Search results
Results from the WOW.Com Content Network
Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer. [3] All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers (): Numbers that correspond to points along a line. They can be positive, negative, or zero.
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.
This polynomial has no rational roots, since the rational root theorem shows that the only possibilities are ±1, but x 0 is greater than 1. So x 0 is an irrational algebraic number. There are countably many algebraic numbers, since there are countably many integer polynomials.
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 ...
The square root of the Gelfond–Schneider constant is the transcendental number = 1.632 526 919 438 152 844 77.... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational ...
300 to 100 BCE [10] Negative one: −1 −1 300 to 200 BCE Cube root of 2: 1.25992 10498 94873 16476 [Mw 6] [OEIS 8] Real root of = 46 to 120 CE [11] Cube root of 3 1.44224 95703 07408 38232 [OEIS 9] Real root of =