Hypercomplex number
hypercomplex one numbers are Verallgemeinerungen of the complex numbers. In this article hypercomplex numbers are regarded as algebraic structure. Sometimes also the quaternions are called the hypercomplex numbers.
Definition |
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remarks 5 related topics 6 literature [ work on] with addition and multiplication. One demands the following characteristics:
- To the addition the commutative law and the associative law apply.
- The addition is invertable.
- The left-sided and the rechtsseitige distributive law apply.
- The multiplication from hypercomplex numbers is bilinear over the real numbers, therefore applies (ax) (by) = off (XY) for all A, b <math> \ in \ mathbb {R}< /math> and x, y hypercomplex numbers.
The following characteristics are not demanded:
- For the multiplication of hypercomplex numbers neither the commutative law nor the associative law need to apply.
- The multiplication does not need to be zero divisor-free .
- The multiplication is not invertable generally .
conjugation
hypercomplex one numbers can be represented as follows as sum:
- <math> A = a_01 + a_1 \ mathbf i_1 + \ cdots + a_n \ mathbf i_n< /math>.
The sizes <math> \ mathbf i_k< /math> are called imaginary units. Too <math> the A< /math> conjugated number develops, as all imaginary units are replaced by their negative (<math> \ mathbf i_k \ mapsto - \ mathbf i_k< /math>). Too <math> the A< /math> conjugates complex numberbecomes through <math> \ without A< /math> or <math> a^*< /math> represented. Their sum representation is
- <math> \ without A = a_01 - a_1 \ mathbf i_1 - \ cdots - a_n \ mathbf i_n< /math>.
The conjugation is a Involution on the hypercomplex numbers, i.e. that
- <math> \ bar {\ without A} = A </math>.
examples
the complex numbers
the complex numbers <math> \ mathbb {C}< /math> a hypercomplex number system, through math
- <the z> = A + b are \ mathrm {i}< /math> with <math> \ mathrm {i} ^2 = -1< /math>
is defined.
the binary numbers
the binary numbers are defined by
- z = A + with <math> E^2 = 1< /math>.
the binary numbers
the binary numbers are defined through
- <math> z = A + b \ omega< /math> with <math> \ Omega^2 = 0 </math>.
It notes that they do not have to do anything with dual numbers.
the quaternions
the quaternions (symbol often <math> \ mathbb {H}< /math> after its discoverer W. R. Hamilton) are four-dimensional hypercomplex numbers with division and associative (however not more commutative) multiplication. It concerns with the quaternions thus an inclined body.
the Oktonionen
the Oktonionen (symbol <math> \ mathbb {O}< /math>, also Oktaven mentioned) is eight-dimensional hypercomplex numbers with division and alternating multiplication.
the Sedenionen
the Sedenionen (symbol <math> \ mathbb {S}< /math>) is sixteen-dimensional hypercomplex numbers. Their multiplication is neither commutatively, associative or alternative. Also they do not possess a division; instead they have zero-divisors.
remarks
with the doubling procedure (also as Cayley Dickson procedures well-known) leave themselves new hypercomplex number systems produce, which have twice as many dimensions like the primary number system.
Each Clifford algebra is an associative hypercomplex number system.
number range whole
literature
- I. L. Kantor, A. S. Solodownikow: Hypercomplex numbers. BSG B. G. Teubner publishing house company, Leipzig, 1978.