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In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a fraction a / b with a and b integers and b not zero. This is also known as being incommensurable, or without common measure. The irrational numbers are precisely those numbers whose expansion in any given base (decimal ...
An example of an irrational algebraic number is x 0 = (2 1/2 + 1) 1/3. It is clearly algebraic since it is the root of an integer polynomial, ...
Irrational numbers (): Real numbers that are not rational. Imaginary numbers : Numbers that equal the product of a real number and the imaginary unit i {\displaystyle i} , where i 2 = − 1 {\displaystyle i^{2}=-1} .
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
Foias constant is the unique real number such that if x 1 = ... [OEIS 100] Computed up to 1 011 597 392 terms by E. Weisstein. ... is irrational.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
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ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.