Square root of 5

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as: The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as: It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are: which can be rounded down to 2.236 to within 99.99% accuracy. The approximation 161/72 (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than 1/10,000 (approx. 4.3×10−5). As of December 2013, its numerical value in decimal has been computed to at least ten billion digits. 1. This irrationality proof for the square root of 5 uses Fermat's method of infinite descent: 2. This irrationality proof is also a proof by contradiction: It can be expressed as the continued fraction The convergents and semiconvergents of this continued fraction are as follows (the black terms are the semiconvergents): Convergents of the continued fraction are colored red; their numerators are 2, 9, 38, 161, ... (sequence A001077 in the OEIS), and their denominators are 1, 4, 17, 72, ... (sequence A001076 in the OEIS). Each of these is the best rational approximation of √5; in other words, it is closer to √5 than any rational with a smaller denominator.