Methods of Computing Square Roots - High/low Method

High/low Method

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 42 is 16, and that 52 is 25. As 16 < 20 < 25, the square root of 20 must be between 4 and 5. Guessing 4.5 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. This pattern is continued 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)

4.4722 = 19.998784 (too low, but close)

Now it is known that the square root of 20 is between 4.472 and 4.473, so the decimal representation begins with 4.472...