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The integers arranged on a number line. An integer is the number zero , a positive natural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .). [1] The negations or additive inverses of the positive natural numbers are referred to as negative integers. [2]
Fractions: A representation of a non-integer as a ratio of two integers. These include improper fractions as well as mixed numbers . Continued fraction : An expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of ...
The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form a/10 n, where a is an integer, and n is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.
The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2 n 5 m, where m and n are non-negative integers. Proof: If the decimal expansion of x will end in zeros, or = = = = / for some n, then the denominator of x is of the form 10 n = 2 n 5 n.
Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator. Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m / n represents m parts of a whole divided into n equal parts.
Unlike positive-base systems, where integers and terminating fractions have non-unique representations (for example, in decimal 0.999... = 1) in negative-base systems the integers have only a single representation. However, there do exist rationals with non-unique representations.
The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is . [19] Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since
In the case of fractions of integers, the fractions a / b with a and b coprime and b > 0 are often taken as uniquely determined representatives for their equivalent fractions, which are considered to be the same rational number. This way the fractions of integers make up the field of the rational numbers.
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