A Stoneham number is a number of the formwhere are relatively prime positive integers. Stoneham (1973) proved that is -normal whenever is an odd prime and is a primitive root of . This result was extended by Bailey and Crandall (2003), who showed that is normal for all positive integers and provided only that and are relatively prime.
Pythagoras's theorem states that the diagonal of a square with sides of integral length cannot be rational. Assume is rational and equal to where and are integers with no common factors. Thensoand , so is even. But if is even, then is even. Since is defined to be expressed in lowest terms, must be odd; otherwise and would have the common factor 2. Since is even, we can let , then . Therefore, , and , so must be even. But cannot be both even and odd, so there are no and such that is rational, and must be irrational.In particular, Pythagoras's constant is irrational. Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for (the golden ratio) and using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality of is given by Wiedijk (2006)...
Let be a real number, and let be the set of positive real numbers for which(1)has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,(2)where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :(3)where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality(4)holds for any and all integers and with sufficiently large.The..
An irrational number is a number that cannot be expressed as a fraction for any integers and . Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.There is no standard notation for the set of irrational numbers, but the notations , , or , where the bar, minus sign, or backslash indicates the set complement of the rational numbers over the reals , could all be used.The most famous irrational number is , sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include , , , etc. The Erdős-Borwein constant(1)(2)(3)(OEIS A065442; Erdős 1948, Guy 1994), where is the numbers of divisors of , and a set of generalizations..
In general, an unresolved nth root, commonly involving a radical symbol , is known as a surd. However, the term surd or "surd expression" (e.g., Hardy 1967, p. 25) can also be used to mean a sum of one or more irrational roots. In the mathematical literature, the term arises most commonly in the context of quadratic surds.The term "surd" has a special meaning in the Wolfram Language, where the principal th root of a complex number can be found as z^(1/n) or equivalently Power[z, 1/n]. However, when is real and only real roots are of interest, the command Surd[x, n] which returns the real-valued th root for real odd and the principal th root for nonnegative real and even can be used.