Doubling procedure

the doubling procedure, also as Cayley - Dickson - procedure admits, is a procedure for the production of hypercomplex numbers. The new number system has thereby doubles as many dimension like the output system.

The meaning of the doubling procedure lies in the fact that it from the real numbers successivelythe complex numbers, the quaternions, which Oktonionen and the Sedenionen bring out.

Table of contents

the Oktonionen 3,4 and

, definition

is <math> A< /math> a hypercomplex number and <math> a^*< /math> complex number conjugates. Weregard now pairs over the hypercomplex numbers with following addition and multiplication

< math> (A, b) + (C, D) = (a+c, b+d)< /math>

<math> (A, b) (C, D) = (AC - d^*b, there + bc^*)< /math> 

With the multiplication the sequence of the factors is importantly, there the commutative lawnot to apply needs.

The pairs with in such a way defined addition and multiplication form system of hypercomplex numbers.

alternative description

another description of the doubling procedure looks in such a way: Add to the hypercomplexPay a new unit <math> E< /math> in addition and regard now sums <math> A + </math> with following addition and multiplication

< math> (A +) + (C + dE) = (A + C) + (b + D) E< /math>

<math> (A +) (C + dE) = (AC - d^*b) + (there + bc^*) E< /math>

In this description one sees easy that

  • <math> E^2 = -1< /math>

and that <math> E< /math> with the imaginary units <math> \ mathbf i_k< /math> of theOutput system anti- kommutiert:

  • <math> E \ mathbf i_k = - \ mathbf i_k E< /math>.

the first steps

of the real to the complex numbers

if <math> A< /math> a real number is, is <math> a^* = A< /math>. In addition is the multiplication the real numbers commutatively. Thus the equations are simplified too:

<math> (A +) + (C + dE) = (A + C) + (b + D) E< /math>

<math> (A +) (C + dE) = (AC - bd) + (ad + UC) E< /math>

One sets <math> E=i< /math> one recognizes the complex numbers again.

from the complex numbers to the quaternions

the complex numbers lose the characteristic compared with the real numbers, to theirconjugated number to be alike. The multiplication is further commutative. Thus we receive:

<math> (A +) + (C + dE) = (A + C) + (b + D) E< /math>

<math> (A +) (C + dE) = (AC - d^*b) + (there + bc^*) E< /math>

One sets <math> E=j< /math> and <math> iE=k< /math> one recognizes the quaternions again.

The multiplication of the quaternions is not commutative any longer, but the associative law applies further.

from the quaternions to the Oktonionen

from now on one needs the formula in its full beauty. With the step to the Oktonionen goes also still the associative law of the multiplication lost. The Oktonionen nevertheless forms an alternative body.


one continues to double the Oktonionen, then one receives the Sedenionen. The Sedenionen loses the characteristic to be a division algebra and also the Alternativität of the multiplication is lost. The Sedenionen are power associative. This characteristic is not lost even with further application of the doubling procedure.


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

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