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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.

 Guess Square of guess High/low 2.3 5.29 Too high 2.2 4.84 Too low 2.25 5.0625 Too high but closer 2.24 5.0176 Too high but closer 2.235 4.995225 Too low, so between 2.235 and 2.24 2.236 4.999696 Too low 2.237 5.004169 Too high, so between 2.236 and 2.237 2.2365 5.0019 Too high, so just below 2.2365 2.2363 5.00103 Too high, so between 2.2363 and 2.2360 2.2361 5.00014 Too high, so between 2.2361 and 2.2360 2.23605 4.9999196 Too 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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