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calc_roots

# calc_roots - Calculating square roots cube roots and n th...

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Calculating square roots, cube roots and n th roots of a real number Calculating square roots . We consider the problem of calculating = 2 1 2 2, that is the number which, when multiplied with itself, gives 2. 2 must lie between 1 and 2, because = 1 2 1, which is less than 2, while = 2 2 4, which is greater than 2. Try squaring the number 3 2 which is mid-way between 1 and 2. Since 3 2 2 = 9 4 is greater than 2, we see that 3 2 is greater than 2. If r represents the square root of 2, then = r 2 2 and = 2 r r , that is, dividing 2 into 2 gives 2. More concisely, = 2 2 2, because 2 x = 2 2. An alternative way to see that that 3 2 is greater than 2 is from the fact that when 3 2 is divided into 2, the resulting number = 2 3 2 4 3 is less than 3 2 . In fact, since 3 2 is greater than 2, = 2 3 2 4 3 is less than 2, that is 4 3 < 2 < 3 2 . Since 4 3 < 2 < 3 2 , we could try taking the number mid-way between 4 3 and 3 2 , namely + 3 2 4 3 2 = 17 12 , and check whether 17 12 is equal to, or close to, 2. In fact 17 12 is a slightly greater than 2, since 17 12 2 = 289 144 , which is 2 + 1 144 . Dividing 17 12 into 2 gives = 2 17 12 24 17 , which must be slightly less than 2. Note that 24 17 ~ 1.411764706 and 17 12 ~ 1.416666667. Since 24 17 < 2 < 17 12 , the number mid-way between 24 17 and 17 12 , namely 577 408 , is likely to be closer to 2 than either 24 17 or 17 12 .

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The number mid-way between 24 17 and 17 12 is 1 2 + 24 17 17 12 = 1 2 + 24 . 12 17 . 17 204 = + 288 289 408 = 577 408 . We can continue this process to calculate 2 as accurately as we want.
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