Approximate Square Root of Two

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 .

  2. Add to both sides.

  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.