You can multiply pairs of real numbers using the rules of complex numbers. Complex numbers have all the algebraic structure of the real numbers, i.e. they form a field.

There is a general process, the Cayley-Dickson construction, that let’s you bootstrap multiplication from 1 real number to 2, from 2 to 4, from 4 to 8, etc. You can repeat the process as many times as you like, defining multiplication on lists of 2ⁿ numbers, but you lose structure as you go.

Quaternions

Multiplication for 4-tuples gives the quaternions. The quaternions retain most of the structure of the real and complex numbers. Multiplication is associative. Non-zero elements have a multiplicative inverse, i.e. you can divide. And multiplication plays well with the norm:

|| xy || = || x || · || y ||.

But multiplication is not commutative: in general, xy ≠ yx,

Octonions

Multiplication of 8-tuples produces the octonions . It’s still true that non-zero elements have a multiplicative inverse, and multiplication still plays well with the norm as above. But now, not only is multiplication not commutative, it’s not even associative: in general, (xy)z ≠ x(yz). It’s the “in general” part that this post wants to elaborate on.

The subalgebra generated by any two elements is associative. That means, for example, that (xy)x = x(yx). If you fix x and y, and look at all the octonions you can form by taking adding, multiplying, conjugating, and inverting these elements, as well as multiplying them by a real number, you get a set of octonions for which multiplication is associative.

In fact, the subalgebra generated by two octonions is isomorphic to either the real numbers, the complex numbers, or the quaternions, depending on the two octonions you start with.

This was brought to my attention by a common on a post on octonions from a few years ago. Someone pointed out that an equation I had written

x* = − (x + (e₁ x) e₁ + … + (e₇ x) e₇) / 6

could be written more simply as

x* = − (x + e₁ x e₁ + … + e₇ x e₇) / 6.

because each term only involves two distinct octonions.

Sedenions

The next step, multiplying 16-tuples of real numbers, gives the sedenions [1]. Now we lose even more structure. Multiplication is not commutative, not associative, and it’s possible for two non-zero numbers to have a zero product. That means the norm property

|| xy || = || x || · || y ||

goes out the window since the left size can be zero when the right side is not.

Sedenions, and indeed all Cayley-Dickson algebras, are flexible, which means (xy)x = x(yx). But it’s not true more generally that the algebra generated by two sedenions is associative.

Trigintaduonions

The next rung in the Cayley-Dickson ladder is the family of 32-tuples known as the trigintaduonions [2]. The sedenions are a mess, and they’re a subset of the trigintaduonions, so the trigintaduonions are a mess. But at least they’re flexible.

Summary of properties

More octonion posts

• Error correcting code based on octonions
• Python code for multiplying octonions and friends
• How close are octonions to being associative?

[1] From the Latin word sedecim for 16.

[2] From the Latin triginta for 30 and duo for 2.

The previous post discussed complex numbers, dual numbers, and double numbers. All three systems are constructed by adding some element to the real numbers that has some special algebraic property. The complex numbers are constructed by adding an element i such that i² = −1. The dual numbers add an element ε ≠ 0 with ε² = 0, and the double numbers are constructed by adding j ≠ 1 with j² = 1.

If adding special elements seems somehow illegitimate, there is an alternative way to define these number systems that may seem more concrete using 2 × 2 matrices. (A reader from 150 years ago would probably be more comfortable with appending special numbers than with matrices, but now we’re accustomed to matrices.)

The following mappings provide isomorphisms between complex, dual, and double numbers and their embeddings in the ring of 2 × 2 matrices.

Because the mappings are isomorphisms, you can translate a calculation in one of these number systems into a calculation involving real matrices, then translate the result back to the original number system. This is conceptually interesting, but it could also be useful if you’re using software that supports matrices but does not directly support alternative number systems.

You can also apply the correspondences from right to left. If you need to carry out calculations on matrices of the special forms above, you could move over to complex (or dual, or double) numbers, do your algebra, then convert the result back to matrices.

Functions of a matrix

The previous post looked at variations on Euler’s theorem in complex, dual, and double numbers. You could verify these three theorems by applying exp, sin, cos, sinh, and cosh to matrices. In each case you define the function in terms of its power series and stick in matrices. You should be a little concerned about convergence, but it all works out.

You should also be concerned about commutativity. Multiplication of real numbers is commutative, but multiplication of matrices is not, so you can’t just stick matrices into any equation derived for real numbers and expect it to hold. For example, it’s not true in general that exp(A + B) equals exp(A) exp(B). But it is true if the matrices A and B commute, and the special matrices that represent complex (or dual, or double) numbers do commute.

Related posts

• Quaternion product as a matrix product
• Continued fractions as matrix products
• Fractional linear and linear
• Power method and Fibonacci numbers

The complex numbers are formed by adding an element i to the real numbers such that i² = − 1. We can create other number systems by adding other elements to the reals.

One example is dual numbers. Here we add a number ε ≠ 0 with the property ε² = 0. Dual numbers have been used in numerous applications, most recently in automatic differentiation.

Another example is double numbers [1]. Here we add a number j ≠ ±1 such that j² = 1. (Apologies to electrical engineers and Python programmers. For this post, j is not the imaginary unit from complex numbers.)

(If adding special numbers to the reals makes you uneasy, see the next post for an alternative approach to defining these numbers.)

We can find analogs of Euler’s formula

for dual numbers and double numbers by using the power series for the exponential function

to define exp(z) in these number systems.

For dual numbers, the analog of Euler’s theorem is

because all the terms in the power series after the first two involve powers of ε that evaluate to 0. Although this equation only holds for dual numbers, not real numbers, it is approximately true of ε is a small real number. This is the motivation for using ε as the symbol for the special number added to the reals: Dual numbers can formalize calculations over the reals that are not formally correct.

For double numbers, the analog of Euler’s theorem is

and the proof is entirely analogous to the proof of Euler’s theorem for complex numbers: Write out the power series, then separate the terms involving even exponents from the terms involving odd exponents.

Related posts

• Osborn’s rule
• Relativistic addition
• Relativity and gyrovectors

[1] Double numbers have also been called motors, hyperbolic numbers, split-complex numbers, spacetime numbers, …

Yesterday I wrote about the fact that quaternions, unlike complex numbers, can form conjugates via a series of multiplications and additions. This post will show that you can do something similar with octonions.

If x is an octonion

x = r₀ e₀ + r₁ e₁ + … + r₇ e₇

where all the r‘s are real numbers. The conjugate flips the signs of all the components except the real component:

x* = r₀ e₀ − r₁ e₁ − … − r₇ e₇

The conjugate theorem says

x* = − (x + (e₁ x) e₁ + … + (e₇ x) e₇) / 6

which is analogous to the theorem

q* = − (q + iqi + jqj + kqk) /2

for quaternions.

The internal parentheses are necessary because multiplication in octonions is not associative:

xyz

is ambiguous because it could mean

(xy)z

or

x(yz)

and the two are not necessarily equal.

Update: It is true in general that (xy)z ≠ x(yz) for octonions. However, the subalgebra of octonions generated by any two elements is associative. In particular, (xy)x = x(yz) and so we can write the equation for x* above as simply

x* = − (x + e₁ x e₁ + … + e₇ x e₇) / 6.

Thanks to Martin Erik Horn for pointing this out.

The proof is also analogous to the proof given in the earlier post for quaternions. First work out what the effect is of multiplying on the left and right by one of the imaginary units, then add everything up. You’ll find that the real component is multiplied by -6 and the rest of the components are multiplied by 6.

This post will present a way of multiplying octonions that’s easy to remember.

Please note that there are varying conventions for how to define multiplication for octonions [1].

Octonions

The complex numbers have one imaginary unit i, and the quaternions have three: i, j, and k. The octonions have seven, and so it makes sense to switch over to subscripts rather than venturing further out into the alphabet. Besides, numerical indices will be useful for reasons we’ll see shortly.

Let e₀ = 1 and let e₁ through e₇ be the imaginary units for octonions. Then the e‘s form a basis for ℝ⁸. The octonions are ℝ⁸ with a product that distributes over addition, but the product is not commutative or associative.

Multiplication is determined by how the basis elements multiply, and there are multiple ways of presenting these multiplication rules, the simplest being to write down a multiplication table. Here I’ll present the way I find easiest to remember [2].

Multiplication rules

The squares of each of the imaginary units e₁ through e₇ are equal to −1, as you’d expect based on experience with complex numbers and quaternions. You can derive the rest of the multiplication facts from

e₁ e₂ = e₄

and two simple rules.

The first rule is that the equation above holds when you shift the indices mod 7.

e₁ e₂ = e₄
e₂ e₃ = e₅
e₃ e₄ = e₆
e₄ e₅ = e₇
e₅ e₆ = e₁
e₆ e₇ = e₂
e₇ e₁ = e₃

The second rule is that each of the triplets of units above behave analogous to i, j, and k in quaternions. That is, when you take a triple like (e₂, e₃, e₅), rotations don’t flip the sign but swapping adjacent elements does. That is,

e₂ e₃ = e₅
e₅ e₂ = e₃
e₃ e₅ = e₂

and

e₃ e₂ = −e₅
e₂ e₅ = −e₃
e₅ e₃ = −e₂

So we have seven rotations of the equation

e₁ e₂ = e₄

and six permutations of each rotation, for a total of 42 rules. We already had 7 rules, saying each unit squares to -1, so we have the 49 rules we need to multiply all the non-real units by each other.

More octonion posts

• How close is octonion multiplication to being associative?
• Python code for multiplying octonions and friends

[1] There are multiple conventions for defining octonion multiplication. For example, according to a common definition,

e₁ e₂ = e₃

but according to our definition

e₁ e₂ = e₄

In fact, there are 480 ways to define the multiplication rules for e₁ through e₇. However, all the ways of defining octonion multiplication are isomorphic. You can translate results from one convention to another by renumbering the basis elements. For example, you can convert from the first convention above to our convention by relabeling

(e₁, e₂, e₃, e₄, e₅, e₆, e₇)

as

(e₁, e₂, e₄, – e₇, e₃, e₆, e₅).

I picked the convention used here because it makes the multiplication rules easy to remember: the mod 7 rule doesn’t hold for some other ways of defining multiplication.

[2] There’s a neat way of representing octonion multiplication rules in terms of a Fano plane diagram, but to use it you have to remember how to label the diagram. The presentation in this post is easier to remember, in my opinion. You might remember the rules given here, then use them to fill in a Fano diagram.

Quaternion multiplication is associative but not commutative. An earlier post looked at the average size of the commutator xy − yx as a measure of how far quaternion multiplication is from being commutative.

This post looks at an analogous question for octonions. Octonion multiplication is neither commutative nor associative. So in this post we look at the associator of three octonions (xy)z − x(yz) as a measure of how far octonion multiplication is from being associative.

(A post from yesterday looked at how close octonion multiplication comes to being associative in an algebraic sense, looking at weak forms of associativity. This post looks at how close multiplication comes to being associative in an analytical sense, looking at norm distances rather than algebraic identities.)

We will use a simulation to look at the average norm of the octonion associator over the octonions of unit length, analogous to the earlier post that looked at the commutator of the quaternions. We developed code for octonion multiplication in the previous post and will reuse that code here. We also developed code for generating random unit-length octonions in the same post.

Here’s code to find the average norm of the associator and plot a histogram of its values.

    
    import matplotlib.pyplot as plt

    # omult is octonion multiplication. 
    # See previous post.
    def associator(x, y, z):
        return omult(omult(x, y), z) - 
               omult(x, omult(y, z))

    N = 100000
    s = 0
    h = np.zeros(N)
    for n in range(N):
        x = random_unit_octonion()
        y = random_unit_octonion()
        z = random_unit_octonion()    
        t = norm(associator(x, y, z))
        s += t
        h[n] = t

    print(s/N)
    plt.hist(h, bins=50)
    plt.show()

This gives an average of 1.095 and the histogram below.

Update: Greg Egan calculated the exact mean value for the norm of the associator to be 147456/(42875 π) ≈ 1.0947336. Here are the details.