How do you figure the square root of 135?

Answer 1

The square root of 135 is an irrational number, so it can not be calculated exactly. You can, however, use several methods to estimate it.

Here is the Babylonian method:

We start with an initial guess, x0 = 11.5

I chose this number because 11^2 = 121 and 12^2 = 144, so 135^(1/2) must be between the two numbers.

We then use the formula:

xn+1 = 1/2 * (xn + S / xn)

Where S is the number we are trying to find the square root of

So since x0 = 11.5 and S = 135, we have

x1 = 1/2 * (11.5 + 135 / 11.5)

= 1/2 * (11.5 + 1350 / 115)

= 2672.5/230

= 26725/2300

Which is approximately 11.619565

Using a calculator, we have sqrt(135) = 11.6189, so our estimate is good to 2 decimal places after a single iteration.

We can improve the accuracy as much as we like by plugging each new approximation into the formula again.

As you can see, approximating square roots without the help of a calculator is time consuming.