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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.


---------- FOLLOW-UP ----------

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.

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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