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- What is the square root of Pi?

The first step is to pick a value you know to be too small the be the square root of Pi. Call this value S (for small guess). The next step is to pick a value you know to be too large to be the square root of Pi. Call this value L (for large value).

Then since Pi = 3.14159…. we’ll say S = 1, and L = 2.

S = 1

L = 2

Then find S^2 and L^2 (S and L squared):

S^2 = 1

L^2 = 4

Which of these values is farther away from Pi? S^2 is farther away from Pi then L^2, so replace S with the average of S and L. Now we have:

S = 1.5

L = 2

Continue the process:

S^2 = 2.25

L^2 = 4

Replace the small value again since its square is farther away.

S = 1.75

L = 2

Goes to:

S^2 = 3.0625

L^2 = 4

Replace the large value now:

S = 1.75

L = 1.875

Goes to:

S^2 = 3.0625

L^2 = 3.515625

Replace the large value again:

S = 1.75

L = 1.8125

Goes to:

S^2 = 3.0625

L^2 = 3.28515625

At which point we know to replace the large value again since its square is farther away then the small value’s square.

So

S = 1.75

L = 1.78125

At this point we know the square root of Pi lies between 1.75 and 1.78125. Continuing this process we can get finer and finer bounds for the value of the square root of Pi. After a while we will see that the square root of Pi equals 1.77245…. Since Pi is irrational we can never get an exact answer and we have to settle for an approximation.

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