Rational number

A rational number is a real number, defined as the ratio (Latin ratio) of two integers can be represented. The set of all rational numbers is denoted by the symbols (of " quotient "). It includes all the numbers that can be represented as a fraction that contains integers in both the numerator and the denominator. The precise mathematical definition is based on equivalence classes of pairs of integers. The rational numbers are - especially in school mathematics - also referred to as fractional numbers, while the term fractional ( decimal, binary fraction, common fraction, mixed fraction ... ) is used for certain spellings of a rational number.

Definition

The set of rational numbers consisting of the amount of negative rational number, the number zero and the amount of positive rational numbers. The definition of rational numbers is based on the representation of rational numbers by fractions, ie pairs of integers. It is structured so that the computing can be carried out with rational numbers as usual with the help of their fraction models, abstracted but also the rational number from its fraction models. The rational numbers are not postulated as completely new things, but attributed to the integers.

The definition starts with the set of all ordered pairs of integers. Important: These couples are not rational numbers.

We define addition and multiplication on this set as follows:

These are the known rules of calculation of fractions. The pairs of numbers can be so interpreted as fractures.

An object of the definition of rational numbers is, for example, that the openings and the same " number", respectively. So you considered breakthroughs that are equivalent to each other ( of equal value ). This is expressed by an equivalence relation which is defined as follows:

Is important that this relation is an equivalence relation in fact, that the total amount (hereinafter called equivalence classes ) into subsets with each equivalent elements disassembled; this can be proved.

For the equivalence classes are defined again calculation rules that are based on the fractions and make sure that what is meant by a rational number, is abstracted from the concrete fracture representation. The addition of the equivalence class, and is defined as follows:

Choose from any element that is an ordered pair of integers (ie it selects a single element of, rather than two). Similarly, one selects the item.

And adding now according to the fractions and get a pair. This element is an equivalence class that is the result of the addition.

It is important that regardless of the specific choice of and always the same result, the equivalence class comes out; this property of addition must and can be proven.

Is defined analogously to the multiplication.

The equivalence classes are now sums up to as elements of a new quantity and call it rational numbers. A single rational number is thus an infinite set of ordered pairs. A notation as so designated in this sense, the equivalence class of all to equivalent pairs.

This can also be interpreted as number range extension of the integers by identifying the integer in each case with the rational number. And two are integers and the sum and product as the calculation rules for fractures are currently designed so that and is. Also by virtue of this identification is in fact a fraction of the ratio of the numerator and denominator.

Properties

The rational numbers contain a subset that is isomorphic to the integers ( choose to break the representation). This is often simplistically expressed so that the integers are contained in the rational numbers.

The rational numbers form a rigid body. is the smallest subfield of the field of real numbers, so its prime field.

A real number is rational if and only if it is algebraic first degree. Thus, the rational numbers themselves are a subset of algebraic numbers.

It can be shown that the smallest body that contains the natural numbers. is the quotient of the integers body.

The rational numbers are dense on the number line, which means that every real number (clearly: every point on the number line ) can be approximated arbitrarily closely by rational numbers.

Despite the tightness of in there can be no function that is only on the rational numbers steadily (and thus discontinuous at all remaining irrational numbers - conversely, has been going ).

Between two rational numbers a and b is always another rational number c (and thus any number ). Just take the arithmetic mean of these two numbers:

What initially sounds surprising, is the fact that the set of rational numbers is equal to the powerful set of natural numbers. In other words, there are bijective maps between and that every rational number q n and vice versa assign a natural number. Cantor's first diagonal argument and the star - Brocot tree provide such bijective mappings. The property to a proper subset to be equally powerful by itself, is characteristic of infinite sets.

Decimal expansion

Any real number can be assigned to a decimal expansion. Remarkably, every rational number has a periodic decimal expansion, every irrational number, however, a non-periodic (note: a finitely terminating decimal expansion is a special case of periodic decimal expansion, in which the decimal digit 0 or 9 repeats periodically using the finite sequence of digits ).

The recurring part is indicated by an overline.

Examples are:

In the square brackets are the equivalent developments are given in the binary system. Multi-digit periods are separated by spaces here.

The b- adic fraction developments to other integral number bases are not periodic for all rational numbers periodically and for all irrational numbers.

Related Topics

• Irrational number
• Rational function
• Valuation theory: Rating, integer
• Ordinals