Why is six times nine the same as nine times six?

When I asked my older son this question his immediate reply was "Well, because multiplication is commutative". Which is as good as saying "because that's what it is, why won't you stop asking obvious questions?". He's been aware for a while of how convenient this property of multiplications is: I would ask him "how much is a thousand times three?", and instead of adding three, plus three, plus three a thousand times he would say "the same as three times a thousand, three thousand".

One way to see why these tricks are possible is to draw a graphic representation of the multiplication. Six times nine means taking a group of nine things and replicating them six times, like this:


And nine times six means taking a group of six and stacking them up nine times, like this:


If we only care about how many elements are there, all we are doing is rotating the rectangle: and a nice property of rectangles is that their size does not change when we rotate them.

Another good opportunity for a shortcut is the answer to "how many groups of 24 minutes are there in a day?". The total number of minutes in a day is: 24 hours in a day times 60 minutes in an hour. But regardless of which number this product turns out to be (1440), it is 24 times 60, which is the same as 60 times 24: so 24 periods of 60 minutes is the same as 60 periods of 24 minutes. And if a day has 60 groups of 24 minutes, it also has 120 groups of 12 minutes, which has nothing to do with the original question but is a beautiful thing.

Juan Reyero Barcelona, 2008-04-06

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