Approximate Square Root of Two
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

Multiply both sides by .

Add to both sides.

Factor from the lefthand side of the equation and split into on the righthand side.

Divide both sides by .

Simplify.

Substitute the for on the lefthand side.

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

Simplify the denominator.

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

Simplify denominator.

Repeat for substitution for more accuracy.