One of the things that's endlessly fascinating to me about math and
science is the way that, no matter how much we know, we're constantly
discovering more things that we don't know. Even in simple, fundamental
areas, there's always a surprise waiting just around the corner.
A great example of this is something called the Ulam spiral,
named after Stanislaw Ulam, who first noticed it. Take a sheet of graph paper.
Put "1" in some square. Then, spiral out from there, putting one number in
each square. Then circle each of the prime numbers. Like the following:
If you do that for a while - and…

# numbers

I'm still getting things squared away after my blogging break, but as a step on the way back toward normal programming, here's a Dorky Poll: What kind of numbers do you most like to work with?
What kind of numbers do you like best?online surveys
You can only choose a single answer, which I'm sure will come as a disappointment to many of those favoring the later options. You could always vote a second time from a different computer, though...

tags: nature, numbers, geometry, mathematics, Fibonacci sequence, Golden Ratio, Angle Ratio, Delaunay Triangulation, Voronoi Tessellations, filmmaking, animation, Cristobal Vila, Nature by Numbers, streaming video
In this beautiful video, "Nature by Numbers," filmmaker Cristobal Vila presents a series of animations illustrating various mathematic principles, beginning with a breathtaking animation of the Fibonacci Sequence before moving on to the Golden Ratio, the Angle Ratio, the Delaunay Triangulation and Voronoi Tessellations. The words are scary-sounding, but the math is beautiful and the…

OUR ability to use and manipulate numbers is integral to everyday life - we use them to label, rank, count and measure almost everything we encounter. It was long thought that numerical competence is dependent on language and, therefore, that numerosity is restricted to our species. Although the symbolic representation of numbers, using numerals and words, is indeed unique to humans, we now know that animals are also capable of manipulating numerical information.
One study published in 1998, for example, showed that rhesus monkeys can form spontaneous representations of small numbers and…

I was planning on ignoring this one, but tons of readers have been writing
to me about the latest inanity spouting from the keyboard of Discovery
Institute's flunky, Denise O'Leary.
Here's what she had to say:
Even though I am not a creationist by any reasonable definition,
I sometimes get pegged as the local gap tooth creationist moron. (But then I
don't have gaps in my teeth either. Check unretouched photos.)
As the best gap tooth they could come up with, a local TV station interviewed
me about "superstition" the other day.
The issue turned out to be superstition related to numbers.…

In my Dembski rant, I used a metaphor involving the undescribable numbers. An interesting confusion came up in the comments about just what that meant. Instead of answering it with a comment, I decided that it justified a post of its own. It's a fascinating topic which is incredibly counter-intuitive. To me, it's one of the great examples of how utterly wrong our
intuitions can be.
Numbers are, obviously, very important. And so, over the ages, we've invented lots of notations that allow us to write those numbers down: the familiar arabic notation, roman numerals, fractions, decimals,…

After my post the other day about rounding errors, I got a ton of
requests to explain the idea of significant figures. That's
actually a very interesting topic.
The idea of significant figures is that when you're doing
experimental work, you're taking measurements - and measurements
always have a limited precision. The fact that your measurements - the
inputs to any calculation or analysis that you do - have limited
precision, means that the results of your calculations likewise have
limited precision. Significant figures (or significant digits, or just "sigfigs" for short) are a method of…

Another alert reader sent me a link to a YouTube video which is moderately interesting.
The video itself is really a deliberate joke, but it does demonstrate a worthwile point. It's about rounding.
The overwhelming majority of us were taught how to round decimals back in either elementary or middle school. (I don't even recall exactly when.) The rule that most of us were taught is:
If the first digit after the rounding point is 0, 1, 2, 3, or 4, then round the previous digit down;
If the first digit after the rounding point is 5, 6, 7, 8, or 9, then round the
previous digit up.
Here's…

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.
Ω is my own personal favorite transcendental number. Ω isn't really a specific number, but rather a family of related numbers with bizarre 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…

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.
One 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…

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.
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?
e is a transcendental irrational number. It's roughly 2.718281828459045. It's also the base of…

. 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?
The Roman Numeral…

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 ze ro, 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)?…

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.
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 -…

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…

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…

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!
Yes, I'm serious. At least in theory, using that…

The Surreal Reals
I was reading Conway's Book, book on the train this morning, and found something I'd heard people talk about, but that I'd never had time to read or consider in detail. You can use a constrained subset of the surreal numbers to define the real numbers. And the resulting formulation of the reals is arguably superior to the more traditional formulations of the reals via Dedekind cuts or Cauchy sequences.
First, let's look at how we can create a set of just the real numbers using the
surreal construction. What we want to do is get a notion of the simplest surreal number that…

Today we're going to take our first baby-step into the land of surreal games.
A surreal number is a pair of sets {L|R} where every value in L is less than every value in R. If we follow the rules of surreal construction, so that the members of L sets are always strictly less than members of R sets, we end up with a totally ordered field (almost) - it gives us something essentially equivalent to a superset of the real numbers. (The reason for the almost is that technically, the surreals form a class not a set, and a field must be based on a set. But for our purposes, we can treat them as a…

In my last post on the surreals, I introduced how the surreal numbers are constructed. It's really fascinating to look back on it - to see the structure of numbers from 0 to infinity and beyond, and realize that ultimately, that it's all built from nothing but the empty set!
Today, we're going to move on, and start looking at arithmetic with the surreal numbers. In this post, I'm going to go through the basic definition of addition, subtraction, and multiplication of surreal numbers. Division will have to wait for a later post; division is quite a subtle operation in the surreals.…