Why Negative Numbers Exist

Negative numbers are amounts less than zero. They were invented because ordinary numbers could not describe some real situations: owing money, temperatures below freezing, or floors below ground level. A negative sign simply means "the opposite direction."

The number line puts every number in order, with negatives stretching left of zero. The farther left, the smaller the number. Negative five is smaller than negative two, even though five is bigger than two, because it is five steps in the negative direction.

Adding a negative is the same as subtracting. Five plus negative three equals two. Subtracting a negative flips direction: five minus negative three equals eight. These rules feel strange at first, but they make the number line behave the same everywhere - which is exactly what mathematicians wanted.