Basics:
I haven't written a basics post in a while, because for the most part, that well has run dry, but once in a while, one still pops up. I got an email recently asking about proofs by contradiction and...
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Posted on November 14, 2007 11:26 AM • 78 Comments •
This is something that came up in some of the comments on the recent "nimbers" post, and I thought it was worth promoting to the front, and getting up under an easy-to-find title in the "basics" series. In a...
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Posted on May 7, 2007 2:58 PM • 36 Comments •
I've used the term innumeracy fairly often on this blog, and I've had a few people write to ask me what it means. It's also, I think, a very important idea. Innumeracy is math what illiteracy is to reading....
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Posted on April 16, 2007 5:01 PM • 47 Comments •
For the basics, I wrote a bunch of stuff about sorting. It seems worth taking a moment to talk about something related: binary search. Binary search is one of the most important and fundamental algorithms, and it shows up...
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Posted on April 7, 2007 12:08 PM • 14 Comments •
This came up in a question in the post where I started to talk about π-calculus, but I thought it was an interesting enough topic to promote it up to a top-level post. If you listen to anyone talking...
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Posted on March 22, 2007 6:33 PM • 6 Comments •
Today's bit of basics is inspired by that bastion of shitheaded ignorance, Dr. Michael Egnor. In part of his latest screed (a podcast with Casey Luskin of the Discovery Institute), Egnor discusses antibiotic resistance, and along the way, asserts...
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Posted on March 14, 2007 11:29 AM • 73 Comments •
I've been getting so many requests for "basics" posts that I'm having trouble keeping up! There are so many basic things in math that non-mathematicians are confused about. I'm doing my best to keep up: if you've requested a "basics"...
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Posted on March 13, 2007 3:17 PM • 27 Comments •
I've received a request from a long-time reader to write a basics post on modal logics. In particular, what is a modal logic, and why did Gödel believe that a proof for the existence of God was more compelling...
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Posted on March 12, 2007 4:51 PM • 26 Comments •
In math and computer science, we have a tendency to talk about "going meta". It's actually a pretty simple idea, which tends to crop up in other places, as well. It's also one of my favorite concepts - the...
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Posted on March 11, 2007 9:20 AM • 18 Comments •
Today's basics topic was suggested to me by reading a crackpot rant sent to me by a reader. I'll deal with said crackpot in a different post when I have time. But in the meantime, let's take a look...
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Posted on March 7, 2007 3:15 PM • 12 Comments •
One thing that I frequently touch on casually as I'm writing this blog is the distinction between continuous mathematics, and discrete mathematics. As people who've been watching some of my mistakes in the topology posts can attest, I'm much...
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Posted on March 1, 2007 8:57 PM • 59 Comments •
Multiple people have written to me, after seeing yesterday's algorithms basics post, asking me to say more about sorting algorithms. I have to say that it's not my favorite topic - sorting is one of those old bugaboos that...
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Posted on February 28, 2007 3:30 PM • 51 Comments •
A kind reader pointed out that I frequently mention algorithms, but that I haven't defined them in the basics posts. To me, they're so fundamental to the stuff I do for a living that I completely forgot that they're...
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Posted on February 27, 2007 10:02 PM • 18 Comments •
Yet another term that we frequently hear, but which is often not properly understood, is the concept of optimization. What is optimization? And how does it work? The idea of optimization is quite simple. You have some complex situation,...
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Posted on February 17, 2007 12:46 PM • 24 Comments •
In yesterdays basics post, I alluded to the second kind of calculus - the thing that computer scientists like me call a calculus. Multiple people have asked me to explain what our kind of calculus is. In the worlds of...
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Posted on February 15, 2007 9:45 PM • 7 Comments •
Calculus is one of the things that's considered terrifying by most people. In fact, I'm sure a lot of people will consider me insane for trying to write a "basics" post about something like calculus. But I'm not going...
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Posted on February 14, 2007 6:05 PM • 18 Comments •
One of the fundamental branches of modern math - differential and integral calculus - is based on the concept of limits. In some ways, limits are a very intuitive concept - but the formalism of limits can be extremely...
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Posted on February 14, 2007 12:06 PM • 9 Comments •
Basics: Algebra While I was writing the vectors post, when I commented about how math geeks always build algebras around things, I realized that I hadn't yet written a basics post explaining what we mean by algebra. And since it...
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Posted on February 12, 2007 9:06 PM • 24 Comments •
There's another way of working with number-like things that have multiple dimensions in math, which is very different from the complex number family: vectors. Vectors are much more intuitive to most people than the the complex numbers, which are...
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Posted on February 11, 2007 9:43 PM • 26 Comments •
When we think of numbers, our intuitive sense is to think of them in terms of quantity: counting, measuring, or comparing quantities. And that's a good intuition for real numbers. But when you start working with more advanced math,...
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Posted on February 7, 2007 9:45 PM • 66 Comments •
Many people would probably say that things like computability and the halting program aren't basics. But I disagree: many of our basic intuitions about numbers and the things that we can do with them are actually deeply connected with...
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Posted on February 6, 2007 10:05 PM • 36 Comments •
What are the real numbers? Before I go into detail, I need to say up front that I hate the term real number. It implies that other kinds of numbers are not real, which is silly, annoying, and frustrating....
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Posted on February 5, 2007 8:30 AM • 40 Comments •
As long as I'm doing all of these basics posts, I thought it would be worth explaining just what a Turing machine is. I frequently talk about things being Turing equivalent, and about effective computing systems, and similar things,...
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Posted on February 3, 2007 6:38 PM • 18 Comments •
Sets are truly amazing things. In the history of mathematics, they're a remarkably recent invention - and yet, they're now considered to be the fundamental basis on which virtually all of mathematics is built. From simple things (like the...
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Posted on January 31, 2007 9:52 PM • 54 Comments •
Another great basics topic, which came up in the comments from last fridays "logic" post, is the difference between syntax and semantics. This is an important distinction, made in logic, math, and computer science. The short version of it is:...
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Posted on January 29, 2007 8:08 PM • 12 Comments •
This is another great basics topic, and it's also one of my pet peeves. In general, I'm a big science fiction fan, and I grew up in a house where every saturday at 6pm, we all gathered in front...
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Posted on January 27, 2007 2:14 PM • 45 Comments •
Correlation and Causation Yet another of the most abused mathematical concepts is the concept of correlation, along with the related (but different) concept of causation. Correlation is actually a remarkably simple concept, which makes it all the more frustrating to...
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Posted on January 25, 2007 2:35 PM • 39 Comments •
Time for another sort-of advanced basic. I used some recursive definitions in my explanation of natural numbers and integers. Recursion is a very fundamental concept, but one which many people have a very hard time wrapping their head around....
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Posted on January 23, 2007 9:05 PM • 8 Comments •
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...
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Posted on January 23, 2007 9:50 AM • 47 Comments •
The margin of error is the most widely misunderstood and misleading concept in statistics. It's positively frightening to people who actually understand what it means to see how it's commonly used in the media, in conversation, sometimes even by other...
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Posted on January 22, 2007 8:30 AM • 20 Comments •
When we look at a the data for a population+ often the first thing we do is look at the mean. But even if we know that the distribution is perfectly normal, the mean isn't enough to tell us...
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Posted on January 18, 2007 1:58 PM • 30 Comments •
In general, when we gather data, we expect to see a particular pattern to the data, called a normal distribution. A normal distribution is one where the data is evenly distributed around the mean in a very regular way,...
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Posted on January 15, 2007 8:46 PM • 16 Comments •
Statistics is something that surrounds us every day - we're constantly bombarded with statistics, in the form of polls, tests, ratings, etc. Understanding those statistics can be an important thing, but unfortunately, most people have never been taught just...
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Posted on January 15, 2007 8:00 AM • 43 Comments •
