# Why a negative times a negative is positive

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Since the other thread is closed, I thought I create one with a simple example of water in a tank that most people will understand.

Imagine a tank designed to hold water with markers every inch from bottom to top. The marker halfway up the tank is chosen to be labeled zero. The next marker above is +1, and the next marker above that is +2. The first marker below the zero marker is labled -1, and the marker below that is -2, and so on.

The tank was and is being filled at a rate of 2 markers per minute. Currently, the water level in the tank has just reached the zero marker. At what marker will the water level be in the tank 3 minutes from now: answer +3 minutes times +2 markers per minute = +6 markers. Where was the water level at 3 minutes before the water reached the zero marker: answer -3 minutes times +2 makers per minute = -6 markers.

In the mean time there is a second tank, marked in the same way with the zero marker at the halfway point, that started off filled. It was and is being drained so that that water level is decreasing by 2 markers per minute. The water level is currently at the zero marker. At what marker will the water level be in the tank 3 minutes from now: answer +3 minutes times -2 markers per minute = -6 markers. Where was the water level at 3 minutes before the water reached the zero marker: answer -3 minutes times -2 makers per minute = +6 markers.

As another example, sand flowing through an marked hour glass. The upper half is being drained, while the bottom half is being filled, same questions and answers as above.

These are just two examples of common real world objects that are easily witnessed. A video tape could be used to record and play time "forwards" (positively) or "backwards" (negatively) to help in understanding.

Anything involving easily viewed constant motion and time are good candidates for demonstrating multiplication of signed numbers.

Last edited:

mfb
Mentor
This sounds very similar to the idea of just continuing the multiplication tables:
-3*2=-6
-3*1=-3
-3*0=0
-3*-1=3
...
There is nothing that special about the zero in that sequence.