Multiplying and Dividing Real Numbers

Learning Outcomes

  • Multiply and divide real numbers
    • Multiply two or more real numbers
    • Divide real numbers

Multiply Real Numbers

With whole numbers, you can think of multiplication as repeated addition. The product (or result of multiplication) [latex]3 \cdot 5[/latex] can be interpreted as

[latex]3 \cdot 5=5+5+5=15[/latex]

So, to multiply [latex]3(-5)[/latex], can be found as follows.

[latex]3(-5)=(-5)+(-5)+(-5)= -15[/latex]

We can multiply two numbers in any order and the result is the same, so

[latex](-5)(3)=15[/latex]

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

The Product of a Positive Number and a Negative Number

To multiply a positive number and a negative number, multiply their absolute values, or their distance from zero. The product is negative.

Example

Find [latex]-2(4)[/latex]

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

The Product of Two Numbers with the Same Sign (both positive or both negative)

To multiply two positive numbers, multiply their absolute values. The product is positive.

To multiply two negative numbers, multiply their absolute values. The product is positive.

Example

Find [latex](-12)(-3)[/latex]

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.

Multiplying More Than Two Negative Numbers

If there are an even number ([latex]0, 2, 4[/latex], …) of negative factors to multiply, the product is positive.
If there are an odd number ([latex]1, 3, 5[/latex], …) of negative factors, the product is negative.

Example

Find [latex](-4)(-2)(-5)[/latex].

The following video contains examples of multiplying more than two signed integers, or counting numbers (1, 2, 3, etc.).

Try It

 

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). 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.

Rules of Division

When one number is positive and the other is negative, the quotient is negative.

When both numbers are negative, the quotient is positive.

When both numbers are positive, the quotient is positive.

Example

Find [latex]24 \div (-3)[/latex]

Example

Find [latex](-36) \div (-9)[/latex]