Dear Susana del Castillo,
Your question is: How can you solve for the square root of a given number (ex: 153) without using a calculator?
Let me reinterpret your question, since a given number could be 4, and you know its square root. How can you solve for the square root of a number that is not a perfect square* (ex: 153) without using a calculator?
This question was actually born in Egypt, where they extracted the first square roots around 1650 BC. It may also rise when you take the ICFES, since no calculators are allowed.
I will use your example: 153.
Process:
1. Determine the two perfect squares that are closest to 153. They are 144 and 169.
2. Since the square root of 144 is 12 and that of 169 is 13, the square root of 153 is a number 13>x>12. Make a guess. Let our guess be 12.5.
3. Divide the number you are taking the square root of by this guess. 153/12.5=12.24
4. Average your guess and the result of the division. (12.5+12.24)/2=12.37.This average is an approximation of the square root of 153. The exact answer rounded to the nearest thousandth is 12.369.
5. To obtain an even more accurate approximation, repeat steps 3 and 4 as many times as you wish, but each time letting your guess be the immediately preceding approximation. In the case exposed above, 12.37 would be your new guess. The division that follows holds 153 as its numerator.
* Perfect square: an integer which is the square of some other integer, i.e. can be written in the form n^2 for some integer n.
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3 comments:
Luis DP,
You have set a high standard for your responses and
have given quality help to anyone who sincerely wants to use it.
Thanks for your efforts.
Thanks for the response, its very helpful. I'll have to practice a bit with it since finding what the square root of 169 or 144 is turns out complicated for me, without using a calculator. Your answer is very complete for the question you proposed in exchange of my original. I would appreciate further more an answer which included perfect squares. But again, thank you very much.
Luis, this is a really great response. I did understand how to solve for squares, but your answer helped me practice. Thank You
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