Square Root of 2 - Computation Algorithms

Computation Algorithms

There are a number of algorithms for approximating, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots. It goes as follows:

First, pick a guess, ; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:

The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better approximation of the square root of 2 is achieved. Each iteration approximately doubles the number of correct digits. Starting with a₀ = 1 the next approximations are

• 3/2 = 1.5
• 17/12 = 1.416...
• 577/408 = 1.414215...
• 665857/470832 = 1.4142135623746....

The value of was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team in 1997. In February 2006 the record for the calculation of was eclipsed with the use of a home computer. Shigeru Kondo calculated 200,000,000,000 decimal places in slightly over 13 days and 14 hours using a 3.6 GHz PC with 16 GiB of memory. Among mathematical constants with computationally challenging decimal expansions, only π has been calculated more precisely. Such computations aim to empirically check whether such numbers are normal.