Positive and negative tokens

The lift attendant in Bela's Building of Fun is bored. He keeps a box of positive (green) tokens and negative (red) tokens. Each time he presses '+', he puts a positive token in his pocket; each time he presses '-', a negative token. He starts on the ground floor with an empty pocket. After an hour he finds 5 positive and 3 negative tokens. He pressed '+' five times and '-' three times, so he is on Floor +2: (+5) + (-3) = +2.

A positive token and a negative token cancel each other: together they are worth 0. We call them a zero pair. Remove all the zero pairs and see what is left. With 5 positives and 3 negatives, remove 3 zero pairs and 2 positives are left: +2. To add +5 and -8, remove 5 zero pairs; 3 negatives are left, so (+5) + (-8) = -3.

Tokens can subtract too, by taking tokens away. (+5) - (+4): from 5 positives take away 4, leaving +1. (-7) - (-5): from 7 negatives take away 5, leaving -2. But (+5) - (+6)? There are not enough positives to take away 6. So first put in one zero pair; this does not change the value, because a zero pair is worth 0. Now there are 6 positives and 1 negative. Take away 6 positives, and 1 negative is left: (+5) - (+6) = -1.

Another example: +4 - (-6). Start with 4 positives. We must take away 6 negatives, but there are none, so we put in 6 zero pairs. Now take away the 6 negatives: 4 + 6 = 10 positives are left. So +4 - (-6) = +10.