PrepShorts · Study sheet · Class 9 Mathematics · Chapter 3, The World of Numbers
Chapter 3 · The World of Numbers
Debts and fortunes: negative numbers close subtraction
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Negative numbers were not invented to be strange. They were invented because subtraction was broken.
The idea
Negative numbers were not introduced to be strange; they were introduced because subtraction was a broken operation. If taking 5 from 5 is a legitimate question with the answer zero, then taking 5 from 3 is a legitimate question too, and the only way to answer it is to keep going past zero. Brahmagupta's contribution was to make that continuation mean something a merchant already recognised: a debt is a quantity of which you hold the opposite. The extension is therefore the smallest one that repairs subtraction, and the debt reading shows it is not arbitrary — it names states that were in the ledgers before they were in the mathematics.
What you should be able to do
- Explain why the natural numbers, and even the whole numbers, cannot answer every subtraction question
- State what integers are, and which three ingredients the chapter combines to form them
- Give the symbol for the integers and say where the letter comes from
- Match Brahmagupta's two named states to positive and negative numbers
- Locate positive and negative integers and zero on the number line, with zero as the boundary rather than an endpoint
- Apply the printed rules for adding integers of the same sign, and for subtracting zero from either kind
- Translate a sequence of financial events into a single integer expression and evaluate it
- Explain, using a debt, why removing a negative amount has the same effect as adding a positive one
- Solve a signed-quantity word problem involving temperature change
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| Integers | the whole numbers together with zero and the negatives of the whole numbers | printed in bold in §3.3, p. 45, with the symbol ℤ |
| Negative Numbers | the numbers found on the far side of zero from the positives | printed in bold in §3.3, p. 45 |
| Fortunes (Dhana) | Brahmagupta's name for positive quantities, standing for wealth or assets | printed in bold in §3.3, p. 45, with the Sanskrit set in italics |
| Debts (Ṛiṇa) | Brahmagupta's name for negative quantities, standing for what is owed | printed in bold in §3.3, p. 45, with the Sanskrit set in italics |
| Zahlen | the German word for numbers, the source of the symbol for the integers | printed in italics in §3.3, p. 45 |
| set of integers | the collection formed by combining positives, negatives and zero | printed in §3.3, p. 45 |
| closed under subtraction | the property that subtracting any two members of a set lands back inside it | printed in Exercise Set 3.1, p. 43 |
| number line | the line on which every number has a position, positives right of zero and negatives left | printed throughout §3.3 and §3.4.1, first here at p. 45 |
| śhūnya | zero, marked at the boundary between the two states in the chapter's figure | printed in italics in Fig. 3.2, p. 45, and in §3.2 |
| repair of an operation | an added phrase for extending a number system so that an operation always returns an answer | an added term; the chapter performs the extension and gives it no name |
Where people slip up
- "Negative numbers are numbers less than nothing, which is impossible." They are numbers on the other side of a chosen origin. A temperature of −11 °C and a debt of ₹100 are both perfectly definite; the minus records a direction, not an impossibility.
- "−5 is smaller than −4 because 5 is bigger than 4." Position on the line settles it: −5 sits further left, so it is the smaller. Fig. 3.2 is the evidence.
- "The natural numbers were just missing some answers we later found." Inside the naturals, 3 − 5 has no answer at all. The answer did not exist and was not waiting to be discovered; the set was enlarged so that it would.
- "Adding two negatives should somehow move you towards zero." Rule 2 says it does not, and the debt gloss is why: borrowing more does not reduce what you owe.
- "Zero is where the negatives start." Zero is neither a debt nor a fortune in the chapter's scheme; Fig. 3.2 marks it as the point both arrows leave from. Treating zero as the first negative breaks the symmetry the figure is drawn to show.
- "Two of the trader's three events are losses, so he must end deeply in debt." He ends only ₹100 down. Signed arithmetic does not obey a majority vote, and this exercise is built to catch that.
- "Subtracting a negative is a rule you memorise." It is the removal of an obligation. Students who have the debt picture never misremember the sign.
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Worked answers to this chapter’s exercises · this video explains Exercise Set 3.2 Q1, Exercise Set 3.2 Q2, Exercise Set 3.2 Q4
Transcript1,327 words
Here are two subtractions that look like the same kind of question. Five take away five. That one is settled: the answer is nought, and we spent a whole lesson earning it. Three take away five. Same operation, same size of numbers, and the box where the answer goes stays empty. Not empty because it is hard. Empty because inside the counting numbers there is nothing to put in it.
It is worth measuring how bad that is, rather than shrugging at one case. Take the counting numbers from one to forty and try every subtraction of one by another. That is sixteen hundred questions. Eight hundred and twenty of them have no answer. More than half. Admitting nought helps, and it helps less than you would hope. It fixes every case where a number is taken from itself, and it fixes nothing else. Eight hundred and twenty still refuse.
An operation that answers only when it feels like it is not an operation. It is a habit with exceptions. The repair came from somewhere unexpected, which is to say from somewhere entirely ordinary. Merchants had been keeping two columns for centuries. On one side, what you hold. On the other, what you owe. Brahmagupta gave the two columns names and treated them as two kinds of number. Dhana, fortunes, for what you hold. Rina, debts, for what you owe.
That is the move. Not inventing a strange new object, but noticing that a state everybody already recognised had never been allowed into the arithmetic. People sometimes object that a number below nothing is impossible. But a debt of a hundred coins is not impossible. It is unpleasant, and it is entirely definite, and any merchant could tell you to the coin how big it was. On the line, a fortune is a step to the right of nought and a debt is a step to the left.
So walk left. One past nought, two past nought, three, four, five. Now three take away five has somewhere to land. Start at three, walk five steps left, and you finish at minus two. The box is filled, and it was filled by extending the road rather than by finding a cleverer route along the old one. The line also settles an argument that catches almost everyone. Which is smaller, minus five or minus four? Five is the bigger number, so minus five feels like the bigger debt, and it is. But bigger debt means further left, and further left means smaller. Position decides it, not size.
Look at where nought ended up in that picture, because it is easy to get this wrong. Nought is not the first of the debts. It is not the smallest fortune either. It is the point both directions leave from. Every fortune has exactly one debt facing it across nought, and every debt has exactly one fortune. Match them up and none is left over. That symmetry is the whole reason the line looks the way it does, and it only works if nought is standing in the middle rather than queuing up on one side.
Here is a fair worry. Did we add more than we needed, just to get one answer? So let us add nothing by choice. Start with one, two and three, and admit a number only when subtraction forces it in. One take away one forces nought. Nought take away one forces minus one. Three take away minus one forces four. Each new arrival is the answer to a question the earlier arrivals were already asking.
Run that until it stops growing and you get every whole number in both directions and not one thing besides. Nothing arbitrary went in. The extension is exactly as large as the repair required, and no larger. And the same machinery can come back smaller. Start it from two and four and it forces only the even numbers, and stops. So reaching everything was a finding, not a foregone conclusion.
That set has a name. The whole numbers, their opposites, and nought: together, the integers. Its symbol is a letter Z, written with a doubled stroke. Not for anybody's name. It is the first letter of Zahlen, which is simply the German word for numbers, and it stuck. Subtraction inside the integers refuses nothing. Sixteen hundred questions, and this time all sixteen hundred have an answer. That is what the word closed means when you meet it. A set is closed under an operation when the operation can never take you outside it. The counting numbers were not closed under subtraction. The integers are, and that was the entire point of building them.
Now the arithmetic. Two rules first, and they are almost boring, which is a good sign. A fortune added to a fortune is a fortune. Five and four make nine. A debt added to a debt is a debt. Owe five, then borrow four more, and you owe nine. There is a temptation to think two negatives should somehow head back towards nought, the way two minus signs cancel elsewhere. They do not. Borrowing more does not reduce what you owe.
Check it over every pair of debts from minus one to minus forty. Sixteen hundred pairs. Not one of them lands closer to nought than the larger debt it started from, and not one comes out a fortune. The third rule is about taking nothing away, and it is the one that has to be said out loud precisely because it looks like it needs no saying. Seven, less nothing, is seven. Minus six, less nothing, is minus six.
Both kinds are untouched. Nought is not a small nudge towards the positive side; it is not a nudge at all. Every number in the range, tested: none of them moves. Subtracting one moves things. Subtracting nought does not. Take those rules somewhere real. A high desert plateau, thin air, and a thermometer. At noon it reads four degrees. Overnight the temperature falls by fifteen. Follow it down. Four, three, two, one, and then a place the old numbers had no name for.
The reading passes through nought and keeps going. Sixteen readings from noon to midnight, and the sixteenth is minus eleven. Notice what nought did there. It did not stop the fall or slow it. The thermometer went through it without noticing, which is exactly what a hinge is for. A trader's week, in three events. A loan of eight hundred and fifty coins. The next day, a profit of twelve hundred. The week after, a loss of four hundred and fifty.
Written as one expression: minus eight fifty, plus twelve hundred, minus four fifty. Before you work it out, feel the pull of the wrong answer. Two of the three events are bad. Most people call it a heavy loss. It comes to minus one hundred. The trader ends a hundred coins in debt, and only a hundred. The signs do not vote. Search every triple of that shape and you find plenty that end ahead: minus fifty-eight, minus one, and sixty. Two losses, and you finish in profit. Nothing about the tally decides it. You have to add.
One last thing, and it is the rule most often memorised and least often understood. A minus in front of a minus turns into a plus. Do not memorise it. Do it. Your ledger has a holding of twenty and two debt slips of five each. It comes to ten. Now one of the debts is cancelled. Tear the slip up. Nothing was added to your holdings, nothing arrived; a piece of paper left the pile.
The ledger now comes to fifteen. Ten, take away a debt of five, is fifteen. And that is general. Removing a debt always raises the total, removing a fortune always lowers it, and removing nought does neither. Subtraction did not become strange when we extended it. It finally became honest.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- One-to-one correspondence: counting without number wordsClass 9 · Ch 3, The World of Numbers
- From śhūnyatā to śhūnya: turning nothing into a number you can compute withClass 9 · Ch 3, The World of Numbers
Comes up again in
- Why a debt times a debt is a fortuneClass 9 · Ch 3, The World of Numbers
- What "rational" means, and why the denominator cannot be zeroClass 9 · Ch 3, The World of Numbers
- Placing a rational number on the line, and distance as |a − b|Class 9 · Ch 3, The World of Numbers