two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning...

mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some...

algebraic irrational, and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it...

on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that...

1],\quad T_{\theta }(x)\triangleq x+\theta \mod 1,} where θ is an irrational number. Under the identification of a circle with R/Z, or with the interval...

constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational...

avoid relying on the definition of the length of a curve. The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of...

clockwise to θ = 0. A rose curve specified with an irrational number for k has an infinite number of petals and will never complete. For example, the...

repeat with a longer period. The representations of irrational numbers in any positional number system (including decimal and sexagesimal) neither terminate...

In mathematics, an irrationality measure of a real number x {\displaystyle x} is a measure of how "closely" it can be approximated by rationals. If a function...

Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions...

In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno...

its diameter, which is called π, and is not just an irrational number, but a transcendental number. Also well known is the golden ratio of two (mostly)...

14{\color {red}28}\ldots } is a commonly used approximation of the irrational number π, the ratio of the circumference of a circle to its diameter. 22...

In number theory, specifically in Diophantine approximation theory, the Markov constant M ( α ) {\displaystyle M(\alpha )} of an irrational number α {\displaystyle...

approximations of any irrational number. The constant in this result may not be further improved without excluding some irrational numbers (see below)....

In mathematics, the extended real number system is obtained from the real number system R {\displaystyle \mathbb {R} } by adding two elements denoted +...

rational numbers. Real numbers that are not rational numbers are called irrational numbers. The real numbers are categorised as algebraic numbers (which...

rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational...

rational number and 1 Q ( x ) = 0 {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=0} if x is not a rational number (i.e. is an irrational number). 1 Q (...

1 {\displaystyle \textstyle \varphi ^{2}=\varphi +1} ⁠ and is an irrational number with a value of φ = 1 + 5 2 = {\displaystyle \varphi ={\frac {1+{\sqrt...

found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them...

Pythagorean theorem. It is an irrational number, possibly the first number to be known as such, and an algebraic number. Its numerical value truncated...

represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient...

= p q ( x  is rational), with  p ∈ Z  and  q ∈ N  coprime 0 if  x  is irrational. {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac...

any irrational number and b is any non-zero rational number, then a and b are incommensurable. On the other hand, if both a and b are irrational numbers...

non-integer positional numeral system that uses the golden ratio (the irrational number 1 + 5 2 {\textstyle {\frac {1+{\sqrt {5}}}{2}}}  ≈ 1.61803399 symbolized...

recurring decimal. An irrational number has an infinite non-repeating representation in all integer bases. Whether a rational number has a finite representation...

{\displaystyle F(x)=x+N,} and its rotation number is N {\displaystyle N} (cf. irrational rotation). The rotation number is invariant under topological conjugacy...

which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals...