First, let us note a simple way to square numbers that end in 5. All numbers that end in 5, when squared, will end with 25. Then, take the number to the left and multiply it to itself plus 1.For example:
15 x 15 = (1 x 2) 25 = 225
25 x 25 = (2 x 3) 25 = 625
The figure to the left shows a "proof" for why this occurs. The trick lies in (e) to (f). At this point, you must realize we work in base 10, therefore making the tens place in the answer a 2 for sure.
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Alright, now let us examine the "dragon" way of squaring arbitrary numbers in your head. The story goes that this was independently developed by www.trickshottim.com while he was lying in bed. Let us find the square of 17.Start with the nearest square you know. 15 x 15 = 225. Now, to this, add 15 (light blue), 16 (light green), 16 (dark blue), and 17 (dark green).
15 x 15 + (15 + 16) + (16 + 17) = 17 x 17 = 289
Similarly, you could subtract.
20 x 20 - (20 + 19) - (19 + 18) - (18 + 17) = 289
Those who are bored can work on generalizing this to more dimensions.
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If you are excited for more pictures, here is another way to visualize squaring numbers.
Again, let's calculate 17 x 17. This time, add and subtract to the nearest 10, in this case, 20.(17 + 3) x (17 - 3) = 20 x 14 = 280
From the figure above, we see that we've neglected a couple of squares. To be precise, we're missing a square whose edges are the size of the number we added and subtracted, 3. Hence, the full answer is as follows.
17 x 17 = (17 + 3) x (17 - 3) + (3 x 3) = 20 x 14 + 9 = 289
Tada!
1 comment:
You'll notice for the dragon method, you can simplify the summation of the numbers in between.
For example, for
17^2 = 15^2 + (15 + 16) + (16 + 17)
can be simplified for mental math by seeing that the first and last numbers in the sum add up to the same as the two middle numbers, namely 32. Since there are two sets, you only need to add 64 to 15^2 to get the answer.
For 18^2, there's this summation:
15^2 + 15+16+16+17+17+18
15+18=16+17=33
3 x 33 = 99
15^2 + 99 = 18^2
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