# At-home math lesson: doubling and dividing

#### By Michael Avidon, math editor

You can calculate the square root of a number to any number of places without a calculator. If the original number is not a perfect square, then its square root will be irrational. Therefore, the decimal expansion is non-terminating and non-repeating. This method will give you the best approximation to any number of places.

### Example

Determine the square root of 76 to two decimal places.

First, find the greatest whole number whose square is less than 76. This is 8, as 64 < 76. After this, you will insert two zeros at a time after the decimal in the radicand for every decimal place in the square root. The procedure follows: ? is the greatest digit such that this product (ignoring decimals) does not exceed the remainder. Because and 7 x 167 = 1169 < 1200 and 8 x 168 = 1344 > 1200, ? = 7. Repeat the procedure. This time, you will double 87 to get 174. (Note that 2 x 1742 = 3484 > 3100, but the product is closer to 3100 than 1741.)

The same procedure one more time yields 8.717 (7 x 17,427 = 121,989 < 135,900). So, rounded to two decimal places, √76 = 8.72.

MORE: View a printable version | NEXT LESSON: At-home lessons: the golden ratio

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