Think of the distance from one integer to another as being one step. If you start at zero and take two steps to the right, you get to 2. If you start at 0 and take two steps to the left, you get to -2. If you take two steps forward, we take two steps back. We come together, 'cuz opposites attract. In any case, we go the same distance.
Each integer (except 0) has two pieces of information: its distance from 0, and its direction from 0.
The distance of an integer from 0 is called its magnitude or its absolute value. We indicate "the absolute value of x" by putting vertical bars around x, like this: |x|. Because absolute value is basically a measurement rather than a number, it's never negative. Can you imagine how difficult it would be, for example, to track down drapes that would fit a window measuring -3 feet long? Not even the helpful people at Target would be able to help you out with that one.
The sign of the number is there to tell us which direction from 0 we're stepping. By the way, don't freak out; there's actually very little physical activity involved in solving these problems. If there's no sign in front of the number, it means the number is positive. If there's one negative sign in front of the number, it means we reflected the number in our mirror, so it's negative. Also, it looks like it's getting ready to brush its teeth.
Of course, things can get more complicated than that. We can go crazy with the negative signs and write things like -(-(-3)). Ever slipped up and said something like, "I don't want no socks for Christmas," and then your grammar-stickler uncle gets you socks just to prove a point? What you did was use a double-negative—because you said that you do not want no socks, that must mean that you do want socks.
Same thing with a number like -(-3), which is the same as positive 3. However, with three negative signs as in the earlier example, the number would once again become negative. Now it's like you're saying, "I don't want none of no socks for Christmas." Wow. You really need to spend some more quality time with your uncle.
Remember, one negative sign meant we reflected a number into the mirror once.
Two negative signs means we reflect the negative number back again, so now we're back to the right:
And we could reflect again to get the following:
Be Careful: Taking the negative of a number doesn't always give us a negative number, as the previous examples demonstrate. So before assuming a number is negative just because you see a negative sign, make sure that there's only one of them. If there are multiple negative signs, the number may be negative or positive, depending on how many negative signs there are. We know that you may not don't never can't not want to deal with too many negative signs, but it's just a part of life.
If we want to be extra clear that a number is positive, we can write an extra "+" in front of it. However, if there's no + or - sign, the number is understood to be positive. If a number is preceded by the symbol ©, then it's copyrighted. You may not reproduce, retransmit or rebroadcast this number without the express written consent of Major League Baseball.
Sample Problem
What's the value of -|-5|?
We've got two negative signs and an absolute value sign. Let's work our way from the inside out. Remember, anything inside those absolute value bars is positive.
-|-5| = -(5) = -5
You didn't pull anything doing that exercise, did you? We warned you to stretch first...