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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 ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
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 ...
For example, any irrational number x, such as x = √ 2, is a "gap" in the rationals Q in the sense that it is a real number that can be approximated arbitrarily closely by rational numbers p/q, in the sense that distance of x and p/q given by the absolute value | x − p/q | is as small as desired. The following table lists some examples of ...
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 .
Solutions of the equation are also called roots or zeros of the polynomial on the left side. The theorem states that each rational solution x = p ⁄ q, written in lowest terms so that p and q are relatively prime, satisfies: p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n.
However, these techniques and results can often be used to bound the number of solutions of such equations. Nevertheless, a refinement of Baker's theorem by Feldman provides an effective bound: if x is an algebraic number of degree n over the rational numbers, then there exist effectively computable constants c(x) > 0 and 0 < d(x) < n such that
For example if A and B only contain rational numbers, they can still be cut at by putting every negative rational number in A, along with every non-negative rational number whose square is less than 2; similarly B would contain every positive rational number whose square is greater than or equal to 2.