Squaring Numbers Near 50

Mental Math · Axiom Academy

EXAMPLE Squaring Numbers Near 50 Anchor at 50 , measure the signed distance d , and read the square straight off 2500 + 100d + d^2 . Work out 48^2 in your head. Anchor at 50 — a square you can write down instantly — measure how far 48 sits from it, and let one identity finish the job. Nicely done — two squares in the fifties, and one identity did both of them. Worth holding on to: One identity, both directions: writing n = 50 + d gives n^2 = 2500 + 100d + d^2 . There is no separate rule for below 50 and no sign to flip — d already carries it. That is why the sign goes on d , not on the formula: 48 has d = -2 and 53 has d = +3 , so the middle term comes out -200 and +300 on its own. The middle term is free: , so the whole term is 100d — the distance with two zeros after it. d^2 is always added: a square is never negative, so 48^2 = 2500 - 200 + 4 = 2304 , not 2500 - 200 - 4 . Two routes, one answer: 45^2 is 2025 by the ending-in- 5 rule ( , then write 25 after it) and 2025 by this one ( 2500 - 500 + 25 ). Agreement like that is a free check. The neighbourhood: 48^2 = 2304 , 49^2 = 2401 , 50^2 = 2500 , 51^2 = 2601 , 52^2 = 2704 , 53^2 = 2809 . Nothing in the identity limits d to a step or two — it is exact for every d . What decides whether it stays a mental calculation is only whether you can hold d^2 in your head, which is why the technique feels effortless through the forties and fifties and starts to cost something well outside them.

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