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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.
All rational numbers are algebraic. 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.
In mathematics a rational number is a number that can be represented by a fraction of the form a / b , where a and b are integers and b is not zero; the set of all rational numbers is commonly represented by the symbol Q or , which stands for quotient.
Some programming languages provide a built-in (primitive) rational data type to represent rational numbers like 1/3 and −11/17 without rounding, and to do arithmetic on them. Examples are the ratio type of Common Lisp , and analogous types provided by most languages for algebraic computation , such as Mathematica and Maple .
Example: Let a and b be nonzero real numbers. Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.
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