Rational number
a rational number is a number, which can be expressed as relationship (Latin reason ) of two whole numbers (usually one writes <math> A/b< /math>, reads A divided by b), whereby the denominator (here <math> b< /math>) not equal zerois. Each number, which can be represented as break of two whole numbers, is thus a rational number.
Table of contents |
definition
a real number of x is called rational if one it as quotient (or break) <math> p/q< /math> two whole numbers with <math> q \ neq 0< /math> to represent can. Andernfalls heißt x irrational.The quantity of all rational numbers becomes with <math> \ Bbb Q< /math> designated (the designation <math> \ mathbf {Q}< /math> is also still common).
Besides there is also a constructional definition of the rational numbers under exclusive use of the whole numbers (see further below).
First characteristics
- the rational numbers contain as subset the whole numbers <math> \ of Bbb Z< /math> (select too <math> z \ in \ Z< /math> the break representation <math> z/1< /math>).
- The rational numbers <math> \ Bbb Q< /math> form a body. <math> \ Bbb Q< /math> the smallest subfield of the body is <math> \ R< /math> the real numbers,thus its prime body.
- A number is rational exactly if it algebraically first degree is. Thus the rational numbers themselves are a subset of the algebraic numbers.
representational forms
to decimal fraction development
of each real number leavesa decimal fraction development assign themselves. Remarkably each rational number possesses a periodic decimal fraction development, each surd against it a nonperiodic (consider: a finally breaking off decimal fraction development is a special case of the periodic decimal fraction development, with itself after the finite number sequence the decimal digit 0 or9 periodically repeats).
Examples are:
1/3 | = 0,333333… | = [0.01 01 01…]_{ 2} |
9/7 | = 1.285714 285714… | = [1.010 010,010…]_{ 2} |
1/2 | = 0,50000… | = [0,10000…]_{ 2} |
1 = 1/1 | = 1,0000…= 0,9999… | = [0,1111…]_{ 2} |
in the square brackets is indicated the appropriate developments in the binary system. Periods of several characters are here separated by blanks in each case.
Also the b - adischen break developments to other integral Zahlenbasen <math> b \ neq 10< /math> are for allrational numbers periodically and for all surds nonperiodically.
one can
often divide further representational forms the break form of a rational number into partial fractions so mentioned, whose denominator is whole powers of prime numbers; z. B.
- <math> 5/6 = 1/2 +1/3 </math> , <math> 1/6 = 1/2 - 1/3< /math> , <math> 1/72 = 1/8 - 1/9< /math> , <math> 1/60 = 1/4 - 4/3 + 8/5< /math>.
There are also dismantlings as Egyptian breaks so mentioned (unit fractions), z. B.
- <math> 3/7 = 1/3 +1/11 + 1/231< /math> , <math> 25/31 = 1/2 + 1/4 + 1/18 + 1/1116< /math>,
the old Egyptians only such sums could do and on these counted.
The Zahlentripel <math> (1/5< /math> , <math> 24/35< /math> , <math> 5/7)< /math> is an example of a pythagoreischen break (see also pythagoreisches Tripel), because
- <math> (1/5) ^2 + (24/35) ^2 = (5/7) ^2< /math>.
the rational numbers can construction of the rational
numbers from the whole numbers of mathematical seen be designed as number range extension of the whole numbers, by one breaks as equivalence classes arranged pairs of whole numbers <math> (A, b)< /math> defined, whereby again <math> b< /math> not equal zero actual often become <math> b< /math> additionally also simply presupposed as (nonnegative ) natural number. Then one defines addition and multiplication with these pairs with the help of the following rules:
- <math> (A, b)+ (C, D) = (A \ cdot D + b \ cdot C, \, b \ cdot D)< /math>
- <math> (A, b) \ (C, D) = (A \ cdot C, \, b \ cdot D) /math< cdot>
Accompanying with our expectation that <math> 2/4 = 1/2< /math> to be is, leads we an equivalence relation<math> \ sim< /math> on these pairs with the following rule:
- <math> (A, b) \ sim (C, D)< /math> exactly then if, <math> A cdot \ cdot D = b \ C< /math>.
With the above arithmetic rules the quantity of the equivalence classes modulo forms ~ a body< math> \ for Bbb Q< /math>, itsElements rational numbers to be called. The equivalence class of <math> (A, b)< /math> one writes as <math> a/b< /math>.
it can
characteristics one show that <math> \ mathbb {Q}< /math> the smallest body is, that the natural numbers <math> \ mathbb {N}< /math> contains. <math> \ mathbb {Q}< /math> is the quotient body of the whole numbers <math> \ Z< /math>.
The rational numbers are closely on the number line, i.e.: Each real number (descriptive: each point on the number line) can be approximated arbitrarily exactly by rational numbers.
Between two rational numbers of A and b always lies a furtherrational number of C (and thus as much as desired). One takes simply the arithmetic means of these two numbers:
- <math> C = (A + b)/2< /math>
Which sounds surprising first, is the fact that the quantity of the rational numbers powerfully to the quantity thatnatural numbers is. In other words there is a bijektive illustration between <math> \ mathbb {Q}< /math> and <math> \ mathbb {N}< /math>, each rational number of q a natural number of n assigns and in reverse (possible such bijektive illustration is more near described with Cantor Diagonalisierung). The characteristic, powerfullyto a subset of itself to be to be able, is characteristic of infinite quantities.
rational function
- evaluation theory used
- topics: <math> p< /math> - evaluation, <math> p< /math> - whole number
- of ordinal numbers
- number system
Web on the left of
- http://www.wakkanet.fi/%7Epahio/ohjelmi.html PC program” break “, computes usual, partial, Egyptian, Pythagoreanand dyadische breaks, decimal and binary developments; solves accurately equations of second degree with rational coefficients; concrete rational cauchysche consequences (from the indicated side also in German and French language down loadable:Bruch.exe and/or. fraction.exe)
natural numbers< math> \ mathbb {N}< /math> | Whole numbers<math> \ mathbb {Z}< /math> | Rational numbers< math> \ mathbb {Q}< /math> | Real numbers< math> \ mathbb {R}< /math> | Complex numbers< math> \ mathbb {C}< /math> | Quaternions< math> \ mathbb {H}< /math> | p-adische numbers< math> \ mathbb {Z} _p, \ mathbb Q_p< /math>