Adding and Subtracting Real Numbers

Learning Outcomes

  • Add and subtract real numbers
    • Add real numbers with the same and different signs
    • Subtract real numbers with the same and different signs

The set of real numbers can be thought of as all possible distances from a fixed point, represented by 0 on the number line below.  The distance of points to the right of 0 are represented by positive (+) numbers.  The distance of points to the left of 0 are represented by negative (−) numbers.  The sign of a number represents its direction relative to 0. Numbers are assumed to be positive if no sign is specified: 2 means +2.

The integers are counting numbers and their negatives, as well as zero:

...,−3,−2,−1,0,1,2,3,...

The set of real numbers includes fractions and decimals, as well as the integers.

This figure is a number line with 0 in the middle. Then, the scaling has positive numbers 1 to 4 to the right of 0 and negative numbers, negative 1 to negative 4 to the left of 0.

The absolute value of a real number x, represented by x, is its distance from 0 without regard to direction. Since it represents distance, the absolute value of a number is never negative. For example, since −2 is located 2 units to the left of 0,|−2|=2. Since 2 is located 2 units to the right of 0,|2|=2.

If we add two positive numbers, such as 1 and 2, we can think of beginning at 0 and moving 1 unit to the right, and then 2 more units to the right. So,

1+2=3

If we add two negative numbers, such as −1 and −2, we proceed in the same way but move to the left each time. So,

(−1)+(−2)=−3

To add two numbers with the same sign (both positive or both negative)

  • Add their absolute values (without the + or − sign)
  • Give the sum the same sign.

Suppose we wish to add two numbers with different signs. If we add −2 and 3 we move from 0 to the left 2 units to −2, and then to the right 3 units, ending at 1.

−2+3=1

If we add 2 and −3 we move from 0 to the right 2 units to 2, and then to the left 3 units, ending at −1.

2+(−3)=−1

To add two numbers with different signs (one positive and one negative)

  • Find the difference of their absolute values. (Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one.)
  • Give the sum the same sign as the number with the greater absolute value.

Example

Find 17+(−20).

Try It

One way to think of subtraction is to consider the distance between two numbers. 5−3=2 since we would need to move 2 units to the right of 3 to get to 5. But we can also think of subtracting a number as the addition of its opposite.

5−3=5+(−3)=2

We can rewrite subtraction as the addition of a number’s opposite.

a−b=a+(−b)

Example

Find −13−(−20).

Example

Find 8−17.

The following video explains how to subtract two signed integers.

Try It

In the following video are examples of adding and subtracting signed decimals.