M.E. Irizarry-Gelpí

M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Beyond Complex Numbers

Complex numbers are very useful in pure mathematics and also in science. In particular, they are essential for quantum physics. For the last few months I have been learning a lot about many different ways to generalize the complex numbers. In this post I would like to gather some of this information for future reference.

A complex number \(z\) consist of a real part \(a\) and an imaginary part \(b\). You write this as

$$ z = a + bi, $$

where the imaginary unit \(i\) satisfies the equation

$$ i^{2} = -1. $$

In general, \(a\) and \(b\) are real numbers. For a complex number you have a conjugation operation \(\operatorname{Conj}\) that changes the sign of the imaginary part:

$$ \operatorname{Conj}{(a + bi)} = a - bi. $$

Clearly, \(\operatorname{Conj}\) is an involution. You also have the quadrance of a complex number:

$$ \operatorname{Quad}{(a + bi)} = (a + bi) * \operatorname{Conj}{(a + bi)} = a^{2} + b^{2}. $$

Unless \(a = b = 0\), the quadrance of a complex number is always positive.

Perplex Numbers

Instead of working with the imaginary unit \(i\), if you use a unit \(s\) satisfying

$$ s^{2} = 1, $$

then \(a + bs\) describes a perplex number. Note that \(s \neq 1\). These numbers are also known as split-complex numbers, and also as double numbers. I will use "perplex" due to its symmetry with "complex". Conjugation and the quadrance are defined in the same way, but now you have:

$$ \operatorname{Quad}{(a + bs)} = (a + bs) * \operatorname{Conj}{(a + bs)} = a^{2} - b^{2}. $$

Thus, if \(a^{2} > b^{2}\), the quadrance is positive. But if \(a^{2} < b^{2}\) then the quadrance will be negative. Furthermore, if \(a^{2} = b^{2}\), then the quadrance is zero with \(a\) and \(b\) not necessarily being zero. This is very different from the complex numbers. Put more formally, the perplex numbers have zero divisors. From the point of view of pure mathematics, this is a bit troublesome. But from the point of view of theoretical physics, zero divisors correspond to light-like vectors which are physical.

Another departure from the complex numbers is the existence of non-trivial idempotents:

$$ Z = \frac{1}{2} \left( 1 + s \right), \qquad W = \frac{1}{2} \left( 1 - s \right). $$

That is, \(Z^{2} = Z\), and \(W^{2} = W\).

Nilplex Numbers

Another possibility is to have a unit \(\alpha\) with

$$ \alpha^{2} = 0, $$

but \(\alpha \neq 0\). Then \(a + b\alpha\) is a nilplex number. These numbers are more commonly known as dual numbers, but I will use "nilplex" due to its symmetry with "complex" and "perplex". Again, you can define conjugation and the quadrance in the same way as for complex and perplex numbers, but now you have:

$$ \operatorname{Quad}{(a + b\alpha)} = (a + b\alpha) * \operatorname{Conj}{(a + b\alpha)} = a^{2}. $$

The quadrance is always non-negative, but note that if \(a = 0\), then for any \(b\) that nilplex number will have zero quadrance. Thus, nilplex numbers also have zero divisors.

Cayley-Dickson Constructs

One convenient way to work with complex numbers is to represent them as a pair of real numbers:

$$ z = \begin{pmatrix} a & b \end{pmatrix}. $$

Here \(a\) is said to be the left component of \(z\), and \(b\) is said to be the right component of \(z\). Let

$$ z = \begin{pmatrix} a & b \end{pmatrix}, \qquad w = \begin{pmatrix} c & d \end{pmatrix}. $$

Then the addition operation is

$$ \operatorname{Add}{(z, w)} = \begin{pmatrix} a + c & b + d \end{pmatrix}.$$

That is, component-wise. Similarly, you also have the subtraction operation:

$$ \operatorname{Sub}{(z, w)} = \begin{pmatrix} a - c & b - d \end{pmatrix}.$$

Another helpful operation is multiplication by a scalar \(k\):

$$ \operatorname{Scal}{(z, k)} = \begin{pmatrix} k a & k b \end{pmatrix}. $$

A special case of this operation is the negation operation:

$$ \operatorname{Neg}{(z)} = \begin{pmatrix} -a & -b \end{pmatrix}. $$

The conjugation operation is

$$ \operatorname{Conj}{(z)} = \begin{pmatrix} a & -b \end{pmatrix}. $$

Finally, you have the multiplication operation of two complex numbers:

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} ac - db & da + bc \end{pmatrix}. $$

Note that this multiplication operation is commutative. That is,

$$ \operatorname{Mul}{(z, w)} = \operatorname{Mul}{(w, z)}. $$

Indeed, the ordering above in the definition of \(\operatorname{Mul}\) is a little weird since multiplication of real numbers is also commutative, but this will make sense later.

So far we have learned two things. First, that besides the complex numbers, you can also have two other kinds of two-dimensional number systems: perplex and nilplex numbers. Second, that complex numbers and operations on them can be neatly represented in terms of pairs of real numbers. It should not be a surprise to learn that perplex and nilplex numbers can also be represented as pairs of real numbers. Most operations remain unchanged, except for the multiplication operation. The multiplication operation for perplex numbers is:

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} ac + db & da + bc \end{pmatrix}. $$

I will refer to this operation as the hyperbolic product. Similarly, the multiplication operation for nilplex numbers is:

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} ac & da + bc \end{pmatrix}. $$

I will refer to this operation as the parabolic product. For completeness, the multiplication operation for complex numbers is referred to as the elliptic product.

There is a neat trick to construct other number systems from a given number system. It is called the Cayley-Dickson construction. The way it works is as follows. You start with a seed number system \(\mathcal{S}\) along with a list of operations on the elements of \(\mathcal{S}\):

  • The binary addition operation \(\operatorname{add}\),
  • the binary subtraction operation \(\operatorname{sub}\),
  • the unary dilation operation \(\operatorname{scal}\),
  • the unary negation operation \(\operatorname{neg}\),
  • the unary conjugation operation \(\operatorname{conj}\),
  • the binary multiplication operation \(\operatorname{mul}\),
  • the unary quadrance operation \(\operatorname{quad}\),
  • the unary inversion operation \(\operatorname{inv}\), and
  • the binary division operation \(\operatorname{quo}\).

Consider a pair of elements \(a\) and \(b\) in \(\mathcal{S}\). With this pair, you can make an element \(z\) in the Cayley-Dickson construct number system \(\mathcal{C}\):

$$ z = \begin{pmatrix} a & b \end{pmatrix}. $$

The operations on the elements in \(\mathcal{C}\) are given in terms of the operations on the elements in \(\mathcal{S}\). Let \(z\) and \(w\) be elements in \(\mathcal{C}\) with

$$ z = \begin{pmatrix} a & b \end{pmatrix}, \qquad w = \begin{pmatrix} c & d \end{pmatrix}. $$

Then the binary addition operation is

$$ \operatorname{Add}{(z, w)} = \begin{pmatrix} \operatorname{add}{(a, c)} & \operatorname{add}{(b, d)} \end{pmatrix}; $$

the binary subtraction operation is

$$ \operatorname{Sub}{(z, w)} = \begin{pmatrix} \operatorname{sub}{(a, c)} & \operatorname{sub}{(b, d)} \end{pmatrix}; $$

the unary dilation operation is

$$ \operatorname{Scal}{(z, k)} = \begin{pmatrix} \operatorname{scal}{(a, k)} & \operatorname{scal}{(b, k)} \end{pmatrix}; $$

the unary negation operation is

$$ \operatorname{Neg}{(z)} = \begin{pmatrix} \operatorname{neg}{(a)} & \operatorname{neg}{(b)} \end{pmatrix}; $$

and the unary conjugation operation is

$$ \operatorname{Conj}{(z)} = \begin{pmatrix} \operatorname{conj}{(a)} & \operatorname{neg}{(b)} \end{pmatrix}. $$

The other operations depend on the which product you use. Since there are three kinds of multiplication operations, you can have three kinds of Cayley-Dickson constructs: elliptic, parabolic, or hyperbolic. For an elliptic Cayley-Dickson construct, the multiplication operation is

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} \operatorname{sub}{( \operatorname{mul}(a, c), \operatorname{mul}(\operatorname{conj}{(d)}, b))} & \operatorname{add}{( \operatorname{mul}(d, a), \operatorname{mul}(b, \operatorname{conj}{(c)}))} \end{pmatrix}, $$

and the quadrance operation is

$$ \operatorname{Quad}{(z)} = \operatorname{quad}{(a)} + \operatorname{quad}{(b)} . $$

Note that the quadrance is always real but not necessarily non-negative (it depends on the properties of \(\operatorname{quad}\)). Similarly, for a parabolic Cayley-Dickson construct, the multiplication operation is

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} \operatorname{mul}(a, c) & \operatorname{add}{( \operatorname{mul}(d, a), \operatorname{mul}(b, \operatorname{conj}{(c)}))} \end{pmatrix}, $$

and the quadrance operation is

$$ \operatorname{Quad}{(z)} = \operatorname{quad}{(a)}. $$

Finally, for a hyperbolic Cayley-Dickson construct the multiplication operation is

$$ \operatorname{Mul}{(z, w)} = \begin{pmatrix} \operatorname{add}{( \operatorname{mul}(a, c), \operatorname{mul}(\operatorname{conj}{(d)}, b))} & \operatorname{add}{( \operatorname{mul}(d, a), \operatorname{mul}(b, \operatorname{conj}{(c)}))} \end{pmatrix}, $$

and the quadrance operation is

$$ \operatorname{Quad}{(z)} = \operatorname{quad}{(a)} - \operatorname{quad}{(b)} . $$

The inversion operation can always be written in terms of the conjugation operation, the dilation operation, and the quadrance. With the inversion operation, and the multiplication operation, you can then define the division operation.

In a way, the complex, nilplex, and perplex numbers are elliptic, parabolic, and hyperbolic Cayley-Dickson constructs with the real numbers as the seed number system. In all these cases you go from a one-dimensional number system (the real numbers) to a two-dimensional number system. Repeating the Cayley-Dickson construction with any of these two-dimensional constructs yields a four-dimensional construct. Note that the multiplication operation for each of the two-dimensional constructs is commutative and associative.

Four-Dimensional Constructs

There are five distinct four-dimensional Cayley-Dickson constructs. All of them have something in common: the resulting multiplication operation is noncommutative but still associative. It is convenient to introduce the commutator operation:

$$ \operatorname{Commutator}{(z, w)} = \operatorname{Sub}{(\operatorname{Mul}{(z, w)}, \operatorname{Mul}{(w, z)})}, $$

and the anticommutator operation:

$$ \operatorname{AntiCommutator}{(z, w)} = \operatorname{Add}{(\operatorname{Mul}{(z, w)}, \operatorname{Mul}{(w, z)})}. $$

You also have to distinguish between two division operations: left division (multiplication on the left by the inverse) and right division (multiplication on the right by the inverse).

Hamilton Quaternions

The most familiar four-dimensional construct is the Hamilton quaternion. This is nothing more than the traditional quaternion. This number system corresponds to an elliptic Cayley-Dickson construct with complex numbers as seed. There are three elliptic units, denoted \(i\), \(j\), and \(k\). These units satisfy the relations

$$ \operatorname{Mul}{(i, i)} = -1, \qquad \operatorname{Mul}{(j, j)} = -1, \qquad \operatorname{Mul}{(k, k)} = -1; $$
$$ \operatorname{Commutator}{(i, j)} = 2k, \qquad \operatorname{Commutator}{(j, k)} = 2i, \qquad \operatorname{Commutator}{(k, i)} = 2j; $$
$$ \operatorname{AntiCommutator}{(i, j)} = 0, \qquad \operatorname{AntiCommutator}{(j, k)} = 0, \qquad \operatorname{AntiCommutator}{(k, i)} = 0. $$

Hamilton quaternions are useful in four- and three-dimensional Euclidean space, since

$$ \operatorname{Quad}{(a + bi + cj + dk)} = a^{2} + b^{2} + c^{2} + d^{2}. $$

Note that the quadrance is always non-negative.

Cockle Quaternions

Performing a hyperbolic Cayley-Dickson construct with complex numbers as seed yields the Cockle quaternions. This number system is more commonly known as the split-quaternions. There is one elliptic unit, denoted \(i\), and two hyperbolic units, denoted \(t\) and \(u\). These units satisfy the relations

$$ \operatorname{Mul}{(i, i)} = -1, \qquad \operatorname{Mul}{(t, t)} = +1, \qquad \operatorname{Mul}{(u, u)} = +1; $$
$$ \operatorname{Commutator}{(i, t)} = 2u, \qquad \operatorname{Commutator}{(u, t)} = 2i, \qquad \operatorname{Commutator}{(u, i)} = 2t; $$
$$ \operatorname{AntiCommutator}{(i, t)} = 0, \qquad \operatorname{AntiCommutator}{(t, u)} = 0, \qquad \operatorname{AntiCommutator}{(u, i)} = 0. $$

You can also obtain the Cockle quaternions from an elliptic Cayley-Dickson construct with perplex numbers as seed, or from a hyperbolic Cayley-Dickson construct with perplex numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + ct + du)} = a^{2} + b^{2} - c^{2} - d^{2}. $$

This can be either positive, negative, or zero. Indeed, if the quadrance vanishes, then that element is a zero divisor.

Infra-Complex Numbers

Performing a parabolic Cayley-Dickson construct with complex numbers as seed yields the infra-complex numbers. There is one elliptic unit, denoted by \(i\), and two parabolic units, denoted \(\beta\) and \(\gamma\). These units satisfy the relations

$$ \operatorname{Mul}{(i, i)} = -1, \qquad \operatorname{Mul}{(\beta, \beta)} = 0, \qquad \operatorname{Mul}{(\gamma, \gamma)} = 0; $$
$$ \operatorname{Commutator}{(i, \beta)} = 2\gamma, \qquad \operatorname{Commutator}{(\beta, \gamma)} = 0, \qquad \operatorname{Commutator}{(\gamma, i)} = 2\beta; $$
$$ \operatorname{AntiCommutator}{(i, \beta)} = 0, \qquad \operatorname{AntiCommutator}{(\beta, \gamma)} = 0, \qquad \operatorname{AntiCommutator}{(\gamma, i)} = 0. $$

You can also obtain the infra-complex numbers from an elliptic Cayley-Dickson construct with nilplex numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + c\beta + d\gamma)} = a^{2} + b^{2}. $$

This is always non-negative, but there is a set of non-trivial elements that have zero-quadrance, so this system also has zero divisors.

Infra-Perplex Numbers

Performing a parabolic Cayley-Dickson construct with perplex numbers as seed yields the infra-perplex numbers. There is one hyperbolic unit, denoted by \(s\), and two parabolic units, denoted \(\tau\) and \(\upsilon\). These units satisfy the relations

$$ \operatorname{Mul}{(s, s)} = +1, \qquad \operatorname{Mul}{(\tau, \tau)} = 0, \qquad \operatorname{Mul}{(\upsilon, \upsilon)} = 0; $$
$$ \operatorname{Commutator}{(s, \tau)} = 2\upsilon, \qquad \operatorname{Commutator}{(\tau, \upsilon)} = 0, \qquad \operatorname{Commutator}{(s, \upsilon)} = 2\tau; $$
$$ \operatorname{AntiCommutator}{(s, \tau)} = 0, \qquad \operatorname{AntiCommutator}{(\tau, \upsilon)} = 0, \qquad \operatorname{AntiCommutator}{(\upsilon, s)} = 0. $$

You can also obtain the infra-perplex numbers from a hyperbolic Cayley-Dickson construct with nilplex numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bs + c\tau + d\upsilon)} = a^{2} - b^{2}. $$

This can be either positive, negative or zero, so this system also has zero divisors.

Supra-Real Numbers

Performing a parabolic Cayley-Dickson construct with nilplex numbers as seed yields the supra-real numbers. There are three parabolic units, denoted by \(\alpha\), \(\beta\) and \(\gamma\). These units satisfy the relations

$$ \operatorname{Mul}{(\alpha, \alpha)} = 0, \qquad \operatorname{Mul}{(\beta, \beta)} = 0, \qquad \operatorname{Mul}{(\gamma, \gamma)} = 0; $$
$$ \operatorname{Commutator}{(\alpha, \beta)} = 2\gamma, \qquad \operatorname{Commutator}{(\beta, \gamma)} = 0, \qquad \operatorname{Commutator}{(\gamma, \alpha)} = 0; $$
$$ \operatorname{AntiCommutator}{(\alpha, \beta)} = 0, \qquad \operatorname{AntiCommutator}{(\beta, \gamma)} = 0, \qquad \operatorname{AntiCommutator}{(\gamma, \alpha)} = 0. $$

The supra-real numbers are the parabolic analog of the Hamilton quaternions.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + b\alpha + c\beta + d\gamma)} = a^{2}. $$

This is always non-negative, but there is a set of non-trivial elements that have zero-quadrance, so this system also has zero divisors.

Eight-Dimensional Constructs

There are seven distinct eight-dimensional Cayley-Dickson constructs. All of them have something in common: the resulting multiplication operation is noncommutative and also nonassociative.

Cayley Octonions

The most familiar eight-dimensional construct is the Cayley octonion. This is nothing more than the traditional octonion. This number system corresponds to an elliptic Cayley-Dickson construct with Hamilton quaternions as seed. There are seven elliptic units, denoted \(i\), \(j\), \(k\), \(m\), \(n\), \(p\), and \(q\).

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + cj + dk + em + fn + gp + hq)} = a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} + g^{2} + h^{2}. $$

This is always non-negative.

Zorn Octonions

Performing a hyperbolic Cayley-Dickson construct with Hamilton quaternions as seed yields the Zorn octonions. This number system is more commonly known as the split-octonions. There are three elliptic units, denoted \(i\), \(j\), and \(k\); and four hyperbolic units, denoted \(r\), \(s\), \(t\), and \(u\).

You can also obtain the Zorn octonions from an elliptic Cayley-Dickson construct with Cockle quaternions as seed, or from a hyperbolic Cayley-Dickson construct with Cockle quaternions as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + cj + dk + er + fs + gt + hu)} = a^{2} + b^{2} + c^{2} + d^{2} - e^{2} - f^{2} - g^{2} - h^{2}. $$

This can be either positive, negative, or zero, so there are zero divisors.

Infra-Hamilton Quaternions

Performing a parabolic Cayley-Dickson construct with Hamilton quaternions as seed yields the infra-Hamilton quaternions. There are three elliptic units, denoted \(i\), \(j\), and \(k\); and four parabolic units, denoted \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\).

You can also obtain the infra-Hamilton quaternions from an elliptic Cayley-Dickson construct with infra-complex numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + cj + dk + e\alpha + f\beta + g\gamma + h\delta)} = a^{2} + b^{2} + c^{2} + d^{2}. $$

This is always non-negative, but there are non-trivial elements with zero quadrance, so there are zero divisors.

Infra-Cockle Quaternions

Performing a parabolic Cayley-Dickson construct with Cockle quaternions as seed yields the infra-Cockle quaternions. There is one elliptic unit, denoted by \(i\); two hyperbolic units, denoted by \(t\) and \(u\); and four parabolic units, denoted by \(\rho\), \(\sigma\), \(\tau\), and \(\upsilon\).

You can also obtain the infra-Cockle quaternions from a hyperbolic Cayley-Dickson construct with infra-complex numbers as seed, or an elliptic Cayley-Dickson construct with infra-perplex numbers as seed, or a hyperbolic Cayley-Dickson construct with infra-perplex numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + ct + du + e\rho + f\sigma + g\tau + h\upsilon)} = a^{2} + b^{2} - c^{2} - d^{2}. $$

This can be either positive, negative, or zero, so there are zero divisors.

Supra-Complex Numbers

Performing a parabolic Cayley-Dickson construct with infra-complex numbers as seed yields the supra-complex numbers. There is one elliptic unit, denoted by \(i\), and six parabolic units, denoted \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\epsilon\), and \(\zeta\).

You can also obtain the supra-complex numbers from an elliptic Cayley-Dickson construct with supra-real numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bi + c\alpha + d\beta + e\gamma + f\delta + g\epsilon + h\zeta)} = a^{2} + b^{2}. $$

This is always non-negative, but there are non-trivial elements with zero quadrance, so there are zero divisors.

Supra-Perplex Numbers

Performing a parabolic Cayley-Dickson construct with infra-perplex numbers as seed yields the supra-perplex numbers. There is one hyperbolic unit, denoted by \(s\), and six parabolic units, denoted \(\rho\), \(\sigma\), \(\tau\), \(\upsilon\), \(\phi\), and \(\psi\).

You can also obtain the supra-perplex numbers from an hyperbolic Cayley-Dickson construct with supra-real numbers as seed.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + bs + c\rho + d\sigma + e\tau + f\upsilon + g\phi + h\psi)} = a^{2} - b^{2}. $$

This can be either positive, negative, or zero, so there are zero divisors.

Ultra-Real Numbers

Performing a parabolic Cayley-Dickson construct with supra-real numbers as seed yields the ultra-real numbers. There seven parabolic units, denoted \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\epsilon\), \(\zeta\), and \(\eta\).

The ultra-real numbers are the parabolic analog of the Cayley octonions.

The quadrance of a general element is

$$ \operatorname{Quad}{(a + b\alpha + c\beta + d\gamma + e\delta + f\epsilon + g\zeta + h\eta)} = a^{2}. $$

This is always non-negative, but there are non-trivial elements with zero quadrance, so there are zero divisors.

Sedenions and Beyond

This process can be continued ad-infinitum. Performing a Cayley-Dickson construct with any of the eight-dimensional constructs will yield one of nine distinct sixteen-dimensional constructs:

  • Elliptic sedenions
  • Hyperbolic sedenions
  • Parabolic sedenions
  • Ultra-Complex numbers
  • Ultra-Perplex numbers
  • Supra-Hamilton quaternions
  • Supra-Cockle quaternions
  • Infra-Cayley octonions
  • Infra-Zorn octonions

The most familiar of these are the elliptic sedenions, which are infamous for containing zero divisors, even though they are the sixteen-dimensional analog of the eight-dimensional Cayley octonion and the four-dimensional Hamilton quaternion.

Implementations

I have implemented all two-, four-, and eight-dimensional number systems mentioned in this post in a Go package called rational. Rational numbers were used instead of real numbers due to the fact that arithmetic with floating-point numbers is not exact. However, in the near future I hope to implement types for integers and floating-point numbers. I would also like to re-implement these number systems in Python, C++, and Rust. I feel that this project is a great opportunity to learn more about these languages.