Saturday, December 07, 2013

Whilst we're on the subject of maths ...

Another thing that was never mentioned (as far as I can remember) was the interesting phenomenon in the multiplication square - you know, this thing ...

  X   1   2   3   4   5   6 ...
  1   1   2   3   4   5   6
  2   2   4   6   8  10  12
  3   3   6   9  12  15  18
  4   4   8  12  16  20  24
  5   5  10  15  20  25  30
  6   6  12  18  24  30  36

etc.

If you look down the diagonal axis from top left to bottom right, then you get a list of the square numbers - 1, 4, 9, 16, 25 ... What I noticed was that, if you go "northeast" and "southwest" from those numbers, you always get a number exactly one less. That is, if you take a number, and multiply the number one more and one less than it, then you get one less than the number squared. Or ...

(n - 1) (n + 1) = n2 - 1

It turns out to be pretty trivial once you expand out the expression, of course ...

(n - 1) (n + 1) = n2 - n + n - 1 = n2 - 1

But nobody ever bothered to point it out, and I felt a gram of so of smug when I proved it for myself.

There's actually a more general thing lurking here ...

(n - k) (n + k) = n2 - k2

... which means that if you look at the differences as you continue "northeast" and "southwest" from numbers on the diagonal, you are going to get another series of square numbers.

1 comment:

Robert Fernandez said...

I have recently been sharing with people the amazing number 1089. Was it you who told me about this in the first place Paul?

1089 is 9 x 121 which is 3 squared times 11 squared. Interesting but not earth shattering.

But - it gets better.

1 X 1089 = 1089. 9 X 1089 = 9801.
2X 1089 = 2178. 8 X 1089 = 8712
3 X 1089 etc

But that's only the beginning. Wanna know some more?

By the way I see I need to prove I ma not a robot. I am obviously partially a robot because I often get these wrong. I wish they were not QUITE so difficult.

Robert