Category Archives: Numbers

Roman Numerals and Arithmetic

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. I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to
have time to write while I’m away, I’m taking the opportunity to re-run an old classic series
of posts on numbers, which were first posted in the summer of 2006. These posts are mildly
revised.

I’ve always been perplexed by roman numerals.

First of all, they’re just weird. Why would anyone come up with something so strange as a
way of writing numbers?

And second, given that they’re so damned weird, hard to read, hard to work with, why do
we still use them for so many things today?

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i: the Imaginary Number (classic repost)

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I’m away on vacation this week, taking my kids to Disney World. Since I’m not likely to have time to write while I’m away, I’m taking the opportunity to re-run an old classic series of posts on numbers, which were first posted in the summer of 2006. These posts are mildly revised.

After the amazing response to my post about zero, I thought I’d do one about something that’s fascinated me for a long time: the number i, the square root of -1. Where’d this strange thing come from? Is it real (not in the sense of real numbers, but in the sense of representing something real and meaningful)? What’s it good for?

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Zero (classic repost)

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This post originally came about as a result of the first time I participated in a DonorsChoose fundraiser. I offered to write articles on requested topics for anyone who donated above a certain amount. I only had one taker, who asked for an article about zero. I was initially a bit taken aback by the request – what could I write about zero? This article which resulted from it ended up turning out to be one of the all-time reader-favorites for this blog!

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XKCD and Friendly Numbers

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I’ve been getting mail all day asking me to explain something
that appeared in today’s XKCD comic. Yes, I’ve been reduced to explaining geek comics to my readers. I suppose that there are worse fates. I just can’t
think of any. 🙂

But seriously, I’m a huge XKCD fan, and I don’t mind explaining interesting things no matter what the source. If you haven’t read today’s
comic, follow the link, and go look. It’s funny, and you’ll know what
people have been asking me about.

The comic refers to friendly numbers. The question,
obviously, is what are friendly numbers?

First, we define something called a divisors function over the integers, written σ(n). For any integer, there’s a set of integers that divide
into it. For example, for 4, that’s 1, 2, and 4. For 5, it’s just 1 and 5. And for 6, it’s 1, 2, 3, 6. The divisors function, σ(n) is the sum of all of the divisors of n. So
$ sigma(4)=8, sigma(5)=6, sigma(6)=12.$

For each integer, there is a characteristic ratio, defined
using the divisors function. For the integer n, the characteristic
is the ratio of the divisors function over the the number itself: σ(n)/n. So the characteristic ratio of 4 is 7/4; for 6, it’s
12/6=2.

For any characteristic ratio, the set of numbers that share that characteristic are friendly with each other. A friendly number is,
therefore, any integer that shares its characteristic ratio with at least one other integer. If an integer isn’t friendly, then it’s called a solitary number. 1, 2, 3, 4, and 5 are all solitary numbers. 6 is
friendly with 28 (1+2+4+7+14+28/28 = 56/28 = 2).

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From Sets to Arithmetic

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Even though this post seems to be shifting back to axiomatic set theory, don’t go thinking that we’re
done with type theory yet. Type theory will make its triumphant return before too long. But before
that, I want to take a bit of time to go through some basic constructions using set theory.

We’ve seen, roughly, how to create natural numbers using nothing but sets – that’s basically what
the ordinal and cardinal number stuff is about. Even doing that much is tricky – witness my gaffe about
ordinals and cardinals and countability. (What I was thinking of is the difference between the ε series in the ordinals, and the ω series in the cardinals, not the ordinals and cardinals themselves.) But if we restrict ourselves to sets of finite numbers (note: sets of finite numbers, not finite sets of numbers!), we’re pretty safe.

Of course, we haven’t defined arithmetic – we’ve just defined numbers. You might think it would be
pretty important to define arithmetic on the numbers. If you thought that, you’d be absolutely
Correct. So, that’s what I’m going to do next. First, I’m going to define addition and subtraction – multiplication can be defined in terms of addition. Division can be defined in terms of multiplication
and subtraction – but I’m going to hold off on defining division until we get to rational numbers.

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My Number

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Don’t you dare use the number 271277229129081016424883074559900780951 under any circumstances. It’s mine, mine I tell you, and if you use it, or copy it, I can have you arrested and sent to do hard time in prison. And it doesn’t matter whether you use it in decimal, like I used above, or it’s hexidecimal form, “CC16180895F94705F667F1BB6DB20997”, or any other way of encoding it. It’s my number, and you’re not allowed to use it. In fact, I don’t think I want to allow you to look at it – so I’m going to sue all of you for having read this post!

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The Strangeness of Nimber Addition

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So, today we’re going to play a bit more with nimbers – in particular, we’re
going to take the basic nimbers and operations over nimbers that we defined last time, and
take a look at their formal properties. This can lead to some simpler definitions, and
it can make clear some of the stranger properties that nimbers have.

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Surreal Nimbers: No, that's not a typo!

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(A substantial part of this post was rewritten since it was first posted. I managed to mangle things while editing, and the result was not particularly comprehensible: for example, in the original version of the post, I managed to delete the definition of “mex”, which continuing to use mex in several other definitions. I’ve tried to clear it up. Sorry for the confusion!)

This is actually a post in the surreal numbers series, even though it’s not going to look like one. It’s going to look like an introduction to another very strange system of numbers, called nimbers. But nimbers are a step on the path from
surreal numbers to games and game theory.

Nimbers come from a very old game called Nim. We’ll talk more about Nim later, but it’s one of the oldest strategy games known. The basic idea of it is that you have
a couple of piles of stones. Each turn, each player can take some stones from one of the piles. Whoever is left making the last move loses. It seems like a very trivial game. But it turns out that you can reduce pretty much every impartial game to some variation of Nim.

Analyzing Nim mathematically, you wind up finding that it re-creates the concept of ordinal numbers, which is what surreals are also based on. In fact, creating nimbers can end up re-creating the surreals. But that’s not what we’re going to do here: we’re going to create the nimbers and the basic nimber addition and multiplication
operations.

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Sign Expansions of Infinity

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Finally, as I promised a while ago, it’s time to look at the sign-expanded forms of infinites in the surreal numbers. Once you’ve gotten past the normal forms of surreal numbers, it’s pretty easy to translate them to sign-expanded form.

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Normal Forms and Infinite Surreals

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When I left off yesterday, we’d reached the point of being able to write normal forms
of surreal numbers there the normal form consisted of a finite number of terms. But
typically of surreal numbers. that’s not good enough: the surreals constantly produce
infinites of all sorts, and normal forms are no different: there are plenty of surreal
numbers where we don’t see a clean termination with a zero term.

For me, this is where the surreal numbers really earn there name. There is something distinctly surreal about a number system that not has a concrete concept of infinity, but allows you to have an infinite hierarchy of infinities, resulting in numbers that have, as their simplest representation, and infinite number of terms, each of which could involve numbers which can’t be written in a finite number of symbols. It’s just totally off the wall, insane, crazy, nuts… But fun!

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