Square root of 5

List of numbers Irrational and suspected irrational numbers $γ$ $ζ$ (3) √2 √3 √5 $φ$ $ρ$ $δ$ _$S$ $e$ $π$ $δ$
Binary	10.0011110001101111…
Decimal	2.23606797749978969…
Hexadecimal	2.3C6EF372FE94F82C…
Continued fraction	$2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\ddots }}}}}}}}$

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. 6995430000000000000♠4.3×10⁻⁵). As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.

This irrationality proof for the square root of 5 uses Fermat's method of infinite descent:

Suppose that √5 is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as $m / n$ for natural numbers $m$ and $n$ . Then √5 can be expressed in lower terms as $5 n - 2 m / m - 2 n$ , which is a contradiction. (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives $5 n 2 = m 2$ and $m / n = \sqrt 5$ , which is true by the premise. The second fractional expression for √5 is in lower terms since, comparing denominators, $m - 2 n < n$ since $m < 3 n$ since $m / n < 3$ since $\sqrt 5 < 3$ . And both the numerator and the denominator of the second fractional expression are positive since $2 < \sqrt 5 < 5 / 2$ and $m / n = \sqrt 5$ .)

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