# 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

3,6 the Sedenionen

## 4

remarks 5 related topics 6 literature [ work on] with addition and multiplication. One demands the following characteristics:

• The multiplication from hypercomplex numbers is bilinear over the real numbers, therefore applies (ax) (by) = off (XY) for all A, b [itex] \ 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:

• [itex] A = a_01 + a_1 \ mathbf i_1 + \ cdots + a_n \ mathbf i_n< /math>.

The sizes [itex] \ mathbf i_k< /math> are called imaginary units. Too [itex] the A< /math> conjugated number develops, as all imaginary units are replaced by their negative ([itex] \ mathbf i_k \ mapsto - \ mathbf i_k< /math>). Too [itex] the A< /math> conjugates complex numberbecomes through [itex] \ without A< /math> or [itex] a^*< /math> represented. Their sum representation is

• [itex] \ 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

• [itex] \ bar {\ without A} = A [/itex].

## examples

### the complex numbers

the complex numbers [itex] \ mathbb {C}< /math> a hypercomplex number system, through math

• <the z> = A + b are \ mathrm {i}< /math> with [itex] \ mathrm {i} ^2 = -1< /math>

is defined.

### the binary numbers

the binary numbers are defined by

• z = A + with [itex] E^2 = 1< /math>.

### the binary numbers

the binary numbers are defined through

• [itex] z = A + b \ omega< /math> with [itex] \ Omega^2 = 0 [/itex].

It notes that they do not have to do anything with dual numbers.

### the quaternions

the quaternions (symbol often [itex] \ 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 [itex] \ mathbb {O}< /math>, also Oktaven mentioned) is eight-dimensional hypercomplex numbers with division and alternating multiplication.

### the Sedenionen

the Sedenionen (symbol [itex] \ 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.

] used topics

## literature

• I. L. Kantor, A. S. Solodownikow: Hypercomplex numbers. BSG B. G. Teubner publishing house company, Leipzig, 1978.