Operations with Integers

Before multiplying integers, it helps to remember how addition and subtraction of integers work, using the token model. A green token stands for +1, and a red token stands for -1; one green and one red token together cancel out to make a "zero pair", since +1 + (-1) = 0.

Adding integers with tokens is straightforward: to add 5 and -3, place 5 green tokens and 3 red tokens together, let 3 green-red pairs cancel out as zero pairs, and count what's left — 2 green tokens, meaning the sum is 2. Subtracting an integer works by removing tokens of the matching colour; if there aren't enough tokens of that colour already in play, you first add in extra zero pairs (which doesn't change the total) so that there's enough to remove.

This leads to a very useful equivalence: subtracting a number is always the same as adding its additive inverse. The additive inverse of an integer a, written -a, is the number that makes a + (-a) = 0. So subtracting 18 is the same as adding -18, and subtracting -12 is the same as adding +12 (since the additive inverse of -12 is 12).

This single idea — that subtraction is really just "addition of the opposite" — turns out to be exactly the bridge needed to understand multiplication of negative numbers too.

The token model extends naturally to multiplication, by thinking of multiplication as repeatedly placing (or removing) a group of tokens into an empty bag. For 4 × 2: start with an empty bag, and place 2 green tokens into it, 4 separate times. The bag ends up with 8 green tokens, so 4 × 2 = 8.

For 4 × (-2): the same idea applies, but now you place 2 red tokens into the empty bag, 4 times. The bag ends up with 8 red tokens, meaning 4 × (-2) = -8. In both of these cases, the multiplier (4) was positive, so tokens were placed into the bag.

What happens when the multiplier itself is negative, as in (-4) × 2? A negative multiplier means removing tokens from the bag instead of placing them in. Starting from an empty bag, to remove 2 green tokens you first need some green tokens to remove — so you add in 2 zero pairs (one green, one red each), then remove the 2 green tokens, leaving 2 red tokens behind. Doing this 4 times leaves 8 red tokens in the bag, so (-4) × 2 = -8.

Finally, (-4) × (-2) means removing 2 red tokens from the bag, 4 times. Again starting from empty, you add 2 zero pairs, remove the 2 red tokens, and are left with 2 green tokens each time — giving 8 green tokens after 4 rounds, so (-4) × (-2) = 8. These four small examples — 4×2=8, 4×(-2)=-8, (-4)×2=-8, (-4)×(-2)=8 — already contain the entire pattern for multiplying any two integers.

The token results from the previous section generalise into one simple rule: when the two integers being multiplied have the same sign (both positive, or both negative), the product is positive. When they have different signs (one positive, one negative), the product is negative. Checking this against real numbers: (-9) × 7 = -63 (different signs, negative product), (-6) × (-8) = 48 (same sign, positive product), and 12 × (-5) = -60 (different signs, negative product).

It's worth noticing that the size (magnitude) of the product never depends on the signs at all — only on the sizes of the two numbers being multiplied. The signs only decide whether the final answer is positive or negative; swapping a number for its negative never changes how big the answer is, only whether it has a minus sign in front.

Integer multiplication is also commutative — swapping the order of the two numbers never changes the product, so a × b = b × a for any integers a and b. This remains true however the signs work out: (-6) × 9 = -54 and 9 × (-6) = -54 as well.

These sign rules for multiplication (and division) were first explicitly written down by the Indian mathematician Brahmagupta in his Brahmasphutasiddhanta (628 CE), using the language of "fortune" (positive wealth) and "debt" (negative wealth): the product of two fortunes is a fortune, the product of two debts is a fortune, and the product of a fortune and a debt is a debt. This was the very first time in history that rules for multiplying positive and negative numbers were written down explicitly — centuries before negative numbers were fully accepted elsewhere in the world.

Just as with fractions and whole numbers, division of integers can always be rewritten as a multiplication question. Asking "What is (-63) ÷ 9?" is the same as asking "What number, multiplied by 9, gives -63?" Since 9 × (-7) = -63, the answer is (-63) ÷ 9 = -7.

This reframing means the exact same sign rules from multiplication carry over directly to division: when the dividend and divisor have the same sign, the quotient is positive; when they have different signs, the quotient is negative. Checking this: (-54) ÷ (-6) = 9 (same sign, positive quotient), while 81 ÷ (-9) = -9 (different signs, negative quotient), and (-45) ÷ 5 = -9 (different signs, negative quotient).

A quick way to fill in a missing number in a division or multiplication statement is to work out the size first, ignoring signs, and then apply the sign rule separately. For "(-72) ÷ x = -8, find x": ignoring signs, 72 ÷ 8 = 9; then, since the dividend is negative and the quotient is also negative, x itself must be positive (because a negative divided by a positive gives a negative) — so x = 9. Checking: (-72) ÷ 9 = -8, confirming the answer.

Integer multiplication keeps the same key properties as whole-number multiplication. It is associative: grouping three integers differently in a multiplication never changes the final product, so (a × b) × c = a × (b × c) for any integers. Checking with numbers: (3 × (-4)) × (-5) = (-12) × (-5) = 60, and 3 × ((-4) × (-5)) = 3 × 20 = 60 — the same answer either way.

It is also distributive over addition: a × (b + c) = (a × b) + (a × c), even when b or c is negative. Checking (-3) × (5 + (-7)): directly, 5 + (-7) = -2, so (-3) × (-2) = 6. Distributing instead: (-3 × 5) + (-3 × -7) = -15 + 21 = 6 — matching exactly.

These properties make real-world integer problems much easier to handle. A room's temperature starts at 10°C and drops by 4°C every hour for 6 hours; the total change is 6 × (-4) = -24°C, so the final temperature is 10 + (-24) = -14°C. A mining elevator starts 20 m below ground (at -20 m) and rises 5 m every minute for 8 minutes; the total rise is 8 × 5 = 40 m, so its final position is -20 + 40 = 20 m, meaning it ends up 20 m above ground.

In an exam with +4 marks for a correct answer and -2 marks for a wrong one, a student with 15 correct and 8 wrong answers scores 15 × 4 + 8 × (-2) = 60 + (-16) = 44 marks. Each of these situations — temperature, elevator position, exam marks — is really the same underlying arithmetic: a rate (or per-unit value) multiplied by a count, added onto a starting value, with the signs doing all the work of tracking direction or gain-versus-loss automatically.