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Chapter 2 · Operations with Integers
Why a negative times a negative must be positive
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You cannot count your way to (−4) × (−2). There is no such thing as adding something minus four times.
The idea
You cannot count your way to (−4) × (−2). Repeated addition says nothing about what it would mean to add a bag of things minus four times, so the value is not discovered — it is decided, and the chapter's whole job is to show that the decision is forced. It argues twice. Once by fixing the only meaning a negative multiplier can have if the token picture is to stay consistent: a positive multiplier puts things in, so a negative one takes things out, and taking out negatives leaves positives behind. Once by refusing to let a ladder of products break at zero: the steps have been the same size all the way down, so they stay the same size below zero too. Two independent commitments, made for different reasons, land on the identical answer — and that agreement, not an announced rule, is why the answer is not arbitrary.
What you should be able to do
- Interpret 4 × 2 and 4 × (−2) as placing tokens into an empty bag, and state what the multiplier and the multiplicand each control
- Name the multiplier, the multiplicand and the product in a written multiplication, using this chapter's convention
- Explain why a negative multiplier is modelled by removal rather than placing
- Carry out (−4) × 2 and (−4) × (−2) with tokens, inserting zero pairs first, and say why the bag must start empty
- Re-express a removal of positives as an addition of negatives, and get the same product
- Continue a descending multiplication ladder past zero and state the constant step it keeps
- State the four sign outcomes of multiplying two integers, and attribute each to the evidence that produced it
- Use 1 × a = a and −1 × a = −a for any integer a, and connect the second to the additive inverse
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| multiplication | the operation being extended here from whole numbers to integers | printed in the heading §2.2, p.29 |
| multiplier | in this chapter, the first factor — the one that says how many times | printed and labelled on the figure, p.29 |
| multiplicand | in this chapter, the second factor — the one being placed or removed | printed and labelled on the figure, p.29 |
| product | the result of the multiplication | printed and labelled on the figure, p.29 |
| token | the green or red counter standing for +1 or −1 | printed throughout §2.2, pp.29–32 |
| empty bag | the starting state for every new multiplication in this model | printed on pp.29–30 |
| zero pair | a green and a red together, worth nothing, inserted to make a removal possible | printed on p.28 and used again on p.30 |
| additive inverse | the number that adds to a given number to make zero | printed on pp.28–29, used again on p.34 |
| magnitude | the size of an integer, ignoring its sign | printed in bold on p.27, used again on pp.33–34 |
| pattern | the descending ladder of products the section reasons from | printed in the heading "Patterns in Integer Multiplication", p.32 |
| sign rule | the four-case statement of when a product is positive | the explanation's shorthand; the chapter states the four cases but gives them no collective name |
| repeated addition | multiplication read as adding the same amount several times | the explanation's phrase; not printed in this chapter |
Page numbers in the provenance column are Part II's, printed pages 24–46.
Two cautions: First, multiplier and multiplicand are used in this chapter in the order the page labels them: in 4 × (−2), the 4 is the multiplier. Some other books reverse this. Since the whole ladder argument is phrased in terms of the multiplier decreasing, getting the two words the wrong way round makes the printed patterns unreadable. Second, the chapter never abbreviates a product's sign behaviour into a slogan; it lists four cases. An explanation that opens with "minus times minus is plus" has thrown away the lesson.
Where people slip up
- "Multiplying always makes things bigger." 4 × (−2) is further from zero but smaller than 4; (−4) × (−2) is positive but nothing was ever positive to start with. Treat "bigger" and "further from zero" as two different questions from the first section.
- "A negative times a negative is positive because two minuses cancel, like in language." The chapter never argues from grammar, and the grammar analogy fails immediately for a negative plus a negative. The reason is the token model plus the unbroken ladder — say the reason, not the mnemonic.
- "(−4) × 2 must be worked out differently from 4 × (−2), since the roles differ." They are modelled differently on pp.29–30 — one places, one removes — and they still land on the same value. That coincidence is worth pausing on; it is the seed of the commutativity argument in Commutative, associative, and distributive over the integers.
- "You can only remove what is already in the bag." Every new operation starts empty on purpose. Inserting zero pairs first is what makes removal always available, exactly as it did for subtraction on p.28.
- "The answer depends on how you draw the tokens." The p.31 task exists to kill this. Three different-looking sets all stand for −2 and all give the same answer when placed four times.
- "The pattern proves the rule." The ladders are strong evidence, not a deduction from nothing — what they actually show is that keeping the step constant forces these values if the extension is to be consistent. Present them as the chapter does: a reason to accept the extension, alongside the token argument, not instead of it.
- "Since the product's magnitude never changes, the signs must not matter." The magnitude is indeed decided by the magnitudes alone. The sign is decided separately. Splitting a product into those two independent questions is the cleanest thing a student can take from p.33.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 2.2 Q1, Figure it Out · 2.2 Q2, Figure it Out · 2.2 Q3, Figure it Out · 3 Q1, Figure it Out · 5 Q13
Transcript1,332 words
Start with an empty bag. Multiplication, so far, has meant repeated adding. Four times two. Put two green tokens in the bag, and do that four times. Two. Four. Six. Eight. The bag is worth eight. Nothing surprising there. But notice exactly what each of the two numbers just did. The four said how many times. The two said what went in each time. Those are different jobs, and keeping them apart is going to matter enormously.
Because in a moment, one of them is going to be asked to do something impossible. Hold on to the bag. We are going to use it for everything that follows. Now change the colour. Four times minus two. Minus two is two red tokens, and a red is worth minus one. So put two reds in, and do that four times. Same four placings, same rhythm. Minus two. Minus four. Minus six. Minus eight. The bag is worth minus eight.
That needed no new idea at all. The multiplier was still four, still a count. Only the thing being placed changed, from green to red. And notice something already. Eight is further from zero than four. But minus eight is smaller than four. Further from zero, and bigger, are not the same question. Three parts here, and each deserves a name, because what is coming depends on telling them apart.
In four times minus two, the first number is the multiplier. It says how many times. The second is the multiplicand. It is the thing being placed. And minus eight is the product. You will see those first two names swapped elsewhere. Here, the multiplier comes first. That is a convention, not a fact, and it matters only because everything ahead is phrased in terms of the multiplier. So far the multiplier has been four. A count. A number of times.
Now watch what happens when we ask it to be negative. Minus four, times two. Read that the way we have read everything so far, and it says: put two greens in, minus four times. Which is not a sentence. There is no such thing as doing something minus four times. Repeated adding has run out. It does not give a wrong answer here. It gives no answer at all.
And that is the important part, so let me say it plainly. The value of minus four times two is not discovered. It is decided. Nobody finds it by counting. Somebody chooses it, and then has to defend the choice. The rest of this is that defence. And it gets made twice, in two completely different ways. First way. If a positive multiplier means putting things in, a negative one should mean taking things out.
That is the only reading that keeps the picture consistent, so let us commit to it and follow it honestly. Minus four times two now means: take two greens out, four times. But the bag is empty. There is nothing in it to take. So buy yourself some room. Drop in two pairs, each one green and one red. A pair like that is worth nothing, so the bag is still worth nothing.
Now take the two greens out. Two reds are left behind. Do that four times over, and the bag holds eight reds. Minus four times two is minus eight. Now the case all of this has been building towards. Minus four, times minus two. Same reading as before. Take out minus two, four times over. Minus two is two reds. So this time, it is reds we are taking out.
Start empty again. Drop in two pairs. Take the two reds out. Two greens are left standing there. Do it four times, and the bag holds eight greens. Minus four times minus two is eight. And look where those greens came from. Not one of them was ever put in. Every single one arrived as half of a pair worth nothing, and stayed behind when its partner was carried out.
Four results now, and all four came from the same two numbers. A four and a two. Four times two is eight. Four times minus two is minus eight. Minus four times two is minus eight. And minus four times minus two is eight. Four sign cases, one pair of sizes, and only two different answers between them. Now. Should you believe this? It is not nothing. It came out of a picture we committed to and then followed without cheating.
But we did choose that picture. And a picture we chose cannot be the only reason to accept an answer. So, two things before trusting it. Whether the picture is even well behaved. Then a completely separate argument. Well behaved first. Here are three different drawings. Two reds. Two reds, and under them, two more reds and two greens. Four reds, and under them, two reds and four greens. Count each one up. Every one of them is worth minus two.
They look nothing like each other. They are the same number wearing three different coats. Now place each one four times, and see whether the answer cares which coat you used. Minus eight. Minus eight. Minus eight. It does not care, and the picture is well behaved. One more thing before we leave the bag alone. Minus four times two was worked out by taking greens out. That felt like the awkward one.
But taking two greens out does exactly what putting two reds in does. The bag cannot tell them apart. So do it the other way. Put two reds in, four times. Eight reds. Minus eight. The same answer, with nothing ever removed and no pairs ever bought. Which is worth stopping on. Four times minus two placed things. Minus four times two removed them. Two genuinely different procedures. One of them buys eight pairs, the other buys none.
And they land on the same number, every time you try it. Now the second argument. And it does not mention a single token. Write out four times three. Then three times three. Then two, then one, then zero. Twelve. Nine. Six. Three. Zero. Every time the multiplier drops by one, the product drops by three. Not roughly three. Exactly three, every single step, all the way down. There is no reason for a staircase with a constant step to suddenly change its step at the bottom.
So do not let it. Keep walking. Minus one times three. Minus two times three. Minus three times three. Minus three, minus six, minus nine. Now build the whole staircase again, with minus three. Four times minus three is minus twelve. Then minus nine, minus six, minus three, zero. Here the product rises by three each time the multiplier drops by one. Rises by three. Say that carefully, because it is easy to say the wrong thing here.
The step is the size of the multiplicand, three, and not the multiplicand itself. Keep that step, and carry on below zero. Minus one times minus three is three. Minus two times minus three is six. Minus three times minus three is nine. The staircase never changed its step. It just kept walking. So there are two arguments. A bag of tokens, and a staircase. They were built for different reasons, and neither of them knows anything about the other.
And they agree. Everywhere, on every single product. That agreement is the answer to the question we left open. Not a rule handed down, but two separate commitments landing in the same place. Which is also why you should refuse the shortcut. Two minuses cancel, the way they do in language. They do not. Minus four plus minus two is minus six, not six. That story collapses immediately. Line the columns up instead. The size of a product is settled by the two sizes.
The sign is settled by whether the two signs match. Two separate questions. And multiplying by minus one simply turns a number round.
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
- Negative numbers on the number line, and adding and subtracting themClass 7 · Ch 2, Operations with Integers
- Factorise and regroup instead of multiplying head-onClass 7 · Ch 1, Large Numbers Around Us
Comes up again in
- The sign rule for division, and why it follows from multiplicationClass 7 · Ch 2, Operations with Integers
- Magic grids of integersClass 7 · Ch 2, Operations with Integers
- Commutative, associative, and distributive over the integersClass 7 · Ch 2, Operations with Integers
- Evaluating expressions that mix integers and bracketsClass 7 · Ch 2, Operations with Integers