How to solve square roots by hand

as an example, use square root of 5.

show me the steps on how to do it? i've always wondered.

A simple way to compute a square root is the high/low method, similar to the bisection method. This method involves guessing a number based on known squares, then checking if its square is too high or too low and adjusting accordingly.

To find the square root of 20, first note that the square of 5 is 25, and that the square of 4 is 16. As 20 is greater than 16 and less than 25, the square root of 20 must be in between 4 and 5. Guessing 4.5, as the average of 4 and 5, yields 20.25 and is too high. The next step is to guess 4.4, yielding 19.36 and is too low. Therefore, as before, the square root of 20 must be in between 4.4 and 4.5. Continue this pattern until the desired number of decimal places is achieved. For example:

4.452 = 19.8025 (too low)
4.472 = 19.9809 (too low, but close)
4.482 = 20.0704 (too high)
4.4752 = 20.025625 (too high)
4.4732 = 20.007729 (too high, but close)
4.4722 = 19.998784 (too low)

Now it is known that the square root of 20 is between 4.472 and 4.473, so the square root of 20 to the first three decimal places is 4.472.


For 5 you know that the square root of 4 is to, so start with 2
2.5^2 = 6.25 (too high)
2.25^2 = 5.0625 (too high, but closer)
2.225^2 = 4.95 (too low)
2.235^2 = 4.995 (too low, but closer)
2.236^2 = 4.9996 (low, but closer)
2.23605^2 = 4.9999 (very close)

If you keep going, the answer is: 2.236067977499....

first things first all square roots of non square numbers are irrational
so you can only make an approximation
1st make a guess
we know the answer will be between 2 and 3 the square roots of the 2 square numbers closest to 5 on either side.
so try 2.5 2.5*2.5 is 6.25 to high
find the midpoint between 2 and 2.5
2.25 2.25^2 is 5.0625 not close enough?