Number range
a number range is an exactly defined quantity of numbers. They differ in the kind of the mathematical operations, which one can accomplish in these quantities without reservation.
Table of contents |
usual number sets
natural numbers
symbol: <math> \ mathbb {N}< /math>
Natural numbers arose for D from the basic need of humans to count things. h. to determine the number from elements to. By them one understands the quantity of all positive whole numbers. Occasionally also still the neutral number of 0 is added to them, some text books notes this counting rangethen as N_{ 0}. Addition and multiplication are without reservation possible.
Example: 3 + 4 = 7, but 3 - 4 no result in math <\> mathbb {N gives}< to /math>.
The quantity covers the numbers of 1, 2, 3, 4, 5, 6 etc.
An important subset of the natural numbers is the quantity of the prime numbers, those sometimes with <math> \ mathbb P< /math> one designates.
whole one numbers
symbol: <math> \ mathbb {Z}< /math> (“gan z e Z awls ")
the whole numbers extend the natural numbers by negative whole numbers. With them it is possible to subtract without reservation .
Example:3 - 4 = -1
the quantity covers the numbers… -3, -2, -1, 0, 1, 2, 3…
rational one numbers
symbol: <math> \ mathbb {Q}< /math>
The rational numbers cover the quantity of all break numbers. A break number is the quotient of two whole numbers, whereby thoseRestriction applies that the divisor (more =Nenner) may not be 0. With the extension on the rational numbers all four basic operations of arithmetic are including the division executable.
Examples: <math> {1 \ more over 3}< /math>, <math> {7 \ more over 13}< /math>, <math> \, 1< /math>, <math> \, 8< /math>
real numbers
symbol: <math> \ mathbb {R}< /math>
The realNumbers do not form a synthesis from the rational numbers and the surds in such a way specified - infinite, periodic and therefore not numbers representable as break. Pulling the root with positive radicand can be accomplished now clearly.
Examples: <math> \ sqrt [] {2}, \ [3] {17} /math<,> π , e sqrt
Complex numbers
symbol: <math> \ mathbb {C}< /math>
Despite the extension on the real numbers it is not yet possible to solve all equations. Thus the equation x cannot ^{be solved} 2 = -1 still, since the square of real numbers is positive always zero or.In order to work against this problem, a recent extension of the number range was necessary on the complex numbers. Their basis is the introduction of an imaginary number of i, whose square -1 results in (i^{ 2} = -1). Complex numbers consist of a real and an imaginary part. Around complex numbersto multiply one often uses the Gauss' level and the polar form.
Examples:
- 5 + 3i
- in the goniometrischen form< math> 5 {,} 83 \ (\ cos 30 {,} 96^ {\ circ} + \ mathrm {i} \ \ sin 30 {,} 96^ cdot {\ circ}) /math< cdot> or in the exponential form math <5.83> \ {e cdot more briefly} ^ {{\ rm i} \, {30 {,} 96} ^ {\ circ}}< /math> and/or. on pocket calculators
5.83 cis 30,96°
- 4 - 5i
comparison of the number ranges
the number ranges mentioned are number range extensions preceding of in each case:
- <math> \ mathbb {N} \ sub \ mathbb {Z} \ sub \ mathbb {Q} \ sub \ mathbb {R} \ sub \ mathbb {C}. </math>
hypercomplex one numbers
the construction procedure for the production of the complex numbers can be generalized and supplies among other things the following counting ranges.
quaternions or Hamilton numbers
symbol: <math> \ mathbb {H}< /math>
These numbers, which are represented by the elements of the quaternion ring, are the extension of the complex numbers. They form one inclined body in their algebraic structure only, since they are not commutative. Their representation takes place in the form of threeImaginary parts.
Examples: 5 + 3i + 9j + 4k, -8 + 6i - 3j + 9k
the complex numbers can be understood in many different kinds as subset of the quaternions: Math <I=x> is \ mathrm i+y \ mathrm j+z \ mathrm k< /math> with <math> x^2+y^2+z^2=1< /math>, then math <\> mathbb R+ is \ mathbb R \ cdot I< /math> a body,that to the complex numbers is isomorphic. Everything these in such a way received subfields of <math> \ mathbb {H}< /math> are to each other conjugated.
Oktaven or Oktonionen or Cayley numbers
symbol: <math> \ mathbb {O}< /math>
The Oktaven represent a eight-dimensional extension of the real numbers (a two-dimensional element of the quaternion ring). Their multiplicationis no longer associatively separate only alternatively. They are the maximumdimensional number range, in division are possible, them form a division algebra.
Examples: 7 + 8i + 3j - 12k + 4E - 8I - 9J + 12K
there <math> \ mathbb {R}< /math>, <math> \ mathbb {C}< /math>, <math> \ mathbb {H}< /math>,< math> \ mathbb {O}< /math> those only standardized division algebras are, them as numbers are likewise designated, although for instance with <math> \ mathbb {O}< /math> not even more the Assoziativität applies.
hyper+rational numbers
the construction procedures for the production of the real numbers can be generalized and supplied among other things:
p-adische numbers | of Q_{ p} |
hyper+real numbers | *R |
Surreale numbers | of Sω |
further one algebraic structures, those numbers to be sometimes called
Sedenionen | <math> \ mathbb {S}< /math> |
Remainder class body | Z/pZ, p prime number |