Archive for the 'Numbers' category

Rounding and Bias

Mar 01 2009 Published by under Basics, Numbers

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.

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My Favorite Strange Number: Ω (classic repost)

Dec 31 2008 Published by under classics, Computation, Numbers

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 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.

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Continued Fractions (classic repost)

Dec 30 2008 Published by under classics, Numbers

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 because both fractions and decimals can both only describe rational numbers - that is, numbers that are a perfect ratio of two integers. The problem with that is that most numbers aren't rational. Our standard notations are incapable of representing the precise values of the overwhelming majority of numbers!

But it's even more annoying than that: if you use decimals, then there are lots of rational numbers that you can't represent exactly (i.e., 1/3); and if you use fractions, then it's hard to express the idea that the fraction isn't exact. (How do you write π as a fraction? 22/7 is a standard fractional approximation, but how do you say π, which is almost 22/7?)

So what do we do?

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e: the Unnatural Natural Number (classic repost)

Dec 27 2008 Published by under classics, Numbers

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.

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35 responses so far

Roman Numerals and Arithmetic

Dec 26 2008 Published by under classics, Numbers

. 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)

Dec 25 2008 Published by under classics, Numbers

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)? What's it good for?

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31 responses so far

Zero (classic repost)

Dec 24 2008 Published by under classics, Numbers

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

Apr 14 2008 Published by under Numbers

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

Nov 19 2007 Published by under Numbers, Set Theory

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

May 09 2007 Published by under Numbers

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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