These algebras all have a notion of norm and conjugate, with the general idea being that the product of an element and its conjugate should equal the square of its norm.

The surprise is that for the first several steps, besides having a higher dimensionality, the next algebra loses a specific algebraic property.

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2 Another step: the quaternions 3 Yet another step: the octonions 4 And so forth 5 External links |

The complex numbers can be written as ordered pairs of real numbers and , with the addition operator being component-by-component and with multiplication defined by

Another important operation on complex numbers is conjugation. The conjugate of is given by

Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.

Form ordered pairs of complex numbers and , with multiplication defined by

The order of the factors seems odd now, but will be important in the next step. Define the conjugate of by

The product of an element with its conjugate is a non-negative number:

Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space.

The multiplication of quaternions isn't quite like the multiplication of real numbers, though. It isn't commutative, that is, if and are quaternions, it isn't generally true that .

From now on, all the steps will look the same.

This algebra was discovered by Graves in 1844, and is called the octonions or the "Cayley numbers".

Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space.

The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it isn't associative, that is, if , , and are octonions, it isn't generally true that .

The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if is a sedenion, , but loses the property of being an alternative algebra.

The Cayley-Dickson construction can be carried on *ad infinitum*, at each step producing an algebra whose dimension is double that of algebra of the preceding step.

After the octonions, though, the algebras even contain zero divisors, that is, if and are elements of one of these algebras, then no longer implies or .