Wednesday, March 07, 2012

Nine times table on your fingers (and algebraic explanation)

Can't remember that 9 x 7 is 63? Here's the really fast way to do it.

Lay your hands on the table and look at your fingers. Imagine they are numbered 1 to 10 from left to right. Find the 7th finger. There are 6 fingers to the left of it: that's the first digit of 9 x 7. There are three fingers to the right of it: that's the second digit. So the answer is 63.

Same thing works for the rest of the 9 times table (up to 9 x 10).

Here's a quick algebraic explanation of why that works. Suppose we're doing 9 x a where a is a number between 1 and 10. On your fingers you've got (a-1) fingers to the left of finger a and (10-a) fingers to the right. The result is 10 x (a-1) + (10-a) because the (a-1) is in the 10s position.
10 x (a-1) + (10-a)

= 10 x a - 10 + 10 - a

= 10 x a - a

= (10 - 1 ) x a

= 9 x a
Another thing you can spot this way is that the sum of the digits in the 9 times table is always 9. For example, 63 (6 + 3 = 9), 9 x 5 = 45 (4 + 5 = 9). Again, algebra shows why:
  (a-1) + (10-a)

= a - 1 + 10 - a

= 9
PS A reader writes that the 9 times table is also a palindrome (if you add a zero before 9 x 1): 09 18 27 36 45 54 63 72 81 90.

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