Signs First: Multiplying and Dividing Decimals

Start with one fact: multiplying a number by $-1$ gives its opposite. Two numbers are opposites when they add to $0$, like $2.5$ and $-2.5$.

Here is why, using $2.5$. Add the products $(-1)(2.5)$ and $(1)(2.5)$. The distributive property works for negative numbers just as it does for positive numbers. It lets you add $-1$ and $1$ first and then multiply the sum by $2.5$:

$(-1)(2.5) + (1)(2.5) = (-1 + 1)(2.5) = (0)(2.5) = 0$

The two products add to $0$, so they are opposites. Since $(1)(2.5) = 2.5$, the other product is $(-1)(2.5) = -2.5$. These steps do not depend on the value $2.5$. Start with $-2.5$ and they show that $(-1)(-2.5)$ is the opposite of $-2.5$, which is $2.5$.

Now look at what a negative factor does. Every negative number is $-1$ times its absolute value. For example, $-0.4 = (-1)(0.4)$. So each negative factor puts one $-1$ into the product, and each $-1$ flips the product to its opposite.

You may multiply factors in any order, so you can bring the $-1$ factors together:

$(-1.5)(-0.4) = (-1)(1.5)(-1)(0.4) = (-1)(-1)(1.5)(0.4) = (1)(0.6) = 0.6$

Here $(-1)(-1) = 1$ because the opposite of $-1$ is $1$. When no factor is $0$, the product of the absolute values is positive. One flip makes the product negative, and a second flip makes it positive again. So for nonzero factors, the number of negative factors decides the sign.

Quotients follow the same pattern because division undoes multiplication, as long as the divisor is not $0$. For example, $(-36) \div 5$ asks: what number times $5$ gives $-36$? A positive number times $5$ is positive, and $0$ times $5$ is $0$, so the answer must be negative. Since $36 \div 5 = 7.2$, try $-7.2$. It works: $(-7.2)(5) = -36$. So $(-36) \div 5 = -7.2$, a rational number written as a decimal.

With two negative numbers the quotient is positive: $(-36) \div (-5) = 7.2$, because $(7.2)(-5) = -36$.

As long as the divisor is not $0$, a single negative sign can sit in the dividend, in the divisor, or in front of the whole quotient. The value is the same each time:

Now you can find the whole answer. To multiply or divide positive and negative decimals, use three steps:

1. Sign. If any number is $0$, use the zero rules. If no number is $0$, count the negative numbers to get the sign.
2. Size. Multiply or divide the absolute values, which are positive decimals.
3. Combine. Put the sign and the size together.

To find the size, change the decimals into whole numbers. In a product, write each decimal as a whole number divided by a power of ten ($10$, $100$, $1000$ and so on). For example, $2.4 = 24 \div 10$. In a quotient, multiply the dividend and the nonzero divisor by the same power of ten. This is like writing an equivalent fraction: the quotient stays the same, just as $6 \div 2$ and $60 \div 20$ are both $3$.

The big idea: find the sign and the size of a product or quotient separately. Look for $0$ first: any zero factor makes a product $0$, zero divided by a nonzero number is $0$, and you never divide by $0$. When every number is nonzero, count the negative numbers. An odd count gives a negative result and an even count gives a positive result, because each negative number flips the result to its opposite. Then multiply or divide the absolute values to get the size.

Next, you will use the same ideas to multiply and divide positive and negative fractions, and you will apply properties of operations, such as the distributive property, to rational numbers.