Duplication process

The doubling method , also known as Cayley - Dickson method , is a method for generating hyper-complex numbers . The new number system has twice as many dimensions as the original system.

The importance of the doubling method lies in the fact that it successively produces the complex numbers , the quaternions , the octonions and the sedenions from the real numbers .

definition

Let be a hypercomplex number and the conjugate hypercomplex number (this is obtained by reversing the signs of the coefficients of the imaginary units in the notation of the number as a linear combination of its units). We now consider pairs over the hypercomplex numbers with the following addition and multiplication: ${\displaystyle a}$ ${\displaystyle a^{*}}$

${\displaystyle (a,b)+(c,d)=(a+c,b+d)}$

${\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*})}$

The order of the factors is important for multiplication, since the commutative law does not have to apply.

The pairs with the addition and multiplication defined in this way again form a system of hyper- complex numbers .

Alternative description

Another description of the doubling process looks like this: Add a new unit to the hypercomplex numbers and now consider sums with the following addition and multiplication ${\displaystyle E}$ ${\displaystyle a+bE}$

${\displaystyle (a+bE)+(c+dE)=(a+c)+(b+d)E}$

${\displaystyle (a+bE)(c+dE)=(ac-d^{*}b)+(da+bc^{*})E}$

In this description it is easy to see that

${\displaystyle E^{2}=-1}$

and that with the imaginary units of the initial system there is an anti-commutation: ${\displaystyle E}$ ${\displaystyle \mathbf {i} _{k}}$

${\displaystyle E\mathbf {i} _{k}=-\mathbf {i} _{k}E}$.

The first steps

From real to complex numbers

If is a real number , is . In addition, the multiplication of the real numbers is commutative . This simplifies the equations to: ${\displaystyle a}$ ${\displaystyle a^{*}=a}$

${\displaystyle (a+bE)+(c+dE)=(a+c)+(b+d)E}$

${\displaystyle (a+bE)(c+dE)=(ac-bd)+(ad+bc)E}$

If you bet, you can recognize the complex numbers . ${\displaystyle E=i}$

From complex numbers to quaternions

Compared to the real numbers, the complex numbers lose the property of being equal to their conjugate number. The multiplication is still commutative. With this we get:

${\displaystyle (a+bE)+(c+dE)=(a+c)+(b+d)E}$

${\displaystyle (a+bE)(c+dE)=(ac-d^{*}b)+(da+bc^{*})E}$

If you place and , you can recognize the quaternions again. The multiplication of the quaternions is no longer commutative , but the associative law still applies. ${\displaystyle E=j}$ ${\displaystyle iE=k}$

From the quaternions to the octonions

From now on you need the formula in its full beauty. With the step towards the octonions, the associative law of multiplication is also lost. After all, the octonions form an alternative body .

And further

If you double the octonions , you get the sedions . The Sedenions lose the property of being a division algebra and the alternative of multiplication is also lost. The sedions are now only potency-associative . This property is not lost even if the doubling process is continued.

literature

• IL Kantor, AS Solodownikow: Hypercomplex numbers. BSG BG Teubner Verlagsgesellschaft, Leipzig, 1978.