Positive and negative numbers

How to add, subtract, multiply, and divide

All numbers and number expressions used in Algebra and higher levels of mathematics are either positive + or negative -. Because of this, in the "old days" (when your parents or grandparents were in school) they were called signed numbers.

This is true even for numbers that don't seem to have a sign, for example:

5 is really +5 
(positive 5)
7x - 8 is really +7x - 8
(positive 7x minus 8)

To do addition, subtraction, multiplication or division with real numbers you need to be aware of the positive and negative signs, even if positive signs are not always written down. If there is no sign, always imagine an invisible +.


Addition and Subtraction

In elementary school arithmetic classes you learned how to add and subtract. In Algebra, we learn that it is all really just addition. We use the concept of opposites to do subtraction. In Algebra we learn that subtracting is the same as adding the opposite.

For example, 7 - 2 (subtracting 2) is the same as 7 + (-2) (adding the opposite of 2). In either case, the answer is +5. For those too lazy to write down the sign (which is most of us) you can write your answer as just plain old 5

Here are the rules for addition and subtraction when working with two numbers that might be either positive or negative:

Signs of numbers in the problem Action Sign of the Answer Example
Both positive numbers Add them Positive 5 + 3 = +8

6a + 2a = +8a

Both negative numbers Add them Negative - 5 - 3 = -8

-6a - 2a = -8a

One positive and one negative Subtract them Same sign as the biggest number +5 - 3 = +2
-5 + 3 = -2

+6a - 2a = +4a
-6a + 2a = -4a

If you have more than two numbers to add or subtract, take them two at a time. Use the rules above to get an answer for the first two numbers - then use the rules to combine that answer with the next number until everything has been added or subtracted.
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Multiplication and Division

Later in elementary school arithmetic classes you learned how to multiply and divide. In Algebra, we learn that it is all really just multiplication.

For example, 8 2 (divide by 2) is the same as 8 (multiply by ). In either case, the answer is +4. For those too lazy to write down the sign (which is most of us) you can write your answer as just plain old 4

The rules are slightly different from those for adding and subtracting. In general, signs the same answer is positive, signs different the answer is negative. Also notice that if there is no sign for the number after the multiplication symbol, then the sign is really positive:

5 3 = +5 (+3) = +15 = 15

Here are the rules for multiplication and division when working with two numbers that might be either positive or negative.

Signs of numbers in the problem Action Sign of the Answer Example
Both positive numbers Multiply them Positive 5 3 = +15

6 2a = +12a

Both negative numbers Multiply them Positive -5 (- 3) = +15

-6 (- 2a) = +12a

One positive and one negative Multiply them Negative +5 (- 3) = -15
-5 3 = -15

+6 (-2a) = -12a
-6 2a = -12a

Again, if you have more than two numbers to multiply or divide, take them two at a time. Use the rules above to get an answer for the first two numbers - then use the rules to combine that answer with the next number until everything has been multiplied or divided.


Comments, questions, or you just need some help? Send an e-mail to Mrs. Shearer
Last Updated: 1/19/2009
K. A. Shearer 2005-2009

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