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05 September 2003

By the Numbers, Minus Times Minus? Huh?

by Peter Torrione

Our model of numbers as piles of pebbles served us well as we multiplied, say, 2 piles of 3 pebbles each giving 6 pebbles or three piles of two each giving six. But subtraction (minus) forced us to give up the pebble idea because it could not give us a model for the outcome of, say, 3 minus 5. The number line came to the rescue. We extended the line to the left, past zero, and on to minus infinity.

Now it is time to go back to multiplying. Clearly 2 times minus 3 (2 X -3) can be viewed on the number line as taking the line to -3 twice yielding minus 6 (-6).

But what happens when we multiply minus 2 by minus 3? It makes absolutely no sense to take minus 2, minus 3 times.

Yet closure says that any operation done on a number or pair of numbers must yield a number in the mathematics. We need a new model of what multiplication by a negative number means.

Again the number line comes to the rescue! We can interpret the minus sign as requiring a flip of the number on the number line by counterclockwise half a circle (180 degrees) about zero. This way minus three (-3) is obtained by going to three (3) and flipping half a circle to minus three.

Similarly, -3 times 2 starts off as 3 X 2 = 6, but the negative sign requires a flip of half a circle, about zero to -6.

Now we're ready to tackle -3 times -2. Again, 3 X 2 = 6, but the first negative sign requires a flip of half a circle and the second minus sign requires another half circle flip. We got back to positive six. -3 X -2 = 6.

Two negatives multiplied together give two half circle flips and a return to where we started! Three negatives give an added 180 degrees.

The question for next week is which number multiplied by itself gives -1? In other words: What is the square root of -1?

Note that this number, applied twice, must result in a half circle flip!