# Integers and Rationals

### Operations with Negative Numbers

#### 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:

Figure %: Subtracting 6 from 4
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

#### Multiplying and Dividing by Negative Numbers

When a positive number is multiplied by a negative number, the result is always negative. For example, 7×(- 4) = - 28 and -10×5 = - 50 . When two negative numbers are multiplied together, the negative signs cancel each other out, and the result is positive. For example, -4×(- 3) = 12 and -11×(- 7) = 77 .

To multiply two or more numbers when at least one of them is negative, count the total number of negative signs. If the total number of negative signs is even, the result will be positive, and if the total number of negative signs is odd, the result will be negative. Multiply the numbers with their signs removed and make this result positive or negative according to the total number of negative signs.

To divide two or more numbers when at least one of them is negative, follow the same steps, dividing instead of multiplying. Divide the numbers with their signs removed, and make this result positive or negative according to the total number of negative signs (positive if the total number of negative signs is even, negative if it is odd).

Examples. 3×(- 4) = - 12 (1 negative sign)
-5×(- 1) = 5 (2 negative signs)
-11×(- 1)×4×(- 2) = - 88 (3 negative signs)
18/(- 2)/(- 3) = 3 (2 negative signs)
-20/4/5 = - 1 (1 negative sign)
-5×(- 12)/3×(- 4)/(- 20) = 4 (4 negative signs)

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