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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 ...
Any rational number, expressed as the quotient of an integer a and a (non-zero) natural number b, satisfies the above definition, because x = a / b is the root of a non-zero polynomial, namely bx − a. [1] Quadratic irrational numbers, irrational solutions of a quadratic polynomial ax 2 + bx + c with integer coefficients a, b, and c ...
An algebraic number is any complex number that is a solution to some polynomial equation () = with rational coefficients; for example, every solution of + (/) + = (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields , or shortly number fields .
For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. ... Given a solution, the three numbers ... Thus a positive rational ...
For example, 3 is a square root of 9, ... All nth roots of rational numbers are algebraic numbers, ... For example, the solutions of the equation
Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. [93] Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers.
In terms of algebraic geometry, the algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational curve, and it is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions.
The set of all rational numbers includes the integers since every integer can be written as a fraction with denominator 1. For example −7 can be written −7 / 1 . The symbol for the rational numbers is Q (for quotient), also written .
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