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|
|2.25||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|
|2.237||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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