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# Subtracting Integers

Objective Learn how to subtract both positive and negative integers.

In this lesson, you will study how to do arithmetic with integers (both negative and positive). We will be studying subtraction in this lesson. The most difficult part of this topic is the confusion that is sometimes experienced regarding “where to put the minus sign”.

## Opposites

Let's begin by recalling that the opposite of any positive integer is the negative integer that is the same distance from 0 on a number line. So, the opposite of 6 is -6, and the opposite of 25 is -25. Conversely, the opposite of any negative integer is the positive integer that is the same distance from 0 on a number line. Thus, the opposite of -8 is 8, and the opposite of -31 is 31.

## Subtracting Integers

How do we subtract a negative integer from another integer? Have you any ideas about how to subtract a negative integer from a positive integer? And what about subtracting a negative integer from another negative integer?

Remember that subtraction and addition are opposite operations.

Key Idea

Subtracting a negative integer gives the same result as adding the opposite positive integer.

Let's demonstrate why this is true using shaded squares in the following way. Think about collections of gray (positive) and black (negative) squares. Then, subtracting a negative integer from another negative integer can be modeled by taking a group of black squares that model the first negative integer and removing the number of black squares (if there are enough in the model) that represent the second negative integer. When there are not enough black squares in the model, a sufficient number of pairs of gray and black squares need to be added to the model so that the required number of black squares can be removed. When a negative integer is subtracted from a positive integer there will not be any black squares in the model for the first integer. In this case, pairs of gray and black squares must be added to the model to begin the subtraction process.

Let's take a look at the following examples, some using models to find the solution and others done by direct calculation. One example of each type is given here.

Example 1

What is the value of 6 - (-2)?

Solution

To model 6 - (-2), first represent 6 as a collection of 6 gray squares. Since there are not 2 black (negative) squares that can be removed to model subtracting -2, add two pairs of squares, one black and one gray in each pair. (Remind students that we removed pairs like these because they “canceled” each other; therefore, we can also add such pairs because adding them will not change the final result.)

There are now 8 gray squares and 2 black squares in the model. So we are able to show the subtraction of -2 by removing the 2 black squares, leaving 8 gray squares.

Therefore, the model shows that 6 - (-2) = 8.

Example 2

Find  5  (  2).

Solution

Use the fact that subtracting a negative integer gives the same result as adding the opposite positive integer. This means that -5 - (-2) is equivalent to -5 + (+2) = -3.

To see this in terms of a model, we can represent -5 as a collection of 5 black squares and then simply remove 2 black squares. The result is a model showing 3 black squares, which represents -3.

Try to solve subtraction problems involving negative integers by using shaded squares, colored counters, or some other type of manipulatives. Then do the subtractions directly by using the idea that subtracting a negative integer is the same as adding its opposite.