Number systems

Matrix representations of number systems

Posted on by John

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.

\begin{align*} a + ib &\leftrightarrow \begin{pmatrix} a & -b \\ b & a \end{pmatrix} \\ a + \varepsilon b &\leftrightarrow \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \\ a + jb &\leftrightarrow \begin{pmatrix} a & b \\ b & a \end{pmatrix} \\ \end{align*}

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.

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Euler’s formula for dual numbers and double numbers

Posted on by John

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

\exp(i\theta) = \cos(\theta) + i \sin(\theta)

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

\exp(z) = \sum_{k=0}^\infty \frac{z^k}{k!}

to define exp(z) in these number systems.

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

\exp(\varepsilon x) = 1 + \varepsilon x

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

\exp(j x) = \cosh(x) + j \sinh(x)

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.

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

A magical land where rounding equals truncation

Posted on by John

Rounding numbers has a surprising amount of detail. It may seem trivial but, as with most things, there is a lot more to consider than is immediately obvious. I expect there have been hundreds if not thousands of pages devoted to rounding in IEEE journals.

An example of the complexity of rounding is what William Kahan called The Tablemaker’s Dilemma: there is no way in general to know in advance how accurately you’ll need to compute a number in order to round it correctly.

Rounding can be subtle in any number system, but there is an alternative number system in which it is a little simpler than in base 10. It’s base 3, but with a twist. Instead of using 0, 1, and 2 as “digits”, we use −1, 0, and 1. This is known as the balanced ternary system: ternary because of base 3, and balanced because the digits are symmetrical about 0.

We need a symbol for −1. A common and convenient choice is to use T. Think of moving the minus sign from in front of a 1 to on top of it. Now we could denote the number of hours in a day as 10T0 because

1 \times 3^3 + 0 \times 3^2 + (-1)\times 3 + 0 = 24

A more formal way of a describing balanced ternary representation of a number x is a set of coefficients tk such that

x = \sum_{k=-\infty}^\infty t_k 3^k

with the restriction that each tk is in the set {−1, 0, 1}.

Balanced ternary representation has many interesting properties. For example, positive and negative numbers can all be represented without a minus sign. See, for example, Brain Hayes’ excellent article Third Base. The property we’re interested in here is that to round a balanced ternary number to the nearest integer, you simply lop off the fractional part. Rounding is the same as truncation. To see this, note that the largest possible fractional part is a sequence of all 1s, which represents ½:

\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \cdots = \frac{1}{2}

Similarly, the most negative possible fractional part is a sequence of all Ts, which represents −½. So unless the fractional part is exactly equal to ½, truncating the fractional part rounds to the nearest integer. If the fractional part is exactly ½ then there is no nearest integer but two integers that are equally near.

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Fibonacci number system

Posted on by John

Every positive integer can be written as the sum of distinct Fibonacci numbers. For example, 10 = 8 + 2, the sum of the fifth Fibonacci number and the second.

This decomposition is unique if you impose the extra requirement that consecutive Fibonacci numbers are not allowed. [1] It’s easy to see that the rule against consecutive Fibonacci numbers is necessary for uniqueness. It’s not as easy to see that the rule is sufficient.

Every Fibonacci number is itself the sum of two consecutive Fibonacci numbers—that’s how they’re defined—so clearly there are at least two ways to write a Fibonacci number as the sum of Fibonacci numbers, either just itself or its two predecessors. In the example above, 8 = 5 + 3 and so you could write 10 as 5 + 3 + 2.

The nth Fibonacci number is approximately φn/√5 where φ = 1.618… is the golden ratio. So you could think of a Fibonacci sum representation for x as roughly a base φ representation for √5x.

You can find the Fibonacci representation of a number x using a greedy algorithm: Subtract the largest Fibonacci number from x that you can, then subtract the largest Fibonacci number you can from the remainder, etc.

Programming exercise: How would you implement a function that finds the largest Fibonacci number less than or equal to its input? Once you have this it’s easy to write a program to find Fibonacci representations.

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[1] This is known as Zeckendorf’s theorem, published by E. Zeckendorf in 1972. However, C. G. Lekkerkerker had published the same result 20 years earlier.