999 times 999 in your head

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What is 999 × 999? Do it in your head, without any written working, and explain the shortcut so that it also works for 998 × 998 and for 997 × 1003. Say which algebraic identity each of those three calculations uses.

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998,001. The shortcut. 999 squared = (1000 minus 1) squared = 1,000,000 minus 2 × 1000 + 1 = 1,000,000 minus 2,000 + 1 = 998,001. 998 squared. (1000 minus 2) squared = 1,000,000 minus 4,000 + 4 = 996,004. 997 × 1003. This is (1000 minus 3)(1000 + 3) = 1,000,000 minus 9 = 999,991, by the difference of two squares. The general tools. For numbers near a round base B: (B minus a) squared = B squared minus 2aB + a squared, and (B minus a)(B + a) = B squared minus a squared. Both come straight from expanding brackets, and both turn a hard multiplication into a subtraction. A related trick for numbe…

999 times 999 in your head

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