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Multiplying Integers

Objective Learn how to multiply both positive and negative integers.

We will be studying multiplication in this lesson. Multiplication may be the most confusing and nonintuitive of the operations when applied to negative integers. We will try to demonstrate how the idea behind multiplication is applied to examples.

Multiplying with Negative Integers

Take a look at the following rules for multiplying negative integers. Have the definition of opposites in mind.

Key Idea

1. When we multiply a positive integer and a negative integer together, the result is the opposite of the number we would get if both integers had been positive.

For example, 2 × ( -3) = -(2 × 3) = -6 and - 5 × 4 = -(5 × 4) = -20.

2. When we multiply two negative integers, the result is the positive integer we would get if both integers had been positive. For example, -3 × (-4) = 3 × 4 = 12.

In summary, (1) multiplying a positive integer times a negative integer gives a negative integer, and (2) multiplying two negative integers gives a positive integer.

Try to answer the following questions in order to make sure you understood the previous properties.

• Will -3 × (-7) be positive or negative?

positive

• Will 457 × ( -325) be positive or negative?

negative

The rules for multiplying two integers can also be remembered using the phrase same signs, positive product; different signs, negative product.

 

Why Are These Rules True?

Multiplying any integer (positive or negative) by a positive integer is the same as adding “copies” of that integer, where the number of copies is equal to the positive integer. Multiplication by a positive integer can be viewed as repeated addition. That is, 5 × 2 is the same as 2 + 2 + 2 + 2 + 2. Then, 5 × (-2) is then equal to (-2) + (-2) + (-2) + (-2) + (-2). The following example uses this thinking.

 

Example

What is 5 × (-3)?

Solution

The product 5 × (-3) can be found by adding 5 copies of the integer -3. So,

5 × (-3) = (-3) + (-3) + (-3) + (-3) + (-3) = -15.

The same answer can be obtained by applying the first rule in the Key Idea shown earlier.

5 × (-3) = -(5 × 3) = -15

The idea that the product of two negative integers is positive is many times harder to understand.

Why is -5 × (-3) = 15?

You do not need a formal proof of this fact. Simply notice that the Example just showed that 5 × (-3) = -15. It is reasonable for the products -5 × (-3) and 5 × (-3) to be opposite of each other since they cannot be the same. So -5 × (-3) should be the opposite of -15 or 15.

Try to give reasons for why the following fact is true.

Given any number x , -1 · x is the opposite of x. By the rules for multiplying negative integers, if x is positive, then -1 · x is negative and its value is the opposite of the value of 1 · x = x . Similarly, if x is negative, then -1 · x is positive and its value is the opposite of the value of 1 · x .

The only way you will fully understand the concept of multiplication of negative integers is if you become skilled at the calculations.

 
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