or 9,000 sila, had been set aside for the flock. The shortfall was
\[ 10{,}440-9{,}000=1{,}440\text{ sila} \]
or
\[ 4\text{ gur}+4\text{ barig}. \]
The weights made Re’u’s judgment about relative need visible. The weighted calculation also showed that 30 gur would not meet that plan.
Abum-ilum folded his arms. “Your better numbers have made the barley smaller.”
“No,” Nisaba said.
“I know.”
For the first time, the overseer sounded tired rather than angry.
16.4.5 One household
The six gur marked for households stood in sealed jars along the north wall.
Two gur were associated with Re’u’s extended household. When Abum-ilum asked what Re’u would pledge for access to the contingency barley, Re’u looked toward those jars.
“Our two,” he said.
Erishtum turned so slowly that he wished she had moved faster.
“Our?”
“The household’s.”
“Which household?”
“Ours.”
“What did you make one?”
The old stranger lowered his head. Nisaba looked from Erishtum to Re’u with sudden attention.
“And what have you counted?” the stranger asked.
“Two gur,” Re’u said.
“Barley is not a claim.”
Erishtum answered him. “Then let us count the claims.”
Erishtum brought out the small tablet witnessed after her husband’s death. The tablet did not divide every jar and basket as though the people beneath one roof lived apart. Neither did it place every jar under Re’u’s hand. Part of the claim maintained Erishtum and the children. The goods she had brought into the marriage, and what was kept for the children, had not passed into Re’u’s hands when his brother died.
“You may answer with your labor,” she said. “You may answer with wool owed to you. You may not make generosity out of my children’s grain.”
Re’u felt every witness in the storehouse become interested in the floor.
“I meant to preserve the flock that preserves us.”
“Then say that. Do not say the cost is yours when you have put it in my jar.”
Nisaba took the reed stylus from Ili-iddinam and drew a line through the single household entry. Below it she made separate marks for claims that had previously appeared as one.
The total remained two gur. The account became larger.
“The sum still balances because we left out her claim,” Nisaba said. “That did not remove it.”
Erishtum inspected the new lines. “No. Someone still has to pay it.”
The witnesses did not leave the matter inside the storehouse. By evening, people in the work yard knew that Erishtum had stopped Re’u from pledging the household grain. Ibiya said she had shamed him before an overseer. Lamassatum said Re’u had first shamed himself by naming another person’s grain. Before dark, Simat-Adad had heard a version in which Erishtum struck his hand away from an open jar. She repeated it only to explain why it could not possibly be true. None of these accounts described what had happened, but each found an audience prepared to improve it.
Re’u carried water to the wool shelter after dark. Erishtum was sorting fiber by lamplight. Ahu-waqar slept against the wall; Amat-Sin separated tangled locks with solemn inattention. The spindle, baskets, sealed jars, and tablet occupied different parts of the room, but they belonged to one working arrangement. If the barley disappeared, the children would eat less. If the wool work stopped, an exchange the household expected would disappear later. If Re’u lost the flock, a different stream of wool would disappear after that. The jars did not all belong to one person in the same way, though they stood against the same wall.
“I was trying to keep all of it,” he said.
Erishtum did not look up. “You were trying to keep what you see from the field.”
“I see this room.”
“When you enter it.”
He set down the water. Amat-Sin discovered a knot and began working at it with both hands.
“If the flock goes,” Re’u said, “my contract goes with it.”
“I know.”
“Then you know why I offered the grain.”
“You looked at my jars before you named your wool.”
That was not altogether fair. It was fair enough that Re’u could not answer quickly.
Erishtum finally set aside the wool. “I knew the stores while my husband was alive,” she said. “I knew the wool. I knew which obligation belonged to which jar. When he died, men began asking me to prove what I already knew.”
Re’u sat against the wall. “What should the account say?”
“It should not say every jar is mine and none of it serves the house. That would lie in the other direction. It should say which grain keeps the children, which came with my marriage, which depends on your flock, and which you may actually promise. Then we can see what saving the flock costs each of us.”
“Ilum-bani would have offered the jars,” Re’u said.
“Ilum-bani offered anything when other people were watching.”
“He would have kept the flock.”
“He would have come home after dark and asked whether I had found a way to do it.”
Re’u remembered his brother making promises in the yard and Erishtum making them possible indoors. He had mistaken the order before.
“That will be a larger tablet.”
“Then use more clay. Write which grain you mean.”
In the doorway, Nisaba repeated the sentence. She had arrived without either of them hearing her.
“Sometimes,” Erishtum amended.
Nisaba entered and examined the work. She lifted the spindle whorl, turned it once, and returned it precisely where it had lain.
“This makes the thread?” she asked.
“Hands make the thread,” Erishtum said. “The spindle helps the hands keep twist. The whorl helps the spindle keep turning. The wool matters. So does practice. If you write that the whorl made the thread, the tablet will be shorter and wrong.”
Nisaba looked at Re’u.
“Do not ask me,” he said. “I have already made one household.”
Erishtum laughed despite herself. The sleeping child stirred, and all three adults became quiet.
The following morning Ili-iddinam made a new account. He did not divide every object among named owners. He recorded the claims that mattered to the pledge they were discussing and left the rest of the household intact. The tablet could not make a shepherd bound by contract, an overseer, a widow, and a young writer equal. It could make it harder for one of them to conceal a transfer inside a total.
Nisaba read the new lines in the open yard where everyone waiting at the storehouse could hear. Her presence did not settle whether Erishtum had behaved properly. It placed the changed account before witnesses. For the rest of the season, anyone who called the two gur simply Re’u’s grain had to do so against a visible record.
16.4.6 The fourth plan
Re’u walked through the adult pen before dawn.
Some animals would not carry the flock into another season. Twenty-four of the other adults were old, chronically lame, or unlikely to survive the longer range ahead. Keeping them within the feeding plan reduced what could go to the breeding ewes and young stock. Removing them would mean sale where a buyer could be found and slaughter where one could not.
There was no notation that made the choice clean.
After the twenty-four were removed, the planned flock contained:
- 180 breeding or nursing ewes;
- 120 other adults; and
- 120 lambs.
That was 420 animals:
\[ 180+120+120=420. \]
The weighted daily requirement became
\[ 180\times1 +120\times\frac34 +120\times\frac12 =180+90+60 =330\text{ feeding units}. \]
For thirty days:
\[ 330\text{ feeding units}\times30\text{ days} =9{,}900\text{ feeding-unit-days}. \]
At one sila per feeding unit per day, the plan required exactly
\[ 9{,}900\text{ sila}=33\text{ gur}. \]
The jars already set aside for the flock supplied 30 gur. The remaining need was 3 gur.
Abum-ilum could release that amount from the 5-gur contingency and retain 2 gur for other failures. He would do it only against a witnessed obligation and a minimum count at the spring reckoning.
Re’u offered his next wool allotment and, if that did not cover the advance, his labor. If he failed to return the agreed minimum flock, he would lose the contract.
Erishtum made Ili-iddinam read each kind of animal and each promised share again. Her claim and the children’s claim did not appear as security.
“Now it is your sacrifice,” she told Re’u.
She did not say it kindly. She did stand beside him when he made his mark.
16.5 Act IV: Re’u’s Mark
16.5.1 What arithmetic could settle
Two days after Re’u completed the fourth plan, the witnesses gathered in the storehouse at midday.
Ili-iddinam had prepared four tablets. The first preserved the original entrustment of 360 animals. The second recorded births, deaths, and transfers leading to the present count of 444. The third divided the animals by age and feeding claim. The fourth described the proposed division of barley, removal of twenty-four adults, release of 3 contingency gur, and Re’u’s obligation.
Abum-ilum had asked Ili-iddinam to call the release necessary.
“Who will witness that it was?” the young man had asked.
The overseer studied him long enough for the room to become quiet. “Write proposed.”
Ili-iddinam wrote proposed.
Abum-ilum asked Nisaba to confirm the final tablet.
She read each line. She recomputed the totals. Then she set the tablet down.
“Is it correct?” he asked.
“If these are the animals, if these weights describe the intended shares, if the plan lasts thirty days, and if one feeding unit receives one sila each day, then 9,900 sila follow.”
“So it is correct.”
“Only if each of those things is true.”
“Which plan should I allow?”
Nisaba looked at the four tablets.
“The numbers do not choose.”
The overseer stared at her. For months, people had brought Nisaba questions because her answers did not tire, flatter, or forget. Some had begun calling any answer she gave a judgment. Now, when judgment was wanted, she had returned the problem.
“Then what good are you?” Abum-ilum asked.
Several witnesses inhaled at once.
Nisaba did not appear offended.
“I can show what follows from what you have made one.” Nisaba turned the four tablets toward Re’u. The witnesses turned with them.
Re’u picked up the tablet that described the fourth plan. “Come to the pens.”
16.5.2 What Re’u knew
Re’u walked the witnesses through the pens.
He did not describe the twenty-four marked adults as useless. They had produced wool, sired lambs, endured dry seasons, and taught younger animals where to find water. Their history did not alter the barley available to them now. Their removal would give the remaining flock a better chance. It would also make their deaths or sale part of Re’u’s decision rather than something he could later blame on weather.
He showed Abum-ilum the breeding ewes whose loss would narrow the next season before it began. He showed him lambs strong enough to survive if the ration lasted. He showed him two animals he expected to lose under every plan.
“Then why feed them?” the overseer asked.
“I expect to lose them. They are still alive.”
“You told me to remove twenty-four because you expect them not to survive.”
“Yes.”
“What is the difference?”
Re’u looked from one animal to the next. He could state differences in age, teeth, gait, appetite, and breeding value. No mark told him where knowledge ended and choice began.
“I cannot draw a clean line,” he said. “I think removing those twenty-four gives the others a better chance. I may be wrong about which ones to keep.”
Abum-ilum looked again at the two animals. Re’u had no further answer for him.
Erishtum described the household claims. She named what would happen if the sealed barley was treated as spare: who would eat less first, which wool work would stop, and which debt would remain after the grain disappeared. She did not ask them to spare the household merely because it was a household. She named the people, jars, work, and debts that Re’u’s word ours had pressed together.
Ili-iddinam described the record. He could preserve the original total without pretending it was the present count. He could preserve the present count without pretending every present animal carried the same obligation. He could place Re’u’s pledge on the same tablet as the contingency release and keep Erishtum’s claim outside it.
Abum-ilum described what he would have to answer for. Two contingency gur would remain. If another delivery failed, people would ask why he had released the other three. The four tablets could defend the reasoning, but they could not replenish the jars.
Each person knew something the others needed.
16.5.3 The choice
Re’u chose the weighted plan.
Before sunrise he carried twenty-four short red cords into the adult pen. Under the fourth plan, the barley allocated to the flock could provide the planned shares for 420 animals. Re’u had chosen that twenty-four would leave it. The arithmetic had not named them.
He began with the obvious ones: a wether that placed no weight on one foreleg, two animals whose teeth could no longer break the coarse feed, another with a swelling that had returned after every treatment. He tied one cord loosely around each neck.
The fifth animal was an old wether with a split left ear. For years it had stayed near water when the rest of the flock wandered toward better grass. Younger animals had sometimes followed it back when heat made the longer route dangerous. It stood soundly enough now. It would probably lose condition first when the range lengthened.
Re’u held the cord for some time before tying it. The wether lowered its head to nose his wrist, searching for salt. Its history was not another mouth, another feeding unit, or another line on the tablet. Its history also did not fill a jar.
Nisaba followed without advising him. After the twelfth cord, she counted those marked and those not marked. After the twenty-fourth, she counted again.
“Twenty-four,” she said.
The stranger stood outside the hurdle, his stick resting lightly against one leg. “What have you counted?”
“Animals marked to leave the flock,” Nisaba answered.
“And what has he chosen?”
She looked toward Re’u.
“That is not another count,” he said.
The trader accepted thirteen animals. They went toward the city under another man’s staff. The remaining eleven were killed near the work yard, where the meat could be divided promptly and the hides cleaned before the day warmed.
Re’u gave the killing to no one else. He held each animal, kept the blade sharp, and remained through the cleaning. Erishtum directed the division of meat among the households whose labor would preserve it. Shubultum brought broad reed trays. Lamassatum measured salt from a jar she had hoped not to open until winter. Simat-Adad carried water after protesting that she had never called anyone moon-struck, and Ahu-waqar guarded the red cords with such attention that he forgot the knife Erishtum had sent him to fetch. Ili-iddinam recorded eleven slaughtered and thirteen transferred by sale. The two lines produced the same subtraction from the feeding count. They did not describe the same event.
When they reached the split-eared wether, Re’u almost sent it with the trader instead. That would have changed where the death occurred, not necessarily whether it occurred. He kept the animal back and did the work he had chosen.
By midday the red cords lay in a bowl. Nisaba touched one and then withdrew her hand. Re’u could not tell whether the bronze retained anything of what it touched. He knew that he would.
Abum-ilum released 3 gur from the contingency reserve. The barley available to the flock became
\[ 30\text{ gur}+3\text{ gur}=33\text{ gur}=9{,}900\text{ sila}. \]
Two contingency gur remained. The 6 gur under household claims remained sealed.
Ili-iddinam read Re’u’s obligation aloud. Re’u pledged his next wool allotment and, if the value credited to it did not satisfy the advance, the labor required to complete the obligation. If he failed to return at least six sixties of animals at the next spring reckoning, Abum-ilum could place the flock with another shepherd.
“Six sixties,” Re’u said. “Not a sign whose size must be guessed.”
Ili-iddinam added words making the magnitude explicit.
Re’u pressed his mark into the clay.
His mark did not make him Abum-ilum’s equal. It preserved what each man had undertaken before witnesses who might outlast either man’s wish to remember it differently.
Erishtum inspected the line naming the security. Her claim did not appear there.
Abum-ilum sealed the tablet.
Nisaba made no declaration. She watched Re’u’s thumb leave the clay.
16.5.4 Thirty days
The plan did not unfold like its table.
On the fourth day, the weather warmed and the animals found more grazing than Re’u expected. On the seventh, a ewe became sick and consumed less. On the twelfth, wind spoiled part of one open measure before anyone covered it. On the seventeenth, two lambs entered the wrong pen and were nearly counted twice.
The daily total was what they meant to pour, not what every day allowed. Re’u and Erishtum adjusted the shares promised to each kind of animal while Ili-iddinam marked every change large enough to empty the jars too soon.
For the first seven days, Nisaba kept another account beside the clay one, using her own signs:
| Day | Planned sila | Issued sila | Difference from plan |
|---|---|---|---|
| 1 | 330 | 330 | 0 |
| 2 | 330 | 326 | \(-4\) |
| 3 | 330 | 328 | \(-2\) |
| 4 | 330 | 318 | \(-12\) |
| 5 | 330 | 324 | \(-6\) |
| 6 | 330 | 329 | \(-1\) |
| 7 | 330 | 327 | \(-3\) |
| Total | 2,310 | 2,282 | \(-28\) |
The last column compared what had been issued with what they had planned:
\[ \text{difference}=\text{issued}-\text{planned}. \]
A negative result meant that less had been issued than planned. It did not mean that anyone had issued negative barley.
On the fourth day, the large difference came from improved grazing. If Ili-iddinam had simply replaced 330 with 318, no mark would remain of what they had meant to pour. By preserving both columns, he could show the first plan beside what they had actually poured.
“Which number governs tomorrow?” he asked.
“Neither,” Re’u said. “We have not reached tomorrow.”
Nisaba pointed to the remaining barley. “What remains changes what we can pour.”
“Yes.”
She pointed to the daily plan. “Which animals they are changes what should be offered.”
“Yes.”
“And the animals change what they will take.”
“Yes. That is why the table keeps changing.”
At the end of the week, 28 sila remained beyond the planned balance. That did not justify increasing the next day’s issue by exactly 28. The difference told them what remained; it did not tell them what to pour tomorrow. Those 28 sila could answer a later day when wind, sickness, or worse grazing moved use in the other direction.
The measures themselves imposed another limit. No one could pour a fraction with the exactness of a mark on clay. They used vessels, hands, and judgment; grain settled differently, spilled, and clung to the sides. The tablet could show a clean measure even when no jar had been poured so cleanly. Adding decimal places would not repair that mismatch.
When the lambs crossed pens on the seventeenth day, the count on the distribution tablet was two higher than the count made at the outer gate. Nisaba recomputed the additions and found no error. Re’u walked the fences. He found a loosened reed hurdle and two sets of fresh tracks.
“The arithmetic was correct,” Ili-iddinam said when they repaired the entry.
“The addition was right,” Re’u answered. “We counted two lambs twice.”
They added a second count at the gate. From then on, the person who divided the animals among pens did not also give the final number at the outer gate. A second count cost time. It cost less than feeding animals twice on clay and once in the yard.
Nisaba asked for a rule behind every adjustment.
“If this ewe does not eat her share,” she asked, “where does it go?”
“Today, to those two.”
“Why those?”
“They lost condition yesterday.”
“By what measure?”
Re’u showed her the ridge of a spine beneath wool, the hollow beside a tail, the resistance of an animal that still had strength to object.
Nisaba touched the same signs. Her bronze fingers could distinguish shape and distance. Re’u did not know whether she could feel warmth.
“Can this be written?” she asked.
“Some of it.”
“Can all of it be learned?”
“Perhaps. Not before they eat.”
She looked again at the animals. “Then they may have to eat before the whole of it can be put on clay.”
“They usually do.”
She gave no sign that she heard either pride or apology in that answer.
At the end of thirty days, the flock had consumed slightly less than the most they had planned to pour because grazing had briefly improved. That did not mean Abum-ilum had been wrong to open the contingency jars. Without them, Re’u could not have kept the promised shares long enough to respond to what happened. Barley left in a jar did not mean the jar should have remained sealed. The unissued balance remained assigned to the flock for the lean weeks that followed. It was part of the 3-gur advance and did not return to the contingency reserve.
16.5.5 The second spring
At the next spring reckoning, 384 animals passed through the gate.
Re’u counted six circles of sixty and then twenty-four more:
\[ 384=6\times60+24=(6,24)_{60}. \]
The total exceeded the six sixties required by the tablet by 24. It was also 60 below the 444 present before they divided the barley:
\[ 444-384=60. \]
Twenty-four of that reduction had been chosen before the barley was divided. The rest came through the season in deaths, losses, and changes no plan had prevented.
The count was enough for Re’u to retain the flock. It was not enough to erase the obligation. His wool allotment went first against the 3-gur advance, and the remaining balance would be met through labor.
Abum-ilum did not congratulate him. Re’u had not expected congratulations from a storehouse. The overseer did renew the arrangement.
Erishtum received the household barley under the claims the second account had preserved. Ahu-waqar and Amat-Sin ate from it without knowing how close their grain had come to being called Re’u’s sacrifice. That ignorance was not ingratitude. Children should not have to study an account before they are permitted to eat.
Ili-iddinam copied the final count beneath the earlier entries. He did not press the history into one total. The tablets would preserve both counts: the animals first entrusted and the animals present later, along with each birth, death, and transfer, how the barley had been divided, and whose wool and labor stood against it.
Nisaba read it from beginning to end.
“Correct?” the young writer asked.
She looked at him.
“Another hand can follow it,” she said.
He smiled. A year earlier he had objected when Erishtum asked what a correct total was correct for. Now he copied the words beside the final count.
“I will take the signs to Nippur,” he said.
“You intended to do that when you hid them under your hand,” Re’u said.
The young man had the grace to look embarrassed. “I intended to take the marks. Now I will take the account of where I learned them.”
Nisaba looked at the new line naming Reed Gate. “The marks would work without my name.”
“In Nippur they will ask whose account they came from,” Ili-iddinam said.
Then he glanced toward the old stranger. “He said you would answer that.”
Nisaba turned. The stranger had tilted his head toward Ili-iddinam with the attentive stillness he used for an unfamiliar voice.
“Who is speaking?” he asked.
“Ili-iddinam,” the young writer said, more sharply than his name required.
“Yes,” said the stranger. “I will remember.”
“A prophet,” someone whispered behind Re’u.
“Not prophecy,” the stranger said. “Memory.”
No one appeared comforted by the correction.
16.6 Back to the Stones
After the reckoning, Re’u found Nisaba arranging the original counting stones in the yard. She had made six rows of sixty. Then she made ten rows of thirty-six, twelve rows of thirty, fifteen rows of twenty-four, and eighteen rows of twenty.
The arrangements looked different. Each contained 360 stones.
Multiplication described their rectangular structure:
\[ 360=6\times60=10\times36=12\times30=15\times24=18\times20. \]
“How many answers?” Re’u asked.
“One number,” Nisaba said. “Several arrangements.”
The stranger came along the wall, tapping until his stick found the first row. He lowered himself to the ground and ran his fingers over the stones. Nisaba returned the stones to six rows of sixty and drew a line after the third row. Two sets of rows, with three rows in each. Within every row, she marked six groups of ten, then separated those groups into two sets of three. Within each group of ten, she made two piles of five.
Re’u followed the divisions from the whole arrangement to one little pile: two sets of rows, three rows in each, two sets of groups in each row, three groups in each set, two piles in each group, five stones in each pile. All 360 stones remained. The smaller numbers counted the groups within groups; they were multiplied to recover the whole, not added as separate piles.
Nisaba spoke the group counts as she followed the lines:
\[ 360=2\times3\times2\times3\times2\times5. \]
Each count in that chain could make equal groups only as a single whole or as ones. Five stones could become a pile of two and a pile of three, but those piles would not be equal.
Then she brushed away the lines and made eighteen rows of twenty. She divided the rows into two sets of nine, then each set into three bundles of three rows. Within each row, she made two sets of ten, each containing two piles of five. The stones had different neighbors now. The group counts were two, three, three, two, two, and five.
“Again,” Re’u said.
Nisaba tried twelve and thirty. The order changed. The final group counts did not.
Then she drew one of the old circles apart and worked with its sixty stones. She began with six groups of ten:
\[ 60=6\times10=(2\times3)(2\times5)=2\times2\times3\times5. \]
She began again with twelve groups of five:
\[ 60=12\times5=(2\times2\times3)\times5. \]
The path changed again. The last factors did not.
The stranger smiled toward the sound of the stones. “A road may turn at every hill and still be the road to one city.”
“Must it always end there?” Nisaba asked.
“You have watched it happen twice,” the stranger said. “Twice is not always.”
Nisaba began another arrangement. She did not ask why he sounded certain that there was more to learn.
16.7 Coda: The Beginning
The old stranger left before sunrise on the day after the reckoning.
Re’u found him at the western path fastening a strap. The old man’s hands trembled less than they had at the first spring reckoning, but the leather had split and would not hold.
“Who is there?” the stranger asked.
“Re’u.”
The old man repeated the name carefully. He had leaned on Re’u’s shoulder through half the year.
Re’u repaired the strap without asking where the old man intended to go.
Footsteps sounded behind them. Erishtum had brought half a loaf, scorched at one edge. Ili-iddinam and Abum-ilum followed, each perhaps unwilling to let the other become the only witness to the departure.
The old man turned from one voice to the next without recognition. Re’u gave him the introductions again. “Abum-ilum oversees the storehouse. Ili-iddinam writes its tablets. Erishtum is of my household and knows what the wool and jars contain better than I do.”
The stranger repeated each name. At the first spring Re’u had seemed only to confirm what he knew. Now he listened like a traveler learning where he had stopped.
“What is this place called?” he asked.
“Reed Gate.”
Nisaba had followed the stranger from the fold. She stopped one forearm’s length to his left and struck the ground once with one bronze finger. The bronze of her face held no expression Re’u could name.
“Do you know me?” she asked.
He turned toward her without hesitation. “Yes.”
“Where did you learn the question?” she asked.
For a long time, the old man said nothing.
“At the beginning.”
“This is the beginning.”
“Not mine.”
Nisaba was silent after that answer. The stranger turned his face toward the sheepfold. Behind Nisaba, animals shifted against the hurdles, a stylus scratched beneath the awning, and the six circles of stone still lay in the yard. Above them, the rush wreath had dried almost white around the bend in the old carrying pole.
“Perhaps I have grown wiser,” he said. “Or perhaps the world has simply grown simpler.”
Re’u thought of the four tablets required to describe one flock, the barley still owed, the animals that had not returned, and the children eating grain that had nearly vanished inside the word household.
“Or perhaps it has not,” he said.
The old man laughed. “No. Perhaps not.”
He raised one hand toward Nisaba’s bronze brow and stopped, uncertain. She took him by the wrist and completed the motion.
He turned toward the road. He walked until the reeds concealed him.
16.8 Arithmetic Review and Practice
The story has introduced the ideas first. This review now names the ideas, states the rules plainly, and gives you calculations to try.
16.8.1 First Reckoning: What the Count Contains
16.8.1.1 Try before reading
Return for a moment to the stones in the yard. Re’u calls what he sees six circles; Ili-iddinam records 360 sheep. How can both descriptions be correct?
Then consider Nisaba’s new sign in \((6,0)_{60}\). How can a mark for no loose group preserve the place value of another mark? Compare your answers with the explanation below, then revise them where needed.
The stranger’s lesson separates five related things:
- a sheep is a thing;
- the flock has a quantity, an amount that can be counted or compared;
- 360 is the number describing that quantity;
- the written marks \(360\) are a numeral, a representation of the number; and
- one sheep is the unit selected for this count.
The stone is neither the sheep nor the number. It helps Re’u preserve a one-to-one correspondence: one stone for each sheep, with none repeated and none omitted. The question “What did you make one?” identifies the unit and boundary before calculation begins.
In decimal notation, ten ones make one ten, ten tens make one hundred, and ten hundreds make one thousand:
\[ 10\times1=10,\qquad10\times10=100,\qquad10\times100=1{,}000. \]
Read the places from right to left: ones, tens, hundreds, thousands. Each place is worth ten times the place beside it on the right. Thus \(360\) records three hundreds, six tens, and no ones:
\[ 360=3\times100+6\times10+0\times1. \]
And \(1{,}206\) records one thousand, two hundreds, no tens, and six ones:
\[ 1{,}206=1\times1{,}000+2\times100+0\times10+6\times1. \]
The comma in \(1{,}206\) makes the long decimal numeral easier to read; removing it does not change the value. The comma inside our base-sixty parentheses has a different job: it separates whole sexagesimal places, each of which may need more than one decimal digit to write.
Zero may name an empty quantity, as in \(0\) sheep, or hold a place in a numeral. In \(205\), it records that there are no tens between two hundreds and five ones:
\[ 205=2\times100+0\times10+5. \]
The equals sign says that two expressions represent the same value; it does not mean merely that an answer follows. Thus
\[ 6\times60=360 \]
is a claim that can be checked.
The signs \(<\) and \(>\) express order. For example, \(444\text{ animals}>360\text{ animals}\) compares two counts with the same unit. The inequality does not explain why the count changed, whether the same animals are present, or whether the two counts use the same boundary. A comparison such as \(12\text{ days}>8\text{ animals}\) does not make a meaningful claim about the quantities merely because \(12>8\) as numbers.
The signs Nisaba writes from 1 through 9 and then combines in 10 belong to our modern decimal notation. Our usual notation is base ten: moving one place left multiplies its value by ten. Old Babylonian calculation often used base sixty. Each sexagesimal place can hold a value from 0 through 59 before another place is needed. This chapter uses a modern comma and subscript to expose the places:
\[ (6,0)_{60}=6\times60+0=360_{10}, \]
\[ (7,24)_{60}=7\times60+24=444_{10}. \]
For these two-place numerals, the part before the comma counts sixties and the part after it counts loose ones. The small number below the numeral is a subscript naming its base. It tells you to use places organized by sixty or by ten; it is not another amount to add or multiply.
The numeral changes with the base; the number does not. Our explicit zero in \((6,0)_{60}\) is a modern teaching device, not a reproduction of Old Babylonian notation. Sixty is useful because it divides evenly by 2, 3, 4, 5, and 6, among other numbers.
16.8.1.2 Practice and transfer
Try these before checking the responses.
- A tray contains one token for each of 24 completed orders. Identify the things, number, numeral, and unit.
- Convert \((5,12)_{60}\) to decimal notation.
- Explain the different jobs performed by zero in “zero defects” and in \(407\).
- A report says that two departments each completed “12.” What must you know before concluding that their outputs were equal?
- Explain why \(12\text{ days}>8\text{ animals}\) is not a useful comparison of quantities, even though \(12>8\).
- Expand \(2{,}040\) into thousands, hundreds, tens, and ones. What does each zero preserve?
- The completed orders are the things; twenty-four is the number; \(24\) is the numeral; one completed order is the unit.
- \((5,12)_{60}=5\times60+12=312_{10}\).
- “Zero defects” gives the count of defects. The zero in \(407\) is a placeholder showing that there are no tens between four hundreds and seven ones.
- You need at least the unit and boundary: twelve what, completed under what definition, and during what interval?
- The numbers can be ordered, but the measured quantities use incompatible units. A useful inequality must compare quantities made comparable by their units and definitions.
- \(2{,}040=2\times1{,}000+0\times100+4\times10+0\times1\). The first zero preserves the hundreds place and the second preserves the ones place. Omitting them would turn the numeral into \(24\), a different number.
16.8.2 Second Reckoning: What an Account Preserves
16.8.2.1 Try before reading
Without performing the exact calculation, decide whether
\[ 360-24-12+150-30 \]
should be greater than or less than 360. What in the expression supports your prediction?
Now imagine that every removal is recorded only as a negative change. What would the total preserve, and what would it erase about the animals and the decisions that removed them?
Arithmetic operations transform represented quantities according to rules. The rules matter, but an operation acquires its meaning from the situation and units around it.
16.8.2.2 Addition and subtraction
Addition combines compatible quantities:
\[ 324\text{ animals}+120\text{ animals}=444\text{ animals}. \]
Adding 324 animals to 120 days produces no meaningful total even though a calculator can add the numerals. Subtraction may find a remainder, compare two values, or reverse an addition. The expression \(444-360=84\) gives a difference; it does not identify what caused it.
16.8.2.3 Multiplication and division
Multiplication forms equal groups or scales a quantity:
\[ 144\text{ animals} \times\frac34\frac{\text{feeding unit}}{\text{animal}} =108\text{ feeding units}. \]
Division may ask how many groups of a selected size can be formed, as in \(360\div60=6\), or how much belongs in each of a selected number of groups, as in \(360\div6=60\). The same operation answers different questions only because the unit and grouping supply its meaning.
16.8.2.4 Signed quantities describe direction
A positive or negative sign can describe a value relative to a chosen zero or the direction of a change.
\[ (+150)+(-30)+(-24)+(-12)=+84. \]
The final count is 444, not 84; \(+84\) is the change from the starting level of 360. In \(-24\), the minus sign belongs to a negative value. In \(360-24\), it directs subtraction. Neither use creates negative sheep.
16.8.2.5 Expressions, statements, malformed notation, and undefined operations
Arithmetic notation has grammar. That means errors have different kinds.
A well-formed expression specifies a calculation and, when it is defined, represents a value:
\[ 360-24-12+150-30. \]
A statement makes a claim:
\[ 360-24-12+150-30=444. \]
This statement is true.
The statement
\[ 360-24-12+150-30=445 \]
is meaningful and false.
The string
\[ 360+\times24 \]
is malformed under ordinary arithmetic grammar. It has not yet expressed a claim that can be true or false.
Some well-formed expressions are undefined. For example, \(12\div0\) would require a number \(x\) for which \(0\times x=12\). No such number exists in ordinary arithmetic, so the expression is undefined.
Do not confuse an undefined operation with a result that falls outside a selected kind of number. The calculation \(3-5=-2\) is defined, although its result is not a nonnegative whole number. Likewise, \(1\div2=\tfrac12\) is defined, although its result is not an integer. Expanding the available kinds of number admits those results; it does not define \(12\div0\) in the arithmetic used here.
16.8.2.6 Grouping and order
Parentheses tell us what to treat as a unit of calculation.
\[ 360-(24+12)=324. \]
By contrast, \((360-24)+12=348\) removes 24 and then adds 12. The grouping changes the claim.
Use the conventional order of operations:
- evaluate grouped expressions;
- evaluate exponents;
- multiply and divide from left to right; and
- add and subtract from left to right.
The familiar mnemonics are less reliable than understanding what the notation says. Multiplication and division share a level; addition and subtraction share a level. Neither division nor subtraction should be silently moved around as though it were commutative.
16.8.2.7 Properties that permit rearrangement
Combining 24 transferred animals and 12 dead animals gives the same removal total whichever group we name first:
\[ 24+12=12+24=36. \]
Likewise, six rows of ten stones and ten rows of six stones contain the same number:
\[ 6\times10=10\times6=60. \]
Addition and multiplication are commutative: exchanging the order leaves the result unchanged. They are also associative: changing the grouping leaves it unchanged. For example, \((24+12)+30=24+(12+30)=66\).
We can state these patterns generally with letters. Here \(a\), \(b\), and \(c\) stand for numbers; they do not name another kind of operation. Writing \(ab\) means \(a\times b\). With those conventions, the same rules say:
\[ a+b=b+a, \qquad ab=ba. \]
\[ (a+b)+c=a+(b+c), \qquad (ab)c=a(bc). \]
The distributive property lets us split a multiplication into smaller parts. Seventeen groups of twelve are seventeen groups of ten together with seventeen groups of two:
\[ 17(10+2)=17\times10+17\times2=170+34=204. \]
The parentheses collect a quantity, and writing a number beside them means multiplication. Using letters to state the same relationship:
\[ a(b+c)=ab+ac. \]
Adding no animals to twelve leaves twelve: \(12+0=12\). Multiplying twelve by one also leaves twelve: \(12\times1=12\). Zero is therefore the additive identity, and one is the multiplicative identity: \(a+0=a\) and \(a\times1=a\). An opposite reverses addition: \(12+(-12)=0\), or generally \(a+(-a)=0\). A reciprocal is a number that multiplies a nonzero number back to one. For example, \(4\times\tfrac14=1\) and \(\tfrac34\times\tfrac43=1\). In general, \(a\times\tfrac1a=1\) for nonzero \(a\). These inverse relationships support useful checks on subtraction and division.
Subtraction and division are not commutative: \(12-3\neq3-12\) and \(12\div3\neq3\div12\). Rearrange an expression only when a valid property licenses the move.
16.8.2.8 Estimate, calculate, inspect
A machine should enter after you know what you are asking.
Before calculating Re’u’s new flock total, we can estimate:
- the starting count is 360;
- births add 150;
- all removals together are a little more than 60;
- the result should therefore be a little less than 450 and greater than 360.
A calculator returns
\[ 360-24-12+150-30=444. \]
That result lies in the expected range. We then inspect the units and the groups being counted. It is a count of currently present animals, not a count of original entrusted animals, adult animals, or animals owed at the next reckoning.
The calculator checked the arithmetic. It did not tell us which of those counts we needed.
16.8.2.9 Practice and transfer
- A process begins the day with 85 open cases, receives 27, and closes 34. Write an expression for the ending count and estimate whether it will be above or below 85 before calculating.
- Explain the difference between “the backlog is 8” and “the backlog changed by \(-8\).”
- Evaluate \(6+4\times5\) and \((6+4)\times5\).
- Use the distributive property to compute \(17\times12\) as \(17(10+2)\).
- Classify each as an expression, a true statement, a false statement, malformed notation, or an undefined expression: \(12+7\); \(12+7=19\); \(12+7=20\); \(12+\div7\); \(12\div0\).
- A machine reports that \(360-24-12+150-30=4{,}440\). Give two independent reasons to reject the result before recomputing it.
- Explain why \(3-5=-2\) and \(1\div2=\tfrac12\) are defined even though their results are not, respectively, a nonnegative whole number and an integer.
- \(85+27-34=78\). Because 34 departures exceed 27 arrivals by 7, the ending count should be 7 below 85.
- The first gives a level: eight cases are currently in the backlog. The second gives a change: the backlog decreased by eight from some prior level.
- \(6+4\times5=6+20=26\); \((6+4)\times5=10\times5=50\).
- \(17(10+2)=170+34=204\).
- Expression; true statement; false statement; malformed notation; undefined expression.
- The reported value is an order of magnitude too large, and the net change should be only \(+84\), so the result should be near 444 rather than 4,440.
- Each operation has a value when the available numbers expand: the first result is admitted when negative numbers are available, and the second when fractions are available. Leaving the selected number set is not the same as dividing by zero or otherwise failing to define a value.
16.8.3 Third Reckoning: Fractions, Rates, and Honest Approximation
16.8.3.1 Try before reading
The story offers three possible wholes: the 360 animals originally entrusted, the 444 animals presently counted, and a weighted total based on differing feed needs. What question can each whole answer? Why can no one of them replace the other two?
Before calculating, decide whether \(9{,}000/(444\times30)\) should be greater than or less than 1. What does your answer mean in the story’s units?
The reserve problem forces several familiar forms of arithmetic to work together. They are related, but they are not interchangeable.
16.8.3.2 Fractions name numbers
A fraction such as
\[ \frac34 \]
is a number. It may describe three of four equal parts, the result of \(3\div4\), a ratio of three to four, or a scaling factor that makes a quantity three-fourths as large.
The denominator, which cannot be zero, names how many equal parts make the selected whole. The numerator names how many of those parts are under consideration.
The phrase “the selected whole” matters. Three-fourths of a sila, three-fourths of the reserve, and three-fourths of the animals are different quantities even though the fraction is the same.
Equivalent fractions name the same number:
\[ \frac34=\frac68=\frac{75}{100}=0.75. \]
Multiplying numerator and denominator by the same nonzero number changes the representation without changing the value.
16.8.3.3 Arithmetic with fractions
To add fractions with the same denominator, add their numerators:
\[ \frac14+\frac24=\frac34. \]
For different denominators, first express the fractions in compatible parts. Half a measure contains two quarter-measures. Those two quarters and three more quarters make five quarters, or one whole measure and one quarter:
\[ \frac12+\frac34 =\frac24+\frac34 =\frac54 =1\frac14. \]
To find three-fourths of two-thirds of a measure, divide the whole measure into three equal parts and keep two. Now divide each third into four equal parts. Each small part is one-twelfth of the whole, and the two thirds contain eight such parts. Take three of the four small parts in each of those two thirds: six twelfths remain, which is half the original measure.
The denominator counts \(3\times4=12\) equal parts in the whole. The numerator counts \(2\times3=6\) selected parts. That is why multiplying fractions multiplies both their numerators and their denominators:
\[ \frac34\times\frac23 =\frac{3\times2}{4\times3} =\frac6{12} =\frac12. \]
A common error is to multiply the numerators while leaving the first denominator unchanged, producing \(\frac64\). The neatness of the written line does not protect it from that mistake. Inspect the operation, not the typography.
How many half-measures fit into three-fourths of a measure? One half-measure uses two quarters and leaves one quarter. That remaining quarter is half of another half-measure. The answer is one and one-half half-measures.
We can express the same comparison as a fraction divided by a fraction. Multiplying both quantities by two turns the half-measure into one and the three-fourths measure into three-halves. Their ratio stays the same:
\[ \frac{3/4}{1/2} =\frac{(3/4)\times2}{(1/2)\times2} =\frac{3/2}{1} =\frac32. \]
Two is the reciprocal of one-half because their product is one. The same method works for any nonzero fraction: multiply both quantities by the divisor’s reciprocal so the divisor becomes one. Division therefore multiplies by the reciprocal:
\[ \frac34\div\frac12 =\frac34\times\frac21 =\frac32. \]
It does not license division by zero, because zero has no reciprocal.
16.8.3.4 Decimals may terminate or repeat
Decimal places extend the grouping rule to the right of the ones place. The first place after the decimal point counts tenths, the next counts hundredths, and the next counts thousandths. Each is one-tenth the size of the place to its left. Thus \(0.75\) means seven tenths and five hundredths, or seventy-five hundredths:
\[ 0.75=\frac7{10}+\frac5{100}=\frac{75}{100}=\frac34. \]
Some fractions have terminating decimal representations:
\[ \frac34=0.75, \qquad \frac12=0.5. \]
Others repeat:
\[ \frac56=0.8333\ldots, \]
\[ \frac{25}{37}=0.675675675\ldots. \]
A repeating decimal is not inherently approximate. The notation \(0.\overline{675}\) represents the exact repeating value. Writing \(0.68\) instead is an approximation.
Base matters here. A fraction may terminate in one base and repeat in another. One-third can be written as exactly twenty sixtieths because \(\tfrac13=\tfrac{20}{60}\). In decimal notation the same number is \(0.333\ldots\): no finite string of tenths, hundredths, and smaller decimal places gives it exactly. The number has not changed; the representational system has.
16.8.3.5 Ratios, rates, and proportions
A ratio compares two quantities by division. After the removals, the ratio of lambs to all animals is
\[ 120:420 \]
or
\[ \frac{120}{420}=\frac27. \]
A rate is a ratio that compares quantities with different units. The emergency feeding plan uses rates such as
\[ 1\,\frac{\text{sila}}{\text{feeding unit}\cdot\text{day}}. \]
Multiplying by feeding units and days cancels those units:
\[ 330\text{ feeding units} \times30\text{ days} \times1\,\frac{\text{sila}}{\text{feeding unit}\cdot\text{day}} =9{,}900\text{ sila}. \]
Unit cancellation is not merely a classroom ritual. It is a check on whether the calculation represents the proposed relationship.
A proportion states that two ratios are equal. If 330 feeding units require 330 sila for one day at the selected rate, thirty days require thirty times that quantity:
\[ 330\text{ sila per day}\times30\text{ days}=9{,}900\text{ sila}. \]
We can check that the daily rate stays the same by comparing the two ratios:
\[ \frac{330\text{ sila}}{1\text{ day}} =\frac{9{,}900\text{ sila}}{30\text{ days}}. \]
Do not set up a proportion until you have reason to believe the rate remains constant over the range in question. Thirty days of emergency planning are not evidence that animal need remains constant through every season.
16.8.3.6 Percentages and percentage change
A percentage expresses a ratio per hundred. To write a ratio as a percentage, express the same share with a denominator of one hundred.
Of 150 lambs born, 120 survived the first dangerous weeks:
\[ \frac{120}{150}=\frac45=\frac{80}{100}=0.8=80\%. \]
Eighty percent means eighty out of every hundred in the same proportion, not that only eighty lambs survived.
The corresponding early loss percentage was
\[ \frac{30}{150}=0.2=20\%. \]
The percentages add to \(100\%\) because every lamb in this particular group is classified as surviving or not surviving through the defined interval.
Percentage change compares a change with a nonzero starting value:
\[ \text{percentage change} =\frac{\text{new}-\text{old}}{\text{old}}\times100\%. \]
If the old value is zero, ordinary percentage change is undefined because the formula would divide by zero. Report the numerical change and its context instead of calling it an infinite percentage increase.
The flock changed from 360 at entrustment to 444 present animals before the allocation decision:
\[ \frac{444-360}{360}\times100\% =\frac{84}{360}\times100\% \approx23.3\%. \]
That does not mean the entrusted flock reproduced by 23.3 percent. The change also contains transfers, adult deaths, births, and lamb deaths. The arithmetic is correct; the causal description would be wrong.
Always name the denominator. A claim that defects fell “by 20 percent” is incomplete until we know the starting defect count, the time interval, and whether the percentage describes items, opportunities, orders, or something else.
16.8.3.7 Exactness, approximation, and rounding
An exact value preserves the represented relationship without rounding:
\[ \frac{25}{37}\text{ sila per animal per day}. \]
An approximation makes the value easier to communicate or use at an available level of precision:
\[ \frac{25}{37}\approx0.68. \]
The approximation is not a defective version of the truth. It is a different representation with an error small enough for some purpose. The phrase “for some purpose” carries the burden.
Rounding an intermediate rate and then multiplying it across many animals and days can accumulate error. Whenever possible, preserve the exact fraction or additional digits during the calculation and round the final result to a precision justified by the measurement and decision.
Estimation provides an independent check. The weighted plan after removing twenty-four adults uses roughly 400 animals, roughly one feeding unit or less per animal, for 30 days. A result near 10,000 sila is plausible. A result of 990 or 99,000 would demand investigation before anyone opened a jar.
16.8.3.8 Use a machine without surrendering the question
Enter the plan into a calculator or spreadsheet as separate rows:
| Class | Count | Feeding units per animal | Days | Required sila |
|---|---|---|---|---|
| Breeding or nursing ewes | 180 | \(1\) | 30 | 5,400 |
| Other adults | 120 | \(0.75\) | 30 | 2,700 |
| Lambs | 120 | \(0.50\) | 30 | 1,800 |
| Total | 420 | 9,900 |
The machine should calculate the final column as
\[ \text{animals} \times\frac{\text{feeding units}}{\text{animal}} \times\text{days} \times1\,\frac{\text{sila}}{\text{feeding unit}\cdot\text{day}}. \]
Then inspect:
- Do the animal counts sum to 420?
- Do the units cancel to sila?
- Is 9,900 close to the estimate?
- Does the table show the groups and the choices behind their weights?
- Would changing a weight alter a biological fact, a policy choice, or both?
The final question cannot be delegated to the spreadsheet.
16.8.3.9 Practice and transfer
- Convert 7 gur, 2 barig, and 3 ban to sila using \(1\text{ gur}=300\text{ sila}\), \(1\text{ barig}=60\text{ sila}\), and \(1\text{ ban}=10\text{ sila}\).
- A reserve of 9,000 sila is divided equally over 444 animals and 30 days. Without calculating the exact rate, explain why the result must be less than 1 sila per animal per day.
- Compute the exact rate and identify its repeating decimal pattern.
- After the removals, what percentage of the 420 animals are lambs?
- Of 120 completed cases in one cohort, 84 were completed within the promised time and the rest were late. Find the on-time percentage and the late percentage.
- A defect count falls from 50 to 35. Find the change and the percentage change.
- A spreadsheet multiplies 120 lambs by 0.5 feeding unit per lamb but omits the 30-day factor. What unit will its result have, and why is that not the required total?
- Explain why releasing barley according to the 360-animal tablet and releasing it according to the 444 present animals are not merely two methods for answering the same question.
- Compute \(\frac12+\frac34\) and explain why the denominators must name compatible parts before the numerators are added.
- Compute \(\frac34\times\frac23\) and simplify the result.
- Compute \(\frac34\div\frac12\) and explain what the quotient means as a comparison of the two quantities.
- Keep the fourth plan’s counts and relative weights, but propose \(0.9\) sila per feeding unit per day instead of \(1\). Does the feeding-unit total change? How much barley does the revised thirty-day plan allocate, and does that calculation establish that the animals will receive enough?
- \(7(300)+2(60)+3(10)=2{,}100+120+30=2{,}250\) sila.
- One sila per animal per day would require \(444\times30=13{,}320\) sila, which exceeds the 9,000-sila reserve.
- \(9{,}000/(444\times30)=25/37=0.675675\ldots\), with 675 repeating.
- \(120/420=2/7\approx0.2857\), so about \(28.6\%\) are lambs.
- \(84/120=70\%\) within the promise; \((120-84)/120=36/120=30\%\) late.
- The change is \(35-50=-15\) defects, a decrease of 15. The percentage change is \(-15/50\times100\%=-30\%\).
- The result is 60 feeding units. It lacks both the 30-day duration and the rate in sila per feeding unit per day, so it is not the required quantity of barley.
- The first distributes against a historical obligation recorded at entrustment. The second distributes among currently present mouths. The unit boundary and purpose differ before the division begins.
- \(\frac12+\frac34=\frac24+\frac34=\frac54=1\frac14\). Halves and quarters are not the same-sized parts, so the halves must first be expressed as quarters.
- \(\frac34\times\frac23=\frac6{12}=\frac12\).
- \(\frac34\div\frac12=\frac34\times\frac21=\frac32=1\frac12\). Three-fourths is one and one-half times as large as one-half.
- The daily feeding-unit total remains 330 because the counts and relative weights have not changed. The revised plan allocates \(330\times30\times0.9=8{,}910\) sila. That fits within the assigned 9,000 sila, but it does not establish that the smaller ration is adequate. Adequacy depends on the animals, grazing, weather, and the evidence behind the chosen ration.
16.8.4 Fourth Reckoning: Exponents, Factors, and Prime Structure
16.8.4.1 Try before reading
Break 360 apart by starting with \(6\times60\). Then start again with \(18\times20\). If you continue until every factor is prime, what do you expect to remain unchanged?
Would writing the number as \((6,0)_{60}\) instead of \(360_{10}\) change those factors? Make a prediction before reading the theorem that answers both questions.
16.8.4.2 Repeated multiplication
Some factors repeated. Repeated multiplication can be written compactly with an exponent:
\[ 2\times2\times2=2^3, \]
\[ 3\times3=3^2. \]
In \(2^3\), 2 is the base and 3 is the exponent. The expression means three factors of 2:
\[ 2^3=8. \]
This use of the word base differs from the base of a numeral system. In \(2^3\), the base is the repeated factor. In base-ten notation, the base determines place values. Context keeps the two uses apart.
Follow the powers of two downward. Removing one factor of two divides the value by two:
\[ 2^3=8,\qquad2^2=4,\qquad2^1=2. \]
If we continue the same pattern, the next value is \(2\div2=1\). We therefore define \(2^0=1\) so that decreasing the exponent by one continues to divide the value by two. It does not mean \(2\times0\).
The same reasoning works for any nonzero base. For example, \(3^2=9\), \(3^1=3\), and \(3^0=1\): divide by three at each step. Writing \(a\) for the chosen base, \(a^1=a\) and \(a^0=1\) for nonzero \(a\). The pattern cannot make that last step for a zero base because it would require division by zero.
Division of powers now gives another way to check the rule. In \(2^3/2^2\), two factors of two in the numerator cancel the two in the denominator, leaving one:
\[ \frac{2\times2\times2}{2\times2}=2=2^{3-2}. \]
Dividing \(2^3\) by itself cancels all three factors and gives \(1\). Subtracting the exponents gives zero, so \(2^0=1\) preserves both relationships. More generally, for nonzero \(a\):
\[ \frac{a^3}{a^3}=1=a^{3-3}=a^0. \]
16.8.4.3 Factors and primes
A positive integer \(a\) is a factor of a positive integer \(n\) if some positive integer \(b\) satisfies
\[ a\times b=n. \]
Because
\[ 12\times30=360, \]
both 12 and 30 are factors of 360.
A positive integer greater than 1 is prime if its only positive factors are 1 and itself. The first primes are
\[ 2,3,5,7,11,13,17,19,\ldots \]
A positive integer greater than 1 that is not prime is composite. It can be expressed as a product of smaller positive integers.
The number 1 is neither prime nor composite. If 1 were prime, we could insert as many factors of 1 as we liked into every prime factorization and destroy the useful meaning of uniqueness.
Zero is not prime either. It is divisible by every nonzero integer and does not have the factor structure required by the definition.
16.8.4.4 Factor 360
Begin with any nontrivial factor pair:
\[ 360=6\times60. \]
Then factor each composite part:
\[ 6=2\times3, \]
\[ 60=6\times10=(2\times3)(2\times5). \]
Combining all prime factors gives
\[ 360=2\times3\times2\times3\times2\times5. \]
Reorder the factors and use exponents:
\[ \boxed{360=2^3\times3^2\times5}. \]
Check:
\[ 2^3\times3^2\times5 =8\times9\times5 =72\times5 =360. \]
Starting with another factor pair leads to the same prime factors. For example:
\[ 360=18\times20 =(2\times3^2)(2^2\times5) =2^3\times3^2\times5. \]
Both routes produce the same prime factors.
16.8.4.5 The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written as a product of primes, and that this prime factorization is unique apart from the order of the factors. You do not need to prove the theorem here. You need to recognize what it guarantees and be able to check a modest factorization.
For 360, the order can change:
\[ 2^3\times3^2\times5 =5\times3\times2\times3\times2\times2. \]
Those are not competing factorizations. They contain the same prime factors with the same multiplicities.
The theorem is a claim about positive integers, not about the numeral marks used to represent them. The number 360 may appear as
\[ 360_{10} \]
or
\[ (6,0)_{60}. \]
Its representation changes with the numeral base. Its prime factorization remains
\[ 2^3\times3^2\times5. \]
Numerals depend on a representational system; prime factorization belongs to the integer represented.
It also explains why sixty is convenient for division.
\[ 60=2^2\times3\times5. \]
Because its prime factors include 2, 3, and 5, sixty has many divisors:
\[ 1,2,3,4,5,6,10,12,15,20,30,60. \]
That makes halves, thirds, quarters, fifths, sixths, tenths, twelfths, and other common shares fit neatly into sexagesimal work. It is one practical virtue of the system, not a complete history of why people developed and retained it.
16.8.4.6 A practical factoring method
For a modest positive integer:
- divide by the smallest prime that works;
- divide the quotient again by the smallest prime that works;
- continue until the quotient is prime; and
- collect repeated factors with exponents.
Factor 444:
\[ 444=2\times222 =2\times2\times111 =2^2\times3\times37. \]
The number 37 is prime, so the process stops.
Check with multiplication:
\[ 2^2\times3\times37 =4\times3\times37 =12\times37 =444. \]
A machine can check this quickly. It can also generate a prime factorization. You still need to know what the output claims and whether the input was the intended positive integer.
Factoring 444 when the question concerned 420 is flawless work on the wrong number.
16.8.5 Responsible calculation
The arithmetic chapter can now be reduced to a short procedure, although living by it is not always short.
Before computation:
- Name the thing or event. What is being observed?
- Define one. What makes one member, occurrence, interval, or opportunity?
- Set the boundary. Which place, population, process step, and time window belong?
- Preserve the unit. What does each value measure or count?
- Choose the relationship. Does the situation call for a sum, difference, product, quotient, rate, proportion, or percentage?
- Estimate. What sign, order of magnitude, and rough range should a plausible answer have?
After computation:
- Check the arithmetic. Recompute, reverse the operation, or use another tool.
- Check the representation. Did rounding, grouping, missing data, or an incorrect denominator change the claim?
- Interpret narrowly. State what the result actually establishes.
- Disclose what it omits. Name material assumptions and unrepresented costs.
- Decide. Arithmetic can clarify alternatives and consequences; it cannot accept responsibility for the action.
This is not a ceremonial checklist to paste after every spreadsheet. It is a compact description of habits that become faster with use.
16.8.6 A machine-assisted reconciliation
Consider a four-week intake process. The working table contains these counts:
| Week | Applications received | Applications completed | Rework events | Applications first counted with rework |
|---|---|---|---|---|
| 1 | 48 | 42 | 6 | 5 |
| 2 | 52 | 50 | 4 | 3 |
| 3 | 45 | 44 | 5 | 4 |
| 4 | 55 | 49 | 6 | 4 |
| Total | 200 | 185 | 21 | 16 |
The totals are easy for a spreadsheet. Before entering a formula, define the columns.
- One application received means one application assigned a unique identifier and accepted into this process during the week.
- One application completed means one application that met the selected completion rule during the week, whether it arrived that week or earlier.
- One rework event means one recorded return from a later step to an earlier step.
- One application first counted with rework means one distinct application whose first recorded return within this four-week window occurred during that week. Count it in that week’s final column only, even if it has further rework in later weeks.
An application returning in both Week 1 and Week 3 contributes rework events in both weeks, but appears in the final column only in Week 1. That rule makes the weekly application counts nonoverlapping, so their sum is 16 distinct applications across the four weeks. If each week instead counted every application with rework that week, adding the weekly counts would not establish a distinct count for the whole period. We would have to count unique application identifiers across the underlying records.
Those definitions reveal why received applications should not be subtracted from completed applications week by week and called defects. The columns describe flow across a boundary, not two classifications of the same weekly group. A completion may come from an earlier week’s intake.
Start with a rough prediction. Intake is near 50 per week, so the four-week total should be near 200. Completion is a little lower, so its total should be somewhat below 200. If the spreadsheet reports 2,000 received or 18.5 completed, inspect the entries and formulas before interpreting the result.
In a spreadsheet, the total received might be calculated with a formula such as:
=SUM(B2:B5)The leading = tells the spreadsheet to evaluate an expression. SUM applies addition to a selected range. The result is 200 only if the intended data occupy those cells and every row uses the same unit. A correct formula over the wrong range is the spreadsheet equivalent of a flawless count at the wrong gate.
Now reconcile the process balance. Suppose 18 applications were already open at the beginning of Week 1, and a direct count finds 31 open after Week 4. If receipts were the only additions and completions the only removals, the expected closing balance would be
\[ 18+200-185=33. \]
The direct observation says 31. The arithmetic has not failed; the account is incomplete. Investigation finds two applications withdrawn by their submitters and omitted from the first table. The reconciled balance is
\[ 18+200-185-2=31. \]
Write this as a general balance relationship:
\[ \text{closing work} =\text{opening work} +\text{entries} -\text{exits}. \]
The words entries and exits should be replaced by the actual kinds of entry and exit in the process. A balance equation is useful because an unexplained difference becomes visible. It does not tell you whether the difference came from an omitted kind of work, a duplicate record, a changed definition, or a bad observation. That requires investigation.
The same table supports several different ratios. Over four weeks, the process recorded 21 rework events across 200 received applications:
\[ \frac{21\text{ rework events}}{200\text{ received applications}} =0.105\frac{\text{events}}{\text{application}}. \]
That is 0.105 recorded rework events per application received during this window. It is a volume-normalized event rate, not necessarily the average number of rework events experienced by those 200 applications. That interpretation would require the events in the numerator to come from the same application cohort as the denominator. It is also not the percentage of applications with rework because one application can generate more than one event.
The distinct-application count supports a different calculation:
\[ \frac{16\text{ applications with rework}} {200\text{ received applications}} =0.08=8\%. \]
This percentage also needs a cohort rule. If some of the 16 applications arrived before Week 1, while some of the 200 new applications will experience rework after Week 4, the numerator and denominator do not describe the same population. The calculation is numerically correct but conceptually misaligned. A defensible cohort measure might follow the 200 applications received in the four-week window until each completes, then count how many ever experienced rework.
The completed total gives an observed average:
\[ \frac{185\text{ completions}}{4\text{ weeks}} =46.25\frac{\text{completions}}{\text{week}}. \]
That rate summarizes the selected four weeks. It does not mean the process completed one quarter of an application, that every week produced 46.25 completions, or that future capacity is exactly 46.25 per week. The decimal belongs to the average, not to a physical fragment of an application.
A reliable workbook should preserve at least four layers:
- the source observations;
- the definitions and units attached to them;
- the formulas that transform them; and
- the reported values, including any rounding or formatting.
Do not type a displayed total back over its formula. Do not convert a fraction to a rounded percentage and later treat the displayed percentage as the original data. Where the decision matters, keep the count behind the rate so another person can reconstruct it.
Finally, perform a deliberate independent check. Add the four weekly receipt counts in a calculator or reverse the balance equation to solve for openings. If you have the underlying rework records with application identifiers, count the unique identifiers across all four weeks to check the distinct-application total. The aggregate table alone cannot show whether an identifier was counted twice. The purpose is not distrust of machines. It is to avoid asking one unnoticed error to certify itself.
16.8.7 Final practice
- Convert \((8,6)_{60}\) to decimal notation.
- Convert \(444_{10}\) to modern explicit base-sixty notation.
- Factor 420 into primes.
- Factor 384 into primes and verify by multiplication.
- Evaluate \(2^5\) and explain what the exponent means.
- Continue the pattern \(3^3=27\), \(3^2=9\), \(3^1=3\) by dividing by three at each step. What value should \(3^0\) have, and why is that different from \(3\times0\)?
- Explain why the prime factorization of 360 does not change when the number is written in base sixty.
- A process records 72 defects across 600 inspected units. Find the defects-per-unit rate. Explain why this is not necessarily the percentage of defective units.
- A weekly count rises from 80 cases to 100 cases. Find the numerical and percentage changes.
- A calculator displays \(2.666666667\) for \(8/3\). Distinguish the exact value from the displayed approximation.
- A team reports average handling time in minutes but enters one day’s data in seconds. Why can the arithmetic be internally correct while the result is unusable?
- Write a short answer to the question “What did you make one?” for each of these measures: completed orders, defect percentage, waiting time, and customer complaints.
- \((8,6)_{60}=8\times60+6=486_{10}\).
- \(444=7\times60+24\), so \(444_{10}=(7,24)_{60}\).
- \(420=42\times10=(2\times3\times7)(2\times5)=2^2\times3\times5\times7\).
- \(384=128\times3=2^7\times3\). Check: \(128\times3=384\).
- \(2^5=2\times2\times2\times2\times2=32\); the exponent 5 says that five factors of 2 are multiplied.
- The next value is \(3\div3=1\), so \(3^0=1\) continues the pattern. The zero is an exponent; it does not instruct us to multiply by zero. The product \(3\times0\) is zero, a different calculation.
- Numeral base changes the marks and place values used to represent the integer, not the integer’s factors.
- \(72/600=0.12\) defects per inspected unit. A single unit may contain more than one defect, so the count of defective units may be smaller than the count of defects.
- The numerical change is \(100-80=20\) cases, an increase of 20. The percentage change is \(20/80\times100\%=25\%\).
- The exact value is \(8/3=2.\overline6\). The display rounds that repeating decimal to a finite number of digits.
- The machine may correctly combine the entered numbers while combining incompatible units. Convert all values to one unit before aggregation.
- Answers will vary, but each must define the member or event, the boundary, and the time window where relevant. For complaints, for example: one complaint might mean one submitted complaint record associated with one order during the selected month, rather than every issue mentioned inside the record.
16.9 Chapter Summary
- A thing, quantity, number, numeral, and unit are related but distinct.
- Counting requires a criterion for treating something as one member of a collection.
- Zero may describe an empty quantity or preserve a place in positional notation.
- A number can have different numeral representations in different bases.
- Arithmetic operations transform represented quantities; their meaning depends on context and compatible units.
- Signed numbers can describe direction and change without turning physical things into “negative objects.”
- Fractions are numbers and can also express division, ratios, rates, and scaling.
- Percentages are ratios per hundred, and percentage change requires an explicit, nonzero starting value.
- Exact values and approximations serve different purposes; rounding should follow the needed use and precision.
- A machine can perform and check computation but cannot choose the unit, boundary, denominator, or decision.
- Every integer greater than 1 has a prime factorization unique apart from factor order.
- A correct calculation can expose the consequences of a frame without selecting the frame or accepting responsibility for what follows.
16.10 Historical Note and Further Reading
This chapter uses Reed Gate, a fictional settlement in the rural orbit of Nippur around 1800 BCE. Its people and conflict are invented. The following sources support the material and institutional frame and also mark its limits:
- The University of Chicago’s Nippur Neighborhoods treats the domestic, social, and economic world of Old Babylonian Nippur.
- ORACC’s Old Babylonian Model Contracts provides editions and translations of scribal model contracts involving barley, property, loans, witnesses, and related transactions.
- ORACC’s lexical entry for nakamtum, storehouse attests Old Babylonian Nippur terms for a storehouse, its doorkeeper, an overseer of the storehouse, and a storehouse official. “Storehouse overseer” therefore has a historical basis; Abum-ilum’s precise duties, Ili-iddinam’s assignment, and their intersecting authority are narrative constructions.
- ORACC’s lexical entries for shepherd, sheepfold, and lamb document relevant Old Babylonian vocabulary and classifications.
- The University of Chicago’s Assyrian Dictionary, Š, Part III, entry šūši, documents the word for sixty, including Old Babylonian examples. ORACC’s Old Babylonian mathematical glossary records meʾatu, hundred, and its limum entry records thousand, Akkadian līmu, with Old Babylonian attestations. The characters’ familiarity with hundreds and thousands is distinct from Nisaba’s introduction of modern decimal notation.
- ORACC’s account of Nisaba/Nidaba describes the goddess’s associations with grain, writing, and scribal culture and her cultic connection to Nippur.
- ORACC’s account of Mesopotamian metrology and the Cuneiform Digital Library Initiative’s survey of Babylonian mathematics describe the sexagesimal place-value tradition behind Ili-iddinam’s work. Nisaba’s modern decimal digits, her explicit zero placeholder, and the notation \((6,0)_{60}\) are deliberate anachronisms, not reproductions of an Old Babylonian tablet.
- The University of Hamburg’s overview of Mesopotamian calendars and the Cuneiform Digital Library Initiative’s study of the Nippur calendar describe locally variable lunisolar calendars tied to seasons and festivals. The unnamed summer feast and the children’s wreath are narrative constructions, not a reconstruction of an attested Nippur solstice festival; an Old Babylonian term for a garland or tiara is attested.
- The Metropolitan Museum’s Old Babylonian bronze figure provides a material comparison for small period bronze work. Nisaba’s scale, seamless construction, movement, speech, and changes of apparent weight are intentionally impossible.
- Stephanie Dalley’s study of Old Babylonian dowries and the University of Tübingen’s inheritance research illustrate why a widow’s claims must be specified rather than inferred from a generic statement about women’s status.
Reed Gate’s name, the storehouse’s exact relationship to an unnamed Nippur temple household, the feeding-unit weights, division of the reserve, household arrangement, temple decree, pledge, and final outcome are narrative constructions. They should not be treated as a reconstruction of one surviving Old Babylonian practice.