The Root of Infinity: It’s Surreal!

Can we make any sense of quantities like “the square root of infinity”? Using the framework of surreal numbers, we can.

In Part 1, we develop the background for constructing the surreals.
In Part 2, the surreals are assembled and their amazing properties described.

Part 1: Brunswick Schnitzel

The number system has been built up by a series of extensions, necessary to complete, or close, the system under an increasing range of operations.

Extending the Number System

We start with the counting numbers ${\{1, 2, 3, \dots\ \}}$ , also called the natural numbers, denoted N. They are closed under addition: add any two and we get another, e.g., ${3 + 5 = 8}$ . But they are not closed under subtraction: ${8 - 5 = 3}$ , but ${5 - 8}$ is a different animal. To close the system, we extend N to include zero and the negative whole numbers. This yields the integers, denoted Z. This bigger set is closed under addition and subtraction.

Z is also closed under multiplication: ${3 \times 5 = 15}$ , but not under division: ${3\div5}$ is not an integer. We must extend the set again to include all fractions, or ratios of integers. The result is the set of rational numbers, Q, which is closed under all four arithmetic operations: addition, subtraction, multiplication and division. The only restriction is that we cannot divide by zero; this operation is not defined.

So far, so good, but we’re not done yet. The square of a rational is another rational: ${\frac{2}{9}\times\frac{2}{9} = \frac{4}{81}}$ . But the square root is not: ${\sqrt{2/9} = \frac{1}{3}\sqrt{2}}$ . As any introduction to the number system shows, ${\sqrt{2}}$ is not a ratio of integers. This was a source of great anxiety to the Pythagoreans, who aimed to express all quantities in terms of whole numbers.

The way forward is to extend the system once more, to include the solutions of all polynomials with rational coefficients. For example, the solutions of ${x^2 - x - 1 = 0}$ are the golden ratio ${\phi = (1+\sqrt{5})/2}$ and ${\psi = -1/\phi\,,}$ both of which are irrational.

We have now reached the set of algebraic numbers, closed under the four basic operations and also under the action of taking (integer and fractional) powers. But further extension is required. Numbers like ${e}$ and ${\pi}$ are not algebraic (i.e., not the solutions of a polynomial with rational coefficients), but transcendental. When they are included, we reach the set R of real numbers.

The real numbers form what is technically called a complete, ordered field. We can think of the real numbers as corresponding to the points on a line, all in order from left to right. However, while this image is very helpful, it is not without problems: are there “enough” points on the line? Are there any “gaps” without any real number assigned to them? We cannot answer unequivocally.

Dedekind Cuts

The integers and rationals can be constructed by a finite sequence of processes. For the real numbers, things are trickier. The German mathematician Richard Dedekind, a student of Gauss, had a bright idea, now called a Dedekind cut (German Schnitt): any real number divides the rationals into two sets, ${L}$ containing all rationals less than the number and ${R}$ containing all rationals greater than or equal to it. Thus, for example ${\sqrt{2}}$ corresponds to ${(L,R)}$ with

$\displaystyle L = \{x\in \mathbf{Q}: x<0\ \mbox{or}\ x^2<2 \} \,, \qquad R = \{x\in \mathbf{Q}: x>0\ \mbox{and}\ x^2\ge2 \}$

In 1874, Dedekind met Georg Cantor whilst on holiday in Interlaken. Cantor thought that cuts were a “spiffing idea” (I imagine him describing them as a wonderschönes geistesblitz); and, reciprocally, Dedekind became a great admirer of Cantor’s work on infinite sets.

Cantor greatly extended the number system, introducing an unlimited range of new, infinite numbers of two species, cardinals and ordinals. The first ordinal greater than all the natural numbers was written as ${\omega}$ . But the extended system was hugely controversial, and for good reasons. For example, Cantor found that ${1 + \omega \ne \omega +1}$ . His system was not commutative under addition. In fact, ${1+\omega = \omega}$ . We might be tempted to subtract ${\omega}$ from each side, but this would yield the nonsensical result ${1 =0}$ . The controversy was a factor in Cantor’s mental breakdown and ultimate suicide.

The situation was very unsatisfactory: it is not possible to do arithmetic with Cantor’s transfinite ordinals. But no resolution of this difficulty was forthcoming. For the dénouement, read on.

Part 2: The Surreal Numbers

There were real problems with Cantor’s numbers, but it was about a century before a satisfactory definition of transfinite ordinals emerged. Around 1972, the brilliantly inventive mathematician John Conway was analysing the board game Go, when he stumbled upon a new way of constructing all the numbers, finite and infinite, from a few simple rules. The result is the system of surreal numbers.

Conway started with an idea reminiscent of Dedekind’s cuts. He defined each new number by means of two sets of previously defined numbers. To begin with, there are no numbers, so both sets must be empty. Thus, Conway defined zero as a pair of empty sets: ${0 = (\ \emptyset\ |\ \emptyset\ )}$ .

He could then proceed to form new pairs:

$(\{0\}|\ \emptyset\ ) \quad (\{0\}|\{0\}) \quad (\ \emptyset\ |\{0\})$

The centre pair is disallowed, as he required elements of the left set to be strictly less than elements of the right set. So, he had two new numbers:

$1 = (0|\ \emptyset\ ) \quad \mbox{and} \quad -1 = (\ \emptyset\ |0)$

(we omit the curly brackets). The next “generation” yields four new numbers

$-2 = (\ \emptyset\ |-1) \,\quad -\frac{1}{2} = (-1|0) \,\quad \frac{1}{2} = (0|1) \,\quad 2 = (1|\ \emptyset\ ) \,.$

The next step yields eight numbers, the next sixteen and so on.

Ultimately, all the dyadic numbers, that is, rationals whose denominators are powers of 2, emerged. At this stage, any real number ${x}$ could be represented as ${(L_x|R_x)}$ where ${L_x}$ is the set of dyadic rationals less than ${x}$ and ${R_x}$ is the set greater than ${x}$ .

But we can also construct completely new, surreal, numbers ${\omega = ( 0, 1, 2, \dots |\ \emptyset\ )}$ and ${\epsilon = (0 | 1, \frac{1}{2},\frac{1}{4},\dots)}$ ; ${\omega}$ is the simplest surreal number greater than all the reals and ${\epsilon}$ is the simplest surreal larger than zero but less than any positive real number.

For two surreals, ${x=(X_L|X_R)}$ and ${y=(Y_L|Y_R)}$ , Conway defined the arithmetic operations of addition, $x+y=(L_{x+y}|R_{x+y})$ , where

$L_{x+y} = (x + Y_L) \cup (X_L + y) \quad\mbox{and}\quad R_{x+y} = (x + Y_R) \cup (X_R + y)$

and multiplication, $\displaystyle xy = ( L_{xy} | R_{xy} )$ , where

$L_{xy} = (xY_L+X_Ly-X_LY_L) \ \cup\ (xY_R+X_Ry-X_RY_R)$
$R_{xy} = (xY_R+X_Ly-X_LY_R) \ \cup\ (xY_L+X_Ry-X_RY_L)$ .

These operations make sense for the new numbers, so we can form ${(\omega|\ \emptyset\ )=\omega+1 = 1+\omega}$ and ${(\omega+1,\omega+2,\dots |\ \emptyset\ ) = \omega+\omega = 2\omega}$ . Also, ${(1, 2, 3, \dots | \omega) = \omega-1}$ and ${(1, 2, \dots | \omega-1,\omega-2, \dots)=\frac{1}{2}\omega}$ , both of which are less than ${\omega}$ and yet greater than any real number.

We also have ${\epsilon\times\omega = 1}$ and it makes sense to write ${1/\omega = \epsilon}$ . We can proceed to ever vaster numbers ${\omega^2}$ , ${\omega^3}$ , ${\omega^\omega}$ , ${\omega^{\omega^\omega}}$ , et in saecula saeculorum.

Conway’s system of surreal numbers allows us to answer questions like “what is the square root of infinity?” by the construction

$\displaystyle \sqrt{\omega} = (0, 1, 2, \dots | \frac{\omega}{1},\frac{\omega}{2},\frac{\omega}{3},\dots )$

Using the definition of multiplication, we confirm that ${\sqrt{\omega}\times\sqrt{\omega} = \omega}$ .

If the measure of great mathematics is elegance, Conway’s surreal numbers are surely an outstanding example. From just a few simple rules, an entire universe of numbers can be constructed. Cantor discovered transfinite integers with arithmetic properties that were, at best, indifferent. Conway has discovered infinite fractions, roots and more, that have beautiful arithmetic structure.

The surreal numbers form the largest possible ordered field. I believe that the full potential of this system has yet to be realised, and its profundity to be fully appreciated by the mathematical world.

ThatsMaths