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Square root of 2

The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself or squared, equals the number 2. It may be written in mathematics as or . It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.

"Pythagoras's constant" redirects here. Not to be confused with Pythagoras number.

Representations

1.4142135623730950488...

Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.[1] The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.


Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places:[2]

Proofs of irrationality[edit]

Proof by infinite descent[edit]

One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation: it proves the statement " is not rational" by assuming that it is rational and then deriving a falsehood.

Representations[edit]

Series and product[edit]

The identity cos π/4 = sin π/4 = 1/√2, along with the infinite product representations for the sine and cosine, leads to products such as

The square root of two is the of a tritone interval in twelve-tone equal temperament music.

frequency ratio

The square root of two forms the relationship of in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.

f-stops

The celestial latitude (declination) of the Sun during a planet's astronomical points equals the tilt of the planet's axis divided by .

cross-quarter day

List of mathematical constants

√3

Square root of 3

√5

Square root of 5

2√2

Gelfond–Schneider constant

1 + √2

Silver ratio

(2000), "Irrationality of the square root of two – A geometric proof", American Mathematical Monthly, 107 (9): 841–842, doi:10.2307/2695741, JSTOR 2695741.

Apostol, Tom M.

(2007), Analytica priora, eBooks@Adelaide

Aristotle

Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.

Flannery, David (2005), The Square Root of Two, Springer-Verlag,  0-387-20220-X.

ISBN

; Robson, Eleanor (1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context", Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209.

Fowler, David

; Gover, T. N. (1967), "The generalized serial test and the binary expansion of ", Journal of the Royal Statistical Society, Series A, 130 (1): 102–107, doi:10.2307/2344040, JSTOR 2344040.

Good, I. J.

Henderson, David W. (2000), "Square roots in the Śulba Sūtras", in Gorini, Catherine A. (ed.), , Cambridge University Press, pp. 39–45, ISBN 978-0-88385-164-7.

Geometry At Work: Papers in Applied Geometry

Gourdon, X.; Sebah, P. (2001), "Pythagoras' Constant: ", .

Numbers, Constants and Computation

by Jerry Bonnell and Robert J. Nemiroff. May, 1994.

The Square Root of Two to 5 million digits

a collection of proofs

Square root of 2 is irrational

Grime, James; Bowley, Roger. . Numberphile. Brady Haran.

"The Square Root of Two"

2 billion searchable digits of √2, π and e

√2 Search Engine