Integers and Rationals: Operations with Negative Numbers

Integers and Rationals Operations with Negative Numbers

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Subtracting a Larger Number from a Smaller Number

It is easy to compute 6 - 4 = 2 and 14 - 5 = 9, because the answers are positive. But how would one compute 4 - 6? On the number line, moving to the left is equivalent to subtracting from a number, and moving to the right is equivalent to adding to a nu mber. To see what 4 - 6 is equal to, start on the number line at the place marked "4" and count 6 spaces to the left:

Thus, 4 - 6 = - 2.

Notice the similarity between 6 - 4 = 2 and 4 - 6 = - 2. 6 - 4 is the opposite of 4 - 6. To find 4 - 6, simply find 6 - 4 and take its opposite. In fact, this works for any two numbers: to subtract a larger number from a smaller number, reverse the two numbers and take the opposite of the answer. 5 - 14 = - (14 - 5) = - 9.

To subtract a positive number from a negative number, switch both signs, and take the opposite of the answer. -5 - 7 = - (5 + 7) = - 12 and -8 - 11 = - (8 + 11) = - 19.

Adding and Subtracting Negative Numbers

As we just saw, adding 6 to a number is the same as moving 6 spaces to the right on the number line. Adding -6 to a number is the same as moving 6 spaces to the left. Thus, adding -6 to a number is the same as subtracting 6 from the number. In general terms, adding a number is the same as subtracting its opposite.

Thus, to add a negative number, subtract its opposite. To subtract a negative number, add its opposite.

Examples:

5 + (- 2) = 5 - 2 = 3
7 - (- 11) = 7 + 11 = 18
-12 - (- 2) = - 12 + 2 = - (12 - 2) = - 10
17 + (- 23) = 17 - 23 = - (23 - 17) = - 6
11 - (- 2) = 11 + 2 = 13
12 - (- 2) + (- 7) = 12 + 2 - 7 = 7