Squaring Numbers in Range 51-100

If A is between 50 and 100, then A = 50 + x. Compute a = 50 - x. Then A² = a² + 200x. For example,

63² = 37² + 200·13 = 1369 + 2600 = 3969.

Similarly, if A = 37, then x = 37 - 25 = 12, and a = 25 - 12 = 13. Therefore,

37² = 13² + 100·12 = 1200 + 169 = 1369.

Why does this work?

(50 + x)² - (50 - x)²= 100·2x
 = 200x.

So that

(25 + x)²= (25 - x)² + 200x.

Another approach was communicated to me by my late father Moisey Bogomolny.

We are looking to compute A², where A = 50 + a. Instead compute 100·(25 + a) and add a². Example: let A = 57. a = 57 - 50 = 7. 25 + 7 = 32. Append 49 = 7². Answer: 57² = 3249.

Why does this work?

The same algebra as above gives

(50 + x)²= 2500 + 100x + x²
 = 100×(25 + x) + x².

In general, if the number to be squared is close to a number with a known square, yet another approach is available. Assume we wish to compute 57². Observe that 60² = 3600 and 57 is pretty close to 60. Take the difference: 60 - 57 = 3 and also compute 57 - 3 = 54. Then
 57²= 60×54 + 3²
  = 3240 + 9
  = 3249.

Why does this work?

It is still the same algebra as above:

A² = (A + x)(A - x) + x².

The whole trick here is to remember some squares, like 6² = 36. The more your remember the better.


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