Whole number

the whole numbers are an extension of the natural numbers.

The whole numbers cover all numbers

< math> \ ldots, - 2, - 1.0.1.2, \ ldots< /math>

and thereby all natural numbers as well as their negative numbers contain. The quantity of the whole numbers becomes with the symbol <math> \ mathbb {Z}< /math>shortened („Z “stands for „numbers “). The alternative symbol <math> \ mathbf {Z}< /math> is meanwhile less common; a disadvantage of this bold print symbol is the difficult handwritten representability.

Above enumerating of the whole numbers shows in ascending order their natural arrangement also at the same time. Those Number theory is the branch of mathematics, which with characteristics of the whole numbers concerns itself.

Table of contents

characteristics

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A ring concerning the addition and the multiplication, D form ring the whole numbers. h. they can be added, subtracted and multiplied without restriction. Arithmetic rules apply like the commutative law and the associative law to addition and multiplication, in additionapply the distributive laws.

By the existence of subtraction linear equations of the form know

< math> A + x = b< /math>

with natural numbers <math> A< /math> and <math> b< /math> to be always solved: <math> x = b - A< /math>. Limits one <math> x< /math> on the quantity thatnatural numbers, then everyone is not solvable such equation.

Abstractly expressed is called that, the whole numbers forms a commutative unitary ring. The neutral element of the addition is 0, the additive inverse element of <math> n< /math> math <-> n /math< is>, the neutral elementthe multiplication is 1.

arrangement

the quantity of the whole numbers is totally arranged, in the order

< math> \ ldots < -2 < -1 < 0 < 1 < 2 < \ ldots< /math>

i.e. one knows ever two whole numberscompare. One speaks of positive <math> \ {1, 2, 3, \ ldots \}< /math>, nonnegative <math> \ {0, 1, 2, 3,… \}< /math>, negative <math> \ {\ ldots, -2, -1 \}< /math> and not-positive <math> \ {\ ldots, -2, -1, 0 \}< /math> whole numbers. The number 0 is neither positive nor negative. This order is compatiblewith the arithmetic operations, i.e.

is <math> A < b< /math> and <math> C \ leq D< /math>, then is <math> A + C < b + D< /math>,
is <math> A < b< /math> and <math> 0 < C< /math>, then is <math> AC < UC< /math>.

Like the quantity of the natural numbersalso the quantity of the whole numbers is countable.

The whole numbers do not form a body, because z. B. the equation is <math> 2x = 1< /math> not in <math> \ mathbb {Z}< /math> solvable. The smallest body, <math> \ mathbb {Z}< /math> , are the rational numbers contain <math> \ Bbb Q< /math>.

Euclidean ring

an important characteristic of the whole numbers is the existence of a division with remainder. Due to this characteristic there is always a largest common divisor, which one with that for two whole numbers Euclidean algorithm to determine knows. Mathematicians say, <math> \ mathbb {Z}< /math> is a Euclidean ring. From this also the sentence of the clear prime factorization in math <\> mathbb {Z follows}< /math>.

construction from the natural numbers

is given the quantity of the natural numbers, then leavethe whole numbers from it as number range extension design themselves:

We regard the quantity <math> \ mathbb {N} \ times \ mathbb {N}< /math> all pairs natural numbers, and define the following equivalence relation:

<math> (A, b) \ sim (C, D)< /math>, if <math> A + D = C + b< /math>

In addition we definean addition and a multiplication in this quantity:

<math> (A, b) + (C, D) = (A + C, b + D)< /math>
<math> (A, b) \ (C, D) = (AC + bd, ad + UC) /math< cdot>

The quantity of the equivalence classes we call <math> \ mathbb {Z} = \ mathbb {N}\ times \ mathbb {N}/\ sim< /math>, the equivalence class of a pair <math> (A, b)< /math> we write /math <as> math (A-B<)> /math <,> math (0 b<)> we write also as <math> - b< /math>.

The addition and multiplication of the pairs induce now well-defined linkages on <math> \ mathbb {Z}< /math>, with those <math> \ mathbb {Z}< /math> tooa ring becomes. Into this ring can which one the natural numbers in such a way embed:

<math> n \ longrightarrow (n - 0)< /math>

A whole number is called negative if it of the form <math> (0 n) = - n< /math> is with a natural number<math> n > 0< /math>.

This construction functions independently of whether <math> \ mathbb {N}< /math> the 0 contains or not.

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