Cayley–Dickson construction

The doubling process, also known as Cayley Dickson method, is a method for producing hypercomplex numbers. The new numbering system has twice as many dimensions as the original system.

The importance of the doubling process is that it brings out the real numbers one after the complex numbers, the quaternions, the octonions and the Sedenionen.

Definition

Be a hyper complex number and the complex conjugate. We now consider pairs over the hypercomplex numbers with the following addition and multiplication

For multiplication, the order of the factors is important because the commutative need not apply.

The couple with the addition and multiplication as defined make up a system hyper complex numbers.

Alternative description

Another description of the doubling procedure is as follows: Add to the hypercomplex numbers a new unit added and now consider sums with the following addition and multiplication

In this description, it is easily seen that

And that with the imaginary units of the initial system anti- commutes:

The first steps

From the real to the complex numbers

If a real number is. Moreover, the multiplication of real numbers is commutative. Thus, the equations simplify to:

Substituting, we recognize the complex numbers again.

From the complex numbers to quaternions

The complex numbers loose in comparison with the real numbers, the ability to be equal to its conjugate number. The multiplication is also commutative. Thus we obtain:

Substituting and, one recognizes the quaternions again. The multiplication of quaternions is not commutative more, but the associative law still applies.

From quaternions to octonions

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

And more

Doubling the octonions, then you get the Sedenionen. Sedenionen losing the property of being a division algebra and the alternativeness multiplication is lost. The Sedenionen are only a potency - associative. This property is not lost upon further application of the doubling process.