Answer:
- use linear interpolation for a first approximation
- use Newton's method iteration to refine the approximation
Explanation:
Here's one way.
Multiply your number by an even power of 10 so that it is in the range 1–100. Call this power 2n. Call the scaled number N.
Make use of your knowledge of the squares of small integers to determine if the number you now have is a perfect square. If it is, the integer whose square it is is the square root of your scaled number. Skip to the Last Step.
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Based on your determination in the previous step, identify the perfect squares on either side of your (scaled) number. If L² is the lower of these squares, and H² is the higher, compute the value ...
X = (N -L²)/(H² -L²) +L . . . . linearly interpolated approximate root
You can use this as an approximation of √N. If it is not close enough for you, you can refine it by computing an iterated value:
X ← (X² +N)/(2X) or, equivalently, X ← (1/2)(N/X +X)
The first of these approximations will get you within 6%; one iteration will get you within 0.2%. (Each additional iteration will approximately double the number of accurate decimal places.)
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Last step: Using the integer or X (or iterated X) value, multiply by 10^-n to restore the appropriate scale of your root.
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Example:
√199.6 ≈ ?
Multiply your value by 10^-2 to get N = 1.996 (with 2n=-2). The squares on either side are 1² = 1 and 2² = 4. Then the first approximation of the root is ...
X = (1.996 -1)/(4 -1)+1 = 0.996/3 +1 = 1.332
One iteration of our formula gives ...
X = (1.332² +1.996)/(2·1.332) = 1.415
Then the last step is to multiply this value by 10^-n = 10^1:
√199.6 ≈ 14.15 . . . . root of 199.6 within 0.2%
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Additional comments
The iteration described above is also called the "Babylonian method".
If you're familiar with squares of integers between 2 and 20, then better linear interpolation accuracy can be had by doing the scaling so that N is between 4 and 400.
A method for developing square roots digit by digit, suitable for hand calculation, is described in some detail on the NIST web site. It can be found by doing a web search on "square root nist algorithm" without the quotes.