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Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the
A simple example is the set of non-zero rational numbers. Here identity 1 is had, as opposed to groups under addition where the identity is typically 0. Note that with the rationals, zero must be excluded because, under multiplication, it does not have an inverse: there is no rational number that can be multiplied by zero to result in 1.
Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number . [153] One way to construct the real numbers relies on the concept of Dedekind cuts.
A rational number has an indefinitely repeating sequence of finite length l, if the reduced fraction's denominator contains a prime factor that is not a factor of the base. If q is the maximal factor of the reduced denominator which is coprime to the base, l is the smallest exponent such that q divides b ℓ − 1 .
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.
The numbers being multiplied are multiplicands, multipliers, or factors. Multiplication can be expressed as "five times three equals fifteen", "five times three is fifteen" or "fifteen is the product of five and three". Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅).
A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x 2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √ x + 4.
If b is a positive real algebraic number, and x is a rational number, then b x is an algebraic number. This results from the theory of algebraic extensions . This remains true if b is any algebraic number, in which case, all values of b x (as a multivalued function ) are algebraic.
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