Multiplication and division are inverse operations, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations “undo” each other.

Multiply Real Numbers

Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven’t learned what effect a negative sign has on the product.

With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\cdot4[/latex] as [latex]3[/latex] jumps of [latex]4[/latex] units each.

So to multiply [latex]3(−4)[/latex], you can face left (toward the negative side) and make three “jumps” forward (in a negative direction).

The product of a positive number and a negative number (or a negative and a positive) is negative.

The following video contains examples of how to multiply decimal numbers with different signs.

The following video shows examples of multiplying two signed fractions, including simplification of the answer.

To summarize:

• positive [latex]\cdot[/latex] positive: The product is positive.
• negative [latex]\cdot[/latex] negative: The product is positive.
• negative [latex]\cdot[/latex] positive: The product is negative.
• positive [latex]\cdot[/latex] negative: The product is negative.

You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors.

The following video contains examples of multiplying more than two signed integers.

Divide Real Numbers

You may remember that when you divided fractions, you multiplied by the reciprocal. Reciprocal is another name for the multiplicative inverse (just as opposite is another name for additive inverse).

An easy way to find the multiplicative inverse is to just “flip” the numerator and denominator as you did to find the reciprocal. Here are some examples:

• The reciprocal of [latex]\frac{4}{9}[/latex] is [latex]\frac{9}{4}[/latex]because [latex]\frac{4}{9}\left(\frac{9}{4}\right)=\frac{36}{36}=1[/latex].
• The reciprocal of [latex]3[/latex] is [latex]\frac{1}{3}[/latex] because [latex]\frac{3}{1}\left(\frac{1}{3}\right)=\frac{3}{3}=1[/latex].
• The reciprocal of [latex]-\frac{5}{6}[/latex] is [latex]\frac{-6}{5}[/latex] because [latex]-\frac{5}{6}\left( -\frac{6}{5} \right)=\frac{30}{30}=1[/latex].
• The reciprocal of [latex]1[/latex] is [latex]1[/latex] as [latex]1(1)=1[/latex].

When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide real numbers.

Think about dividing a bag of [latex]26[/latex] marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has one half of the marbles.

[latex]26\div 2=26\left( \frac{1}{2} \right)=13[/latex]

Notice that  [latex]2[/latex] and [latex]\frac{1}{2}[/latex] are reciprocals.

Try again, dividing a bag of [latex]36[/latex] marbles into smaller bags.

Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the divisor, the second number in the division problem.)

Now let’s see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn’t change any of the signs, division follows the same rules as multiplication.

The following video explains how to divide signed fractions.

The following video explains how to divide signed fractions.