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Useful Computational Methods


Square-root by guess-and-check

One simple way to find a decimal approximation to, say √5, is to make an initial guess, square the guess, and improve the guess.

Suppose we want to find √5 to four decimal places.

Since 22 = 4 and 32 = 9, we know that √5 is between 2 and 3. Let's make a guess of it being 2.3. Squaring that we get 2.32 = 5.29. That's too high, so make the guess a little less, say 2.2. To find approximation to four decimal places we need to do this till we have five decimal places, and then round the result.

GuessSquare of guessHigh/low
2.35.29Too high
2.24.84Too low
2.255.0625Too high but closer
2.245.0176Too high but closer
2.2354.995225Too low, so between 2.235 and 2.24
2.2364.999696Too low
2.2375.004169Too high, so between 2.236 and 2.237
2.23655.0019Too high, so just below 2.2365
2.23635.00103Too high, so between 2.2363 and 2.2360
2.23615.00014Too high, so between 2.2361 and 2.2360
2.236054.9999196Too low, so between 2.23605 and 2.2361

We stop since we now know it would be rounded to 2.2361 (and not to 2.2360).

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Last updated - December 14, 2003