numbers
goodmath | February 5, 2007 What are the real numbers?
Before I go into detail, I need to say up front that I hate the termreal number. It implies that other kinds of numbers are not real,
which is silly, annoying, and frustrating. But we're pretty much stuck with it.
There are a couple of ways of describing the real numbers. I'm going to take you through a couple of them: first, an informal intuitive description; then an axiomatic definition, and finally, a constructive definition.
The Reals, Informally
The informal, intuitive description is the basic number line. Think about
a line, that goes on forever in both…
goodmath | January 23, 2007 One of the interestingly odd things about how people understand math is numbers. It's
astonishing to see how many people don't really understand what numbers are, or what different kinds of numbers there are. It's particularly amazing to listen to people arguing
vehemently about whether certain kinds of numbers are really "real" or not.
Today I'm going to talk about two of the most basic kind of numbers: the naturals and the integers. This is sort of an advanced basics article; to explain things like natural numbers and integers, you can either write two boring sentences, or you can go a…
goodmath | December 7, 2006Tons of folks have been writing to me this morning about [the BBC story about an idiot math
teacher who claims to have solved the problem of dividing by zero][bbc-story]. This is an absolutely *infuriating* story, which does an excellent job of demonstrating what total innumerate idiots reporters are.
[bbc-story]: http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_…
What this guy has done is invent a new number, which he calls "nullity". This number is not on the number line, can't be compared to other numbers by less than or greater than, etc. In other words, he's given…
goodmath | November 15, 2006While I was researching yesterdays post on Archimedes integration, one of the things I read reminded me of one of the stranger things about Greek and earlier math. They had a notion that the only valid fractions were *unit* fractions; that is, fractions whose numerator is 1. A fraction that was written with a numerator larger than one was considered *wrong*. Even today, if you look in a lot of math books, they use the term "vulgar fraction" for non-unit fractions.
Obviously, there *are* fractions other that *1/n*. The way that they represented them is now known as *Egyptian fractions*. An…
goodmath | October 16, 2006After my binary fingermath stuff, a few people wrote to me to ask about just how binary really works. For someone who does the kinds of crazy stuff that I do, the idea of different number bases is so fundamental that it's easy to forget that most people really don't understand the idea of using different bases.
To start off with, I need to explain how our regular number system works. Most people understand how our numbers work, but don't necessarily understand *why* they're set up that way.
Our number system is *positional* - that is, what a given digit of a number means is dependent on its…
goodmath | August 21, 2006Since we talked about the surreals, I thought it would be interesting to take a *very* brief look at an alternative system that also provides a way of looking at infinites and infinitessimals: the *hyperreal* numbers.
The hyperreal numbers are not a construction like the surreals; instead they're defined by axiom. The basic idea is that for the normal real numbers, there are a set of basic statements that we can make - statements of first order logic; and there is a basic structure of the set: it's an *ordered field*.
Hyperreals add the "number" ω, the size of the set of natural numbers, so…
goodmath | August 21, 2006Last thursday, I introduced the construction of John Conway's beautiful surreal numbers. Today I'm going to show you how to do arithmetic using surreals. It's really quite amazing how it all works! If you haven't read the original post introducing surreals, you'll definitely want to [go back and read that][surreals] before looking at this post!
Transfinite Induction and ≤
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I'm going to start off by working through the way that the recursive
definition of the surreals and the "≤" operator work. It's based on
something called *transfinite induction*.
Transfinite…
goodmath | August 17, 2006Surreal numbers are a beautiful, simple, set-based construction that allows you to create and represent all real numbers, so that they behave properly; *and* in addition, it allows you to create infinitely large and infinitely small values, and have *them* behave and interact in a consistent way with the real numbers in their surreal representation.
The surreals were invented by John Horton Conway (yes, *that* John Conway, the same guy who invented the "Life" cellular automaton, and did all that amazing work in game theory). The name for surreal numbers was created by Don Knuth (yes, *that*…
goodmath | August 16, 2006I'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?
The Roman Numeral System
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I expect most people already know this, but it never hurts to be complete. The roman numeral system is non-positional. It assigns numeric values to letters. The basic system is:
1. "I" stands for 1.
2. "V" stands for 5.
3. "X" stands for 10.
4. "L"…
goodmath | August 14, 2006How can you talk about interesting numbers without bringing up π?
History
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The oldest value we know for π comes from the Babylonians. (Man, but those guys were impressive mathematicians; almost any time you look at the history of fundamental numbers and math, you find the Babylonians in the roots.) They tended to work in ratios, and the approximation that they used 25/8s (3.125), which is not a terribly bad approximation. Especially when you realize *when* they came up with this approximation: 1900BC!
The next best approximation came from Egypt, around the time of Pharaoh Amenemhat…
goodmath | August 12, 2006In the comments onmy post about φ, Polymath, (whose [blog][polymath] is well worth checking out) provided a really spectacular [link involving φ][desert]. It's an excerpt from a book called "[Mathematical Gems 2][gems]", showing a problem that came from John Conway, called the "Sending Scouts into the Desert" problem.
The problem is:
Suppose you're given a checkerboard with all of the squares on the bottom filled. You're allowed to do standard checks jumps (jump over a man and remove it), but you can't jump diagonally, only up, left, or right. How far *up* can you get a man? How many men…
goodmath | August 10, 2006In email, someone pointed me at an automated proof system called [Metamath][metamath]. Metamath generates proofs of mathematical statements using ZF set theory. The proofs are actually relatively easy to follow, which is quite unusual for an automated theorem prover. I'll definitely write more about Metamath some other time, but I thought it would be interesting today to point to [metamaths proof of the fifth axiom of Peano arithmetic][meta-peano], the principle of induction. Here's a screenshot of the first 15 steps; following the link to see the whole thing.
[metamath]: http://us.metamath…
goodmath | August 9, 2006Sorry for the delay in the category theory articles. I've been busy with work, and haven't had time to do the research to be able to properly write up the last major topic that I plan to cover in cat theory. While doing my research on closed cartesian categories and lambda calculus, I came across this little tidbit that I hadn't seen before, and I thought it would be worth taking a brief diversion to look at.
Category theory is sufficiently powerful that you can build all of mathematics from it. This is something that we've known for a long time. But it's not really the easiest thing to…
goodmath | August 8, 2006Lots of folks have been asking me to write about φ, the golden ratio. I'm finally giving up and doing it. I'm not a big fan of φ. It's a number
which has been adopted by all sorts of flakes and crazies, and there are alleged sightings of it in all sorts of strange places that are simply *not* real. For example, pyramid loons claim that the great pyramids in Egypt have proportions that come from φ - but they don't. Animal mystics claim that the ratio of drones to larvae in a beehive is approximately φ - but it isn't.
But it is an interesting number. My own personal reason for thinking it's…
goodmath | August 7, 2006Quaternions
Last week, after I wrote about complex numbers, a bunch of folks wrote and said "Do quaternions next!" My basic reaction was "Huh?" I somehow managed to get by without ever being exposed to quaternions before. They're quite interesting things.
The basic idea behind quaterions is: we see some amazing things happen when we expand the dimensionality of numbers from 1 (the real numbers) to 2 (the complex numbers). What if we add *more* dimensions?
It doesn't work for three dimensions But you *can* create a system of numbers in *four* dimensions. As with complex numbers, you need a…
goodmath | August 7, 2006Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizzare properties. It's the one real transcendental number that I know of that comes from the theory of computation, that is important, and that expresses meaningful fundamental mathematical properties. It's also deeply non-computable; meaning that not only is it non-computable, but even computing meta-information about it is non-computable. And yet, it's *almost* computable. It's just all around awfully cool.
So. What is it Ω?
It's sometimes called the *halting…
goodmath | August 4, 2006If you look at the history of math, there've been a lot of disappointments for mathematicians. They always start off with an idea of math as a clean, beautiful, elegant thing. And they seem to often wind up disappointed.
Which leads us into todays strange numbers: irrational and transcendental numbers. Both of them were huge disappointments to the mathematicians who discovered them.
So what are they? We'll start with the irrationals. They're numbers that aren't integers, and that aren't a ratio of any two integers. So you can't write them as a normal fraction. If you write them as a continued…
goodmath | August 3, 2006One of the annoying things about how we write numbers is the fact that we generally write things one of two ways: as fractions, or as decimals.
You might want to ask, "Why is that annoying?" (And in fact, that's what I want you to ask, or else there's no point in my writing the rest of this!)
It's annoying because both fractions and decimals can both only describe
*rational* numbers - that is, numbers that are a perfect ratio of two integers. And *most* numbers aren't rational.
But it's even more annoying than that: if you use decimals, then there are lots of rational numbers that you can't…
goodmath | August 2, 2006Looks like I've accidentally created a series of articles here about fundamental numbers. I didn't intend to; originally, I was just trying to write the zero article I'd promised back during the donorschoose drive.
Anyway. Todays number is *e*, aka Euler's constant, aka the natural log base. *e* is a very odd number, but very fundamental. It shows up constantly, in all sorts of strange places where you wouldn't expect it.
What is e?
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*e* is a transcendental irrational number. It's roughly 2.718281828459045. It's also the base of the natural logarithm. That means that by…