Multiplying and Dividing Integers

A positive times a negative

In the foundations chapter you read $3 \times 4$ as “three groups of four” and found the answer by adding:

Nothing about that meaning depends on the thing in each group being positive. If each group is a negative number, you simply add a negative number repeatedly. Take $3 \times (-4)$, three groups of $-4$:

Three leftward steps of four units each pile up, the way two same-sign steps always do. Together those three steps land you twelve units to the left of zero:

So $3 \times (-4) = -12$. A positive count of negative groups gives a negative total, for the same reason that adding several negatives gives a negative. In both cases every step goes the same way, to the left.

That reasoning never mentioned the particular numbers. However many equal leftward steps you take, they pile up instead of cancelling. So the total always lands left of zero, and its size is the number of steps times the length of each.

Positive times negative: multiply the two sizes as usual, and the answer is negative.

There is a catch, and it is worth seeing now. That argument needs the positive factor to be the one counting the groups. Write the factors the other way round, as $(-3) \times 5$, and “negative three groups of five” describes nothing you could count out. From the foundations chapter, multiplication is commutative: the two factors can swap places without changing the answer. So the order that has a meaning settles the one that does not. Five groups of $-3$ is $-15$, and therefore:

Either way the rule is the same: when one factor is negative and the other is positive, the product is negative.

A negative times a negative is positive

Repeated addition has now run out. Commutativity rescued $(-3) \times 5$ by turning it into a product you could count out. Nothing rescues $(-3) \times (-5)$ that way, because neither order describes groups. So we cannot read the answer off a picture. We have to work out what the answer is forced to be.

First, watch the pattern. Hold the first factor at $-3$ and walk the second factor down one step at a time, using only products you already have:

Each step down in the second factor adds $3$ to the product: $-6$, then $-3$, then $0$. Keeping that same step going past zero gives $3$ and then $6$. So the pattern predicts $(-3) \times (-1) = 3$ and $(-3) \times (-2) = 6$, both positive.

Now see why arithmetic forces it. A pattern is a strong hint, not a reason. Here is the reason, and it uses only rules you already have. Start with a sum that is worth nothing:

Now work out that same expression the other way. The distributive property from the foundations chapter lets a factor outside a pair of parentheses multiply each term inside:

Both lines describe one expression, so they must agree. The first line says it is $0$. The first product in the second line has one negative factor, so it is $-15$. That leaves

Only one number brings $-15$ up to zero, and that is $15$. So $(-3) \times (-5) = 15$, not because anyone chose it, but because any other value would break the distributive property.

Negative times negative: multiply the two sizes as usual, and the answer is positive.

Nothing about the argument depended on $3$ and $5$. Run it yourself on a different pair.

The same steps work for any two negatives, so the result is general. A short way to remember it is that each negative factor flips the sign once. One negative flipped $15$ down to $-15$, and the second negative flipped it back up to $15$:

The sign rules, gathered

For factors that are not zero, the sign of a product depends on nothing but how many of them are negative. Take $(-3) \times (-5) = 15$, which has two negative factors and lands positive. Multiply that by one more negative, $15 \times (-2) = -30$, and now there are three negative factors and the answer lands negative.

Every case above reduces to one idea, for factors that are not zero: multiply the sizes, then count the negative factors. An even number of negative factors gives a positive product; an odd number gives a negative product, because each negative flips the sign once.

The two rows that feel new, positive times negative and negative times negative, are the two you just saw. Among nonzero factors, notice the pleasant symmetry: the product is positive exactly when the two factors share a sign, and negative when they disagree.

Zero appears nowhere in the chart, because zero is neither positive nor negative, so it does not need a row. It settles a product a different way: if any factor is zero, the product is zero and you can stop there, as $0 \times (-7) = 0$ shows. Count negative factors only when every factor is nonzero.

Division undoes multiplication

Ask yourself what $12 \div 4$ really means. It asks: what number, multiplied by $4$, gives $12$? The answer is $3$, and the reason it is $3$ is that $3 \times 4 = 12$. Every division is a multiplication with one factor missing, and the quotient is that missing factor. Nothing in that idea mentions signs, so it keeps working when the numbers go negative.

Take $(-12) \div 4$. What number times $4$ gives $-12$? Not a positive one, since a positive times a positive stays positive. The missing factor is negative, with size $3$:

Now take $(-12) \div (-4)$. What number times $-4$ gives $-12$? Here $3 \times (-4) = -12$ does the job, so the missing factor is positive:

That second answer is easier to believe in a real situation. A submarine drops $6$ feet every minute, a change of $-6$ feet per minute, and over the whole dive its depth changes by $-24$ feet. How long did the dive last? It takes four of those changes to make $-24$, so $(-24) \div (-6) = 4$ minutes. The quotient counts minutes, and a count of minutes cannot be negative.

Because every division question is secretly a multiplication question, the sign rule is identical.

Division signs: when neither number is zero, divide the sizes as usual. The quotient is positive when the two signs match and negative when they differ, exactly as for multiplication.

Why you cannot divide by zero

The “reverse of multiplication” definition also explains the one division that is forbidden: dividing by zero. Ask what $6 \div 0$ would mean. By the definition, it is the number that you multiply by $0$ to get $6$. But anything times zero is zero, never $6$, so no such number exists. The question has no answer, which is why $6 \div 0$ is undefined.

What about $0 \div 0$? Now we want a number that times $0$ gives $0$, and every number does that, so there is no single answer to give. Where the zero sits is what decides the outcome:

A quotient has to be one definite number. Dividing by zero never gives exactly one, whether nothing fits or everything does, so it is undefined in both cases. Dividing zero by a nonzero number is ordinary and gives $0$.

Common mistakes