Calculate Square Numbers

Calculate Square Numbers

This article shows a simple method for calculating square numbers in your head. You don't need a calculator!

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Example:  23² = 529

The rule: Find the nearest multiple of ten for 23, which is 20

               Now calculate: ((23-3) * (23+3)) + 3² =
                                        20 * 26 + 9 =
                                        520 + 9 =
                                        529

 

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Example:  46² = 2116

The rule: Find the nearest multiple of ten for 46, which is 50

Now calculate: ((46+4) * (46-4)) + 4² =

               Now calculate: ((46+4) * (46-4)) + 4² =
                                        50 * 42 + 16 =
                                        100 * 42 / 2 + 16 =
                                        2100 + 16 =
                                        2116

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 General Rule for x

The rule: Find the nearest multiple of ten for x, which is x+a or x-a

Now calculate: ((x+a) * (x-a)) + a²

Proof:  ((x+a) * (x-a)) + a² =
            (x² + x*a - a*x - a²) + a² =
            x² - a² + a² =
            x²