Surreal numbers

Surrealism. That’s Dali and Magritte, right?

Well, yes… but surrealism is not just in art. It also pops up in mathematics — mathematics of all things, much to the surprise of those who are not in the world of Donald Knuth and the late, great John Horton Conway.

So, do surreal numbers really exist? Well, that’s a more complicated question than you might imagine. Starting more simply, we ask do negative numbers exist? (After all, can you go outside and point to a –2?) Do irrational numbers exist? (Can you go outside and point to √2?) Do complex numbers exist? (Can you go outside and point to 3+4i?) For that matter, do numbers exist at all — or are they just figments of our imagination?

Perhaps they exist as the ultimate reality: abstract Platonic forms that we can’t see, since all we can perceive is shadows of their images. Ironically, that means that they are simultaneously the most human and the least human of objects: the most human since they live only in our heads, the least human since they are “out there” regardless of our knowledge of them. Mathematical truths are truths, regardless of our individual knowledge.

But enough philosophy — let’s turn to a children’s book. Or at least what’s ostensibly a children’s book: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happinessby the great computer scientist Don Knuth. Before you read the book, you should watch the short video about it. Before you watch the video, you should ponder a couple of questions:

  • What is the largest positive number?
  • What is the smallest positive number?

In a totally unscientific sample of a small number of teens, I learned that the most common answers are infinity and one, respectively. The first answer engenders an argument about whether infinity is a number; the second answer always results in a classmate’s observation that 1/2 and 0.0000001 and so forth are all positive numbers less than the offered answer of one. So where does that leave us?

By now I’m sure you’ve guessed… It leaves us with surreal numbers, of course.

But what in the world is a surreal number? (Or not in the world, as the case may be.) So now is the time to watch the video. Before you do so, let’s start with a quick review of a few types of numbers:

  • Positive numbers are the numbers to the right of zero on the number line. Examples: 42 and π. (But that’s not starting at the beginning: where did π come from?)
  • Of those, the counting numbers are 1, 2, 3, 4, etc. Examples: 27 and 1000.
  • Throw in 0 as well, and we get the whole numbers. Examples: 7 and 0.
  • The reflections of the positive numbers across 0 are the negative numbers. Examples: –27 and –π.
  • The whole numbers and their reflections are collectively the integers.
  • The ratio of two integers is a rational number. No, not “rational” in that they are logical, but “rational” in that they are ratios. Examples: 42 and –2/13. (42, of course, is just 42/1, so it’s rational. √2 is not.)
  • All the numbers on the number line are collectively called the real numbers. (Another terrible term, since there’s nothing unreal about the numbers that are not on the number line.)
  • All the numbers on the plane (whether on the number line or not) are collectively called the complex numbers. Examples: 2+13and √13. And 1. And 0.

Note that this list is not a logically ordered development of types of numbers, but note also that the real numbers themselves are ordered: if we pick two distinct real numbers, we can always say that one is greater than the other. It might take some work to figure it out in a particular case, like √10 and π, but we still know that one is greater than the other. Non-real complex numbers are not ordered: which is smaller, 3+4or 4+3i?

Now go watch the video. Since it’s in the form of an interview of Don Knuth by Brady Haran, you’ll a useful perspective that a lecture or paper by a single person is unlikely to give you. After you finish the video, go read Conway’s book! It’s available from Addison-Wesley and Amazon; it used to be available online at, but no longer. Copyrights, you know.


Categories: Math