# Negative Numbers
## Why Use Negative Numbers?
Have you seen negative numbers somewhere in your life? Think about some instance for a
minute â€“ I would bet it has something to do with direction, along with the value.
Negative numbers generally occur when a change is measured from some reference.
## What Are Negative Numbers
Negative numbers are numbers less than (to the left of) zero. The number line below
demonstrates this idea. Notice that both positive and negative numbers go on forever.
Numbers get larger as you go to the right and smaller as you go to the left. Notice for example
that â€“3 is smaller than â€“2.
Two numbers are **opposites** if they are the same distance from zero on a number line, but on
opposite sides of zero. The number -5 is read â€œnegative fiveâ€ or â€œthe opposite of fiveâ€.
Is +0 the same as â€“0? These are two different names for the same point on the number line.
They mean the same thing. Adding zero is identical to subtracting zero.
Addition and subtraction with negative numbers is particularly convenient, since there is nothing
special about the number zero. There is no extra â€œspecial caseâ€ to handle when you combine
numbers on either side of zero. You simply add and subtract them as usual.
## Negatives on Your Calculator
Every calculator seems to have a different way to enter negative numbers. The +/- key on your calculator gives the opposite of the number (changes the sign) on the
display.
## A Few Words About Parentheses
A parenthesis is a symbol like â€œ(â€œ or the matching symbol â€œ)â€. The plural of parenthesis is
parentheses.
Parentheses in mathematics are used to group things together. They tell you that items inside
them belong together; they are slightly separated from things outside them. Operations inside
parentheses must be done before other operations.
We use parentheses with negative numbers to avoid confusion with other operations such as
addition or subtraction. For example, â€œ3 + (-5)â€ means, â€œthree plus negative fiveâ€ and tells you
the minus sign is working on the five. Otherwise, you would see the â€œ+ -â€ together and it would
be confusing.
## Adding: a + (-b)
This is just another way to write regular subtraction! a + (-b) = a - b
Start with the first number you are given and move:
Â· to the **left** if you are adding a negative number;
Â· to the **right** if you are adding a positive number.
Why would you ever write subtraction that way? First, because there is no longer any
subtraction. That is, all your subtraction problems are merely addition, and you just happen to
have some negative numbers thrown into the mix. Second, because addition problems let you
easily swap the order of the numbers. Sometimes it is handy to write a + (-b) in another form
such as (-b) + a.
## Subtracting: a - (-b)
This is just another way to write regular addition The two negative signs cancel each other out!
Start with the first number you are given and move:
Â· to the left if you are subtracting a positive number;
Â· to the right if you are subtracting a negative number.
## Multiplying and Dividing: (-a) Ã—Â·(-b) or (-a)/(-b)
To multiply or divide positive or negative numbers
Â· Ignore the sign (positive or negative) and multiply or divide as usual.
Â· The answer is positive if both numbers have the same sign.
Â· The answer is negative if the numbers have opposite signs.
**Examples: **
6 Ã— 3 = 18 |
6 Ã— (-3) = -18 |
-6 Ã— 3 = -18 |
-6 Ã— (-3) = 18 |
10 / 2 = 5 |
10 / (-2) = -5 |
-10 / 2 = -5 |
-10 / (-2) = 5 |
Do parentheses first, then exponents, then multiplication and division, and addition and
subtraction last.
Just as two negatives in a sentence mean positive, so a negative times a negative equals a
positive: -3 Ã— -4 = 12 |