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An example of an irrational algebraic number is x0 = (2 1/2 + 1) 1/3. It is clearly algebraic since it is the root of an integer polynomial, ( x3 − 1) 2 = 2, which is equivalent to x6 − 2 x3 − 1 = 0.
Algebraic expression. In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number ). [1] For example, 3x2 − 2xy + c is an algebraic expression.
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are and . Algebraic fractions are subject to the same laws as arithmetic fractions . A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials.
An algebraic number is a number that is a root of a non-zero polynomial (of finite degree) in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero.
Approximation of algebraic numbers, Liouville's result In the 1840s, Joseph Liouville obtained the first lower bound for the approximation of algebraic numbers: If x is an irrational algebraic number of degree n over the rational numbers, then there exists a constant c(x) > 0 such that holds for all integers p and q where q > 0 .
Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to . The numerical value of an infinite continued fraction is irrational; it is defined from its infinite ...
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