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Concept
Square root of 5
Summary
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:
2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089... .
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. 4.3e-5). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.
The square root of 5 can be expressed as the continued fraction
The successive partial evaluations of the continued fraction, which are called its convergents, approach :
Their numerators are 2, 9, 38, 161, ... , and their denominators are 1, 4, 17, 72, ... .
Each of these is a best rational approximation of ; in other words, it is closer to than any rational with a smaller denominator.
The convergents, expressed as x/y, satisfy alternately the Pell's equations
When is approximated with the Babylonian method, starting with x0 = 2 and using xn+1 = 1/2xn + 5/xn, the nth approximant xn is equal to the 2nth convergent of the continued fraction:
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial . The Newton's method update, , is equal to when . The method therefore converges quadratically.
The golden ratio φ is the arithmetic mean of 1 and . The algebraic relationship between , the golden ratio and the conjugate of the golden ratio (Φ = –1/φ = 1 − φ) is expressed in the following formulae:
(See the section below for their geometrical interpretation as decompositions of a rectangle.
Official source
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