# Approximate Square Root of Two

## Formula

To approximate the square root of two you can use algebra to derive a recursive formula to come up with an arbitrarily precise number. The formula is an example of recursion and is continues on forever. For practical purposes, one only needs to compute a value accurate enough for their use.

Shown below are the steps to deriving the formula:

## Steps

1. Multiply both sides by .

3. Factor from the left-hand side of the equation and split into on the right-hand side.

4. Divide both sides by .

5. Simplify.

6. Substitute the for on the left-hand side.

7. Recursively substitute the expression for from step number 6 into the denominator.

8. Simplify the denominator.

9. Recursively substitute the expression for from step number 6 into the denominator.

10. Simplify denominator.

11. Repeat for substitution for more accuracy.