Question 509587
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You don't.  Saying *[tex \Large \sqrt{89}\ =\ 9.43] is incorrect.  In fact, saying *[tex \Large \sqrt{89}\ =\ 9.4339811320566038113206603776226], which is the answer the built-in Windows calculator gives, is also incorrect.  Both are approximations, although one is much closer than the other.  Since *[tex \Large \sqrt{89}] is an irrational number it is impossible to write it exactly as a decimal fraction.  On the other hand, 9.43 is probably close enough for most purposes.


Just looking at *[tex \Large \sqrt{89}] and not having a calculator, one way to go about it is to recognize that *[tex \Large 9\ \times\ 9\ =\ 81] and *[tex \Large 10\ \times\ 10\ =\ 100], and since *[tex \Large 81\ <\ 89\ <\ 100], *[tex \Large \sqrt{81}\ =\ 9\ <\ \sqrt{89}\ <\ 10\ =\ \sqrt{100}].  Then 89 is a little less than half-way from 81 to 100, so if you try 9.5 times 9.5 to get 90.25 (too big) and 9.4 times 9.4 to get 88.36 (too small), you know you have to be somewhere in between.  Keep refining to as many decimal points as you like.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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