The Number Line

Mark off equal distances along a line and label the points 0, 1, 2, 3, \ldots in order. This scale can act out both addition and subtraction from arithmetic.

To add 5 to 3, start at 3 and count 5 spaces to the right: you land on 8. (Start at 5 and count 3 spaces right, and you get the same thing.)

Rule for adding. To add the number a to the number b, begin at b and count a spaces to the right.

To subtract 4 from 7, start at 7 and count 4 spaces to the left: you land on 3.

Rule for subtracting. To subtract the number a from the number b, begin at b and count a spaces to the left.

Going past zero

Now try to subtract 5 from 4. Counting 5 spaces left from 4 carries you one space past zero. Arithmetic has no number for that spot; it calls this subtraction impossible. You could subtract 4 of the 5 units, but 1 unit is left over, unsubtracted.

In algebra it's both convenient and necessary to subtract a greater number from a less. We call the part left unsubtracted the remainder, and we show the subtraction was incomplete by writing a minus sign in front: 4 - 5 = -1. So the first point to the left of zero stands for -1. In the same way 3 - 5 = -2 is the second point to the left, 5 - 8 = -3 is the third, and so on.

Numbers like -1, -2, -3 are called negative numbers. The minus sign is never left off a negative number (though a letter like x can stand for a negative number). The ordinary numbers of arithmetic are called positive numbers; a number with no sign, or with a plus sign, is positive.