THE SQUARE ROOT OF 2
The square root of 2 is a number that when multiplied by itself equals 2. It is also the hypotenuse of a right triangle whose legs are 1 unit each.
ADJUSTMENT
Suppose the square root of two could be expressed as the ratio between two numbers A/B. The result would be an equation in which the numerator squared is equal to double the denominator squared.
APPROXIMATING THE SQUARE ROOT OF 2
The square root of 2 is an irrational number, meaning that it cannot be expressed as the ratio of two whole numbers A/B. There is, however, two sequences that when combined together come close to achieving this proportion, always missing the mark by +/- 1.
A COMPOSITE SEQUENCE
Known since ancient times, the series of fractions A/B are a composite of two sequences, the Pell numbers (1,2,5,12…) and the Half-companion Pell numbers (1,3,7,17..). Together they form the closest rational approximations of the square root of 2. Each successive ratio more accurate then the previous, alternating between overestimating and underestimating the square root of 2.