Multiplying and Dividing Integers

Read down the middle column. Each row is 3 groups of the second factor. So when the second factor goes down by 1, the product goes down by 3, even past zero: $9, 6, 3, 0, -3, -6, -9$.

Now read down the right column. Switching the order of the factors explains the top rows: $(-3) \times 2 = 2 \times (-3)$, which is 2 groups of $-3$, or $-6$. Each step down takes away one group of $-3$, so you subtract $-3$. Subtracting a number is the same as adding its additive inverse (its opposite: the number you add to it to get $0$). The additive inverse of $-3$ is $3$, so each product is 3 more than the one above it: $-9, -6, -3, 0$. The last three rows continue that pattern: $3, 6, 9$.

So the pattern suggests $(-3) \times (-1) = 3$. Here is why that must be true. Add the products $(-3) \times (-1)$ and $(-3) \times 1$. Both share the factor $-3$, so the distributive property lets you add the other factors first:

The sum is $0$ because $-1 + 1 = 0$, and zero times any number is zero. Two numbers that add to $0$ are opposites. You know $(-3) \times 1 = -3$, so $(-3) \times (-1)$ must be $3$.

The same reasoning works with any integers: replace one factor with its opposite, and the product becomes its opposite. So when no factor is zero, each negative factor flips the product to the opposite side of zero. Start from $3 \times 4 = 12$. One negative factor flips it: $(-3) \times 4 = -12$. A second negative factor flips it back: $(-3) \times (-4) = 12$. That answers the opening question.

You can now multiply nonzero integers in two parts. Find the size by multiplying the absolute values. A number's absolute value is its distance from zero, so $\lvert -4 \rvert = 4$ and $\lvert 4 \rvert = 4$. Then find the sign, positive or negative, by counting the flips. If any factor is zero, skip the counting, because zero times any number is zero.

Division undoes multiplication. In $(-12) \div 3$, the number being divided, $-12$, is the dividend. The number you divide by, $3$, is the divisor. The result is the quotient. When the divisor is nonzero, the quotient is the one number you multiply by the divisor to get the dividend back. Because $(-4) \times 3 = -12$, you know that $(-12) \div 3 = -4$.

Every division fact comes from a multiplication fact, so you do not need a new way to find the sign. For a nonzero dividend and a nonzero divisor, count the negative numbers just as before:

• Two negative numbers: $(-12) \div (-3) = 4$, because $4 \times (-3) = -12$.
• One negative number: $12 \div (-3) = -4$, because $(-4) \times (-3) = 12$.

Zero needs care:

• $0 \div 5 = 0$, because $0 \times 5 = 0$. Zero divided by any nonzero number is zero.
• $5 \div 0$ is undefined, which means it has no value. No number times $0$ gives $5$, because zero times any number is zero.
• $0 \div 0$ is undefined too. Every number times $0$ gives $0$, so no single quotient fits.

So a divisor must never be zero.

Takeaway: multiplying or dividing integers is two small jobs. Find the size from the absolute values. Then find the sign: when no number is zero, an even count of negative numbers gives a positive result, and an odd count gives a negative result. Remember the zero facts too. Zero times any number is zero, zero divided by a nonzero number is zero, and division by zero is undefined.

Next, you will use these skills to evaluate longer expressions with integers and to find and correct errors in integer work.