This example demonstrates how to approximate the square root of using algebra. The square root of can be visualized as the length of the hypotenuse of a square with sides equal to . The value of is shown below.

Start by representing the square root of with the variable .

Multiply both sides by

Substitute from step 1 in for on the right side.

Add to both sides.

Factor from the left side.

Divide both sides by .

Simplify the left side and split the right side fraction into two parts.

Simplify again.

Substitute from step 1 on the left side.

Recursively substitute the expression from step 8 in for .

Combine like terms.

Recursively substitute the expression from step 8 in for .

Combine like terms.

Repeat the recursive substition for more accuracy. To calculate an approximate value substitute into the expression. This is shown below.

Recursively substituting six times gives us an approximation of that is accurate for 5-digits. This can be repeated for more and more accuracy.