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The conjugate construction
of the quaternion and octonion algebras, etc.

Prerequisites

To understand this, you need a grounding in the complex numbers, as may be had from a high school second-year algebra course.

Introduction

The conjugate construction produces a sequence of higher-dimensional algebras that are like numbers in that they have a norm and a multiplicative inverse. The crux of this construction is the “conjugate” of an element, whose product with the element is the square of the norm of the element.

One surprise is that for the first several steps, besides having a higher dimensionality, the next algebra in the sequence loses a specific algebraic property of the previous algebras.

The construction starts with the real numbers, and proceeds to construct the complex numbers, the quaternions, the octonions, the sedenions, and a limitless chain of other “hypercomplex” algebras. The first few are quite interesting, but due to the loss of algebraic properties, all but these first are considered to be pretty boring algebras.

Complex numbers as ordered pairs

The complex numbers can be written as ordered pairs (a, b) of real numbers a and b, with multiplication defined by

A complex number whose second component is zero is associated with a real number: the complex number (a, 0) is the real number a.

Another important operation on complex numbers is conjugation. The conjugate (a, b)^* of (a, b) is given by

The conjugate has the property that

which is a non-negative number. In this way, conjugation provides the norm: for a complex number z,

Furthermore, for any nonzero complex number z, conjugation gives a multiplicative inverse,

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.

Another step: the quaternions

The next step in the construction is to generalize the multiplication and conjugation operations. This is easy, if not quite obvious.

Form ordered pairs (a, b) of complex numbers a and b, with multiplication defined by

There is more than one way to arrange the conjugates so that everything will work out, and the choice is arbitrary, but will later affect the terminology about “handedness”. The order of the factors seems peculiar now, but will be important in the next step.

Define the conjugate (a, b)^* of (a, b) by

These operators are direct extensions of their complex analogs: if a and b are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

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

As before, the conjugate thus yields a norm and an inverse for all such ordered pairs. So in the sense explained above, these pairs constitute an algebra that shares many properties with the real numbers. They are the quaternions, invented and named by Hamilton in 1843.

Inasmuch as quaternions consist of a pair of complex numbers, they form a 4-dimensional vector space with real scalar multiplication.

The multiplication of quaternions isn’t quite like familiar multiplication of real numbers, though. It is not commutative, that is, for quaternions p and q, it isn’t generally true that pq = qp.

Yet another step: the octonions

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

This time, form ordered pairs (p, q) of quaternions p and q, with multiplication and conjugation defined exactly as for the quaternions.

For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.

So this set can be thought of as a kind of algebra. It was discovered by Graves in 1844, and is called the octonions or the Cayley numbers.

Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were b c rather than c b, the formula for the conjugate wouldn’t yield a real number.

Inasmuch as octonions consist of a pair of quaternions, they form an 8-dimensional vector space with with real scalars.

The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it isn’t associative, that is, if p, q, and r are octonions, it isn’t generally true that (pq)r = p(qr).

And so forth

The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if s is a sedenion, sⁿ s^m = s^n+m, but loses the property of being an alternative algebra, meaning that if s and t are sedenions, the rules s²t = s(st) and st² = (st)t do not always hold.

The conjugate construction can be carried ad infinitum, at each step producing an algebra whose dimension is double that of the algebra of the preceding step. The algebras beyond the complex numbers go by the generic name hypercomplex numbers.

After the octonions, though, the algebras even lose the property of being division algebras, that is, if p and q are elements of one of these algebras, then pq = 0 no longer implies p = 0 or q = 0.

See also