## Negative Numbers: Overview

You've probably run into situations that could be modeled by negative numbers: profits and losses, bank deposits and withdrawals, temperatures above and below freezing, buildings with elevators that go to floors above and below ground level, yardage gained and lost in football, and walking forward and backward.

Mathematicians define *negative numbers* as “the opposites of positive numbers,” since they are on the opposite side of zero from the positive numbers on a number line. Similarly, the opposites of the negative numbers are the positive numbers. The number ^{-}3 (negative three) is the number that when added to ^{+}3 (positive three) equals zero. That is, ^{-}3 is the solution to the equation 3 + *x* = 0. The set of integers consists of all the whole numbers and their opposites. Zero is its own opposite. Those numbers that are greater than zero are positive and those numbers that are less than zero are negative. Zero is neither positive nor negative. Negative numbers are always written with a negative sign, but positive numbers may be written with or without a positive sign.

A common way to visualize integers is to use a number line like the one shown below. Here we can see that the numbers to the left of zero are negative and the numbers to the right of zero are positive.

The number ^{-}6 is 6 units from zero, just as the number ^{+}6 is 6 units from zero. The only difference is that ^{-}6 is to the left of zero and ^{+}6 is to the right of zero. If you folded the number line at zero, each number would match up with its opposite.

Adding and subtracting integers on a number line is very similar to adding and subtracting whole numbers on a number line. The operation of addition on the number line means “go in the direction of the number that follows.” To add 4 and ^{-}5, start at 4 and go to the left five units as shown below.

The operation of subtraction means to “go in the opposite direction of the number that follows.” Thus, to subtract 2 from ^{-}4, start at ^{-}4 and go 2 units to the left as shown below. Another way to do this is to change the number being subtracted to its opposite and add it to the other number. That is, ^{-}4 − ^{+}2 = ^{-}4 + ^{-}2, which equals ^{-}6.