The illustration at the right depicts Log(z), confining the arguments of z to the interval (-π, π]. < 1 Such a number can be visualized by a point in the complex plane, as shown at the right.

Both are defined via Taylor series analogous to the real case. Then √2 is the only number c > 1 for which f(c) = c2. Therefore, m and n cannot be both integers, hence √2 is irrational. z [107] By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals. Privacy Policy | Addition, the simplest of these, is undone by subtraction: when you add 5 to x to get x + 5, to reverse this operation you need to subtract 5 from x + 5. In 1997 the value of √2 was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team. {\displaystyle \sin } The square root of two has the following continued fraction representation: The convergents formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (known as side and diameter numbers to the ancient Greeks because of their use in approximating the ratio between the sides and diagonal of a square). {\displaystyle R'={\frac {L'}{S'}}} [23], Similar in appearance but with a finite number of terms, √2 appears in various trigonometric constants:[24], It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.

odd and This is related to the property of silver ratios. A simple proof is attributed by John Horton Conway to Stanley Tennenbaum when the latter was a student in the early 1950s[18] and whose most recent appearance is in an article by Noson Yanofsky in the May–June 2016 issue of American Scientist. As with the proof by infinite descent, we obtain . Such a locus is called a branch cut. This gives a lower bound of 1/3b2 for the difference |√2 − a/b|, yielding a direct proof of irrationality not relying on the law of excluded middle; see Errett Bishop (1985, p. 18).

The following nested square expressions converge to √2: In 1786, German physics professor Georg Lichtenberg[28] found that any sheet of paper whose long edge is √2 times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. Math. any complex number z may be denoted as. Proof: For the given equation, there are only six possible combinations of oddness and evenness for whole-number values of Unique positive real number which when multiplied by itself gives 2, "Pythagoras's constant" redirects here.