PrepShorts · Study sheet · Class 7 Mathematics · Chapter 2, Operations with Integers
Chapter 2 · Operations with Integers
The sign rule for division, and why it follows from multiplication
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Division of integers gets no picture of its own — no bag, no tokens, no staircase. That absence is the whole point.
The idea
Division of integers gets no model of its own — no bag, no tokens, no ladder — and that absence is the argument. Every division here is rewritten as a multiplication with a hole in it: (−100) ÷ 25 becomes what must 25 be multiplied by to reach −100? Once the question is asked that way there is nothing new to settle about signs, because the multiplication table already decided which factor fills the hole. So the four division cases are not a second rule to be memorised alongside the four multiplication cases; they are the same four facts read from right to left. Brahmagupta wrote them as one statement covering both operations fourteen centuries ago, which is exactly what you would expect of two operations that are one operation asked in two directions.
What you should be able to do
- Restate any integer division as a multiplication with an unknown factor
- Answer that unknown-factor question using a multiplication fact already known
- State the sign of a quotient from the signs of dividend and divisor, in all four cases, and justify it from the matching multiplication
- Read and use the chapter's three printed division identities, including the condition attached to them
- Use dividend, divisor and quotient correctly
- Explain why the answer to the unknown-factor question is unique when it exists
- Recognise that reframing a division does not guarantee an integer answer, and say what the answer is when the divisor does not divide the dividend
- Place Brahmagupta's fortune-and-debt statement alongside the modern sign rules and see them as the same content
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| division | the operation being extended to integers here, by reframing it as multiplication | printed in the sub-heading "Division of Integers", p.38 |
| quotient | the answer to a division | printed in the SUMMARY, p.45, and in the Brahmagupta passage, p.35 |
| dividend | the number being divided | printed in the SUMMARY, p.45 |
| divisor | the number you are dividing by | printed in the SUMMARY, p.45 |
| product | the result of a multiplication — what the reframed question aims at | printed throughout §2.2, pp.29–41 |
| multiplied | the verb the reframe turns every division into | printed on p.38 in the reframing questions |
| Brahmagupta | the seventh-century mathematician whose rules the chapter quotes | printed in the Brahmagupta passage, p.35 |
| fortune / debt | Brahmagupta's words for a positive and a negative quantity | printed in the Brahmagupta passage, p.35 |
| dhana | the Sanskrit term the chapter glosses as fortune | printed in italics in the Brahmagupta passage, p.35 |
| ṛṇa | the Sanskrit term the chapter glosses as debt | printed in italics in the Brahmagupta passage, p.35 |
| unknown factor | the hole in the multiplication that the division is asking you to fill | the explanation's phrase; the chapter poses the question but gives it no name |
| exact division | a division whose answer is itself an integer | an added term; not printed in this chapter |
Page numbers in the provenance column are Part II's, printed pages 24–46.
One caution: Dividend and divisor appear in this chapter only in the SUMMARY on p.45, not in the section that teaches division. Introduce them, because they are examinable and they are how the chapter finally words the rule, but do not imply the teaching section used them.
Where people slip up
- "Division needs its own sign rule, so that's eight rules in total." There are four facts, readable in two directions. The section's entire method is to convert every division into a multiplication before any sign question is asked.
- "Because dividing makes things smaller, a negative divided by a negative should be negative." Size and sign are separate questions, as they were for products. (−100) ÷ (−4) is 25: positive, smaller in magnitude than the dividend, and — this is the half that does the work — larger in magnitude than the divisor. Dividing shrank the answer relative to one of its inputs and grew it relative to the other, so "dividing makes things smaller" is not even a statement about size, let alone about sign.
- "The rules on p.39 use a and b, so a and b could be anything." Read the condition: on p.39 the letters stand for positive integers and the negatives are written as −a and −b. This is precisely where a student who has memorised the shape rather than the statement gets caught.
- "Every integer division has an integer answer." Nothing in the section promises that, and item 11 on p.44 contains two that do not. The reframe only answers the question when the divisor actually divides the dividend; otherwise the hole has no integer filling.
- "Dividing by zero gives zero." The chapter attaches b not equal to zero to its general statement and leaves it there. The reframe explains why: 0 ×? can never reach a non-zero number, so the question has no answer at all.
- "Brahmagupta's lines are a quaint old version; the real rule is modern." They are the same rule, and the chapter says so. The interesting point is that fortune and debt gave the seventh century a reason to accept it, which is the same job the token bag does on p.30.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 4 Q1, Figure it Out · 4 Q4, Figure it Out · 5 Q2, Figure it Out · 5 Q11
Transcript1,285 words
Division of whole numbers you already have. Division of integers is what is left. And here is the surprising thing about it. Nothing new gets built. There is no bag this time. No tokens, no counters, no staircase of products. If you are waiting for a fresh picture to arrive, it never does, and that is not an oversight. It is the whole method. Every division gets turned into a multiplication before anyone asks about a sign.
And multiplication has already been settled. So watch for the moment the question changes shape. Everything else follows from it. Here is the first one. Minus one hundred, divided by twenty five. Do not reach for a rule. Ask a different question instead. Twenty five, multiplied by what, gives minus one hundred? That is the same question. Not a similar one, not an easier one. The same one, written the other way round.
Division has become a multiplication with a hole in it. The divisor sits on the left, the dividend on the right, and the hole is the only thing we do not know. So look at the shape of it. Twenty five, times something, is minus one hundred. You have seen that multiplication before. It was on the board a moment ago. Twenty five times minus four is minus one hundred.
That is a fact from the multiplication we did already. Nothing about division went into making it. So minus four fills the hole. Which means minus one hundred divided by twenty five is minus four. Now notice what did not happen there. Nobody asked what sign a quotient ought to have. The question never came up. It could not, because the multiplication had already answered it before we started. That is the trick, and it is the only trick in this whole topic.
Now change one thing about it, and watch the same trick work again. Same dividend. Minus one hundred, again. But this time, divide it by minus four. Reframe it the same way. Minus four, multiplied by what, gives minus one hundred? And again, you already know. Minus four times twenty five is minus one hundred. So the answer is twenty five. Positive twenty five, out of two negatives. Put those two side by side now, because this is exactly where the pattern shows itself.
A negative divided by a positive came out negative. A negative divided by a negative came out positive. And neither of those was decided just now. Both were read straight off multiplications we had already done. One more, and this time run the whole thing in reverse. Start from a product. Minus twenty five, times minus two, is fifty. That single fact is just sitting there, doing nothing. What divisions does it answer?
Fifty divided by minus twenty five is minus two. And fifty divided by minus two is minus twenty five. Two divisions, out of one multiplication. Every product answers two, unless its factors happen to be equal. So the multiplication table you already have is a division table as well. Nobody needs to print a second one. You have been reading it in one direction all along. Read it in the other.
There is a step in all of this that gets skipped, and it deserves a sentence. We said minus four fills the hole, and then we said the answer is minus four. That only works if nothing else fits. And nothing else does. Twenty five times minus three is minus seventy five. Times minus five is minus one hundred and twenty five. There is no second integer that lands on minus one hundred. There never is.
As long as the divisor is not zero, the hole has at most one filling. That is the hinge the whole method turns on. Answer the multiplication, and you have answered the division. Not one of several answers. The answer. So where does that leave the signs? Positive divided by positive is positive. Positive divided by negative is negative. Negative divided by positive is negative. And negative divided by negative is positive.
Four cases. And you have seen all four of them before, in exactly that order. They are the four multiplication cases. Not similar to them. The same four. So there are not eight rules here to carry around. There are four facts, readable in two directions. Which is worth being slightly annoyed about, because it is very often taught as eight. Every one of these cells is a multiplication you already know, turned around and read backwards.
Written in letters, three of these get stated together, and there is a catch in them. Divide by minus b instead of b, and the answer turns round. Divide minus a instead of a, and it turns round. Divide minus a by minus b, and it turns round twice, so you are back where you started. Here is the catch. In those three statements, a and b are positive. The minus signs are written in. They are not hiding inside the letters.
That matters, because if a is already negative, then minus a is a positive number. A student who memorised the shape instead of the statement gets caught right there. Now something the reframe does not promise. It does not promise the hole can be filled at all. Thirty two divided by six hundred and forty eight. Reframe it. Six hundred and forty eight, times what, gives thirty two? Times one is too big already. There is no integer that works.
The hole simply stays empty. The answer exists, but it is a fraction, four eighty firsts. So the method tells you the sign, and it also tells you honestly when there is nothing to sign. That is not a weakness. A method that always produced an integer would be lying. And zero, which fails more interestingly than you might expect. Divide seven by zero. Reframe it. Zero, times what, gives seven?
Nothing. Zero times anything is zero, so it never reaches seven. The hole cannot be filled. Now divide zero by zero. Zero, times what, gives zero? Everything. One works. Minus fifty works. Every integer there is works. So dividing by zero fails twice over, and for opposite reasons. Too few answers in one case, far too many in the other. And you can only see that because the question was rewritten. The rule alone would just say no.
One more idea to take apart. Dividing makes things smaller. Minus one hundred divided by minus four is twenty five. Twenty five is smaller than a hundred, so that sounds right. But twenty five is bigger than four. Much bigger. So the division shrank the answer next to one of its inputs, and grew it next to the other. Smaller than what, is the question. Until you say, the sentence means nothing.
And none of it touches the sign anyway. Size and sign are separate, as they were for products. The magnitudes settle the magnitude. The signs settle the sign. Last thing, and it is old. In six hundred and twenty eight, an Indian astronomer named Brahmagupta wrote these rules down. He called a positive quantity a fortune and a negative one a debt. Fortune divided by fortune is fortune. Debt divided by debt is fortune.
Fortune divided by debt is debt. And debt divided by fortune is debt. Four lines. And here is the part that matters. The same four lines covered multiplication too. He did not write one set for products and another for quotients, because there was never any need. Fourteen centuries later, that is still the cleanest way to say it. Turn the division into a multiplication, and the sign takes care of itself.
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
- Why a negative times a negative must be positiveClass 7 · Ch 2, Operations with Integers
- Negative numbers on the number line, and adding and subtracting themClass 7 · Ch 2, Operations with Integers
Comes up again in
- Evaluating expressions that mix integers and bracketsClass 7 · Ch 2, Operations with Integers
Either side of this one
- Magic grids of integersClass 7 · Ch 2, Operations with Integers