Advanced Math/Square Roots
QUESTION: Hi!How can I calculate the square root of an arbitrary number without using calculator?
ANSWER: Michael, first of all I assume you mean not using the square root key on the calculator, so artimetic is OK.
My approach is to take the nearest integer that is a square and then use a Taylor series expansion about this value to find the target square root. You may not be famiiar with Taylor series; the first few terms of a Taylor series are
f(x+a) = f(x) + f'(x)a + f''(x)a^2/2! + ...
where f' = df/fx = first derivative with respect to x of the function f, f'' = second derivative, etc. This example is expanding f about f(x); the approximation gets better with smaller a and/or using more terms in the expansion.
For the square root problem, let x^2 = n^2 + a, where n is an integer and n^2 is the square nearest to x^2 and x is the square root we are trying to find. We have
x = (n^2 +a)^(1/2)
Taking derivatives and using the above expressions gives
x = n +a/(2n) - (1/8)a^2/n^3 + ...
Note that all the operations can be done easily with pencil and paper.
As an example, I calculated the results for 3, 4, 5, 6, and 7 using 4 as trhe nearest square (square root = 2) using this formula. The error is only a few percent, except for 7, but this mught be better approximated using 3^ = 9 as the nearest square.
number nearest perfect difference linear approx^2 actual root quad approx quad approx^2 error
3 4 -1 1.75 3.06 1.73 1.749 3.059 0.020
4 4 0 2.00 4.00 2.00 2.000 4.000 0.000
5 4 1 2.25 5.06 2.24 2.249 5.058 0.012
6 4 2 2.50 6.25 2.45 2.496 6.230 0.038
7 4 3 2.75 7.56 2.65 2.741 7.514 0.073.
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QUESTION: Uh...I think your method is too difficult for me coz I can hardly understand....Can you give me other simpler methods?
Here's a method I found online:
Finding square roots of of numbers that aren't perfect squares without a calculator
1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.
2. Divide - divide your number by one of those square roots.
3. Average - take the average of the result of step 2 and the root.
4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
Let me know if you need this explained.