Wednesday, August 13, 2008

The Top 8 Most Important Mathematical Constants

Well, I'm currently studying for an Ordinary Differential Equations final that I have tomorrow, but I figured I'd take a break and HIT UP THE INTERNET with a math-oriented list. I was considering doing something on the best musicians to listen to while doing math, but we've had quite a few pop-culture oriented lists in the last several days, so I'm going to instead do something purely focused on math. But that other's coming...later. (A preview: Battles and Frank Zappa.) And now, the most important numbers. (No physical constants, that shit is for poseurs.)

8. Phi, the golden ratio constant

Chances are that you're probably at least a little familiar with the concept of the golden ratio, less likely directly through math and more likely through things like architecture, art, or nature. The golden ratio is the unique ratio a/b such that a/b = (a+b)/a. Or, with Wikipedia's help:This comes out to about 1.618. This number, especially rectangles with that ratio (1.618:1) show up in tons of places, like Greek architecture, human head shape and in the angles of plant leaves. You can use it to generate the Fibonacci sequence. So why is it so low on the list? Well, two reasons. One, you don't really ever have to use it for much, mathematically. Two, you can solve the equation with a and b above pretty easily and get a fairly simple exact algebraic answer for phi: (1+(5)^(1/2))/2. That looks complicated written out like that, but it really isn't. Basically, the only interesting thing in its composition is the square root of five, and your average irrational square root (like the square roots of two or three) is simple not interesting enough to make this list. Ergo, phi barely makes it.

7. 17

17 is an incredibly important number because it is the amount of money I paid last night at Kim's for the first season of Taxi and the first season of The Bob Newhart Show-COMBINED. Yeah, that's right. Seventeen dollars total for the first season of two classic sitcoms. I was kind of sad that Kim's (the local video and rental store) is closing, but this deal was totally worth all that. I would wish for the closing of all stores in the area, including all restaurants and take-out places, if every time one of them closed, I could buy two seasons of classic sitcoms for $17. I would be hungry for food but filled with laughter and celluloid friendship. Note: this number is not actually very important.

6. Pi, or the Archimedes constant

Pi is probably the most famous constant in the world. Just about anybody who has taken any math classes at all could tell you it's about 3, and most people could probably even produce 3.14. I'm going to go ahead and admit that I've seen pi on calculators so many times that I have memorized 14 decimal digits of it incidentally: 3.14159265358979 (That was from memory and I'm not checking it. Feel free to call me out if I got it wrong.) So, pi: what's the deal? Well, a good amount of people, if pressed, could probably tell you pi*d=c, where d is the diameter and c the circumferance of a circle, or pi*r^2=A, where A is the area of the circle. And it shows up in all sorts of calculations like that which involve circles, cylinders, spheres, the unit circle, my balls, etc. Round things in general. So, what else does it do? Well, lots of things, but basically all of those things are true just because of the original c=pi*d. It's pretty notable and rightfully famous for that, but overall, it's not the most amazing number; you can calculate it a bunch of cool ways, and it gets involved in equations that are legitimately amazing because it relates to circles, and lots of things relate to circles. We'll get to some of those really amazing equations in a second.

5. 1

Number one. Numero uno. I thought about making this one #1, just because it's kind of funny in that not-actually-funny-at-all-gotcha way. It is true, however that you needed this number before beginning to develop any sort of mathmatical system. And from an abstract standpoint, it's pretty cool that humans are able to separate and quantize all of the many sensory inputs to the point where we can point to something and go THAT. There is ONE of that. It is separate from the things around it and wholly self-contained. (Which is, objectively, a concept that doesn't make much sense.) But, from a non-abstract standpoint: it's fucking one. How hard was this shit to come up with, really? We'd be pretty fucked if we couldn't get that far. Oh, it's also the multiplicative identity, which is almost cool but not.

4. 18

18 is the legal age of consent in most of the U.S.A. It is a number that, in certain situations, can get very important very fast.

3. 0

0 is an interesting number because it is, historically, the first entirely abstract number that people had to deal with. As "difficult" as it is to point to something and call it one of whatever it is (seriously, not that friggin' difficult) it is significantly more difficult to take that thing away and declare that there is still an "amount" of it present; that amount is zero. Zero was first used, sensibly enough, as a placeholder. One more one than 9 and you now have one ten and zero ones--10. Then people realized you could just have zero of something without having any of anything else along with it. We don't have much trouble abstracting it now, but it was certainly a mindfuck a long time ago.

2. The imaginary unit, i

And now, to one that's still a bit of a mindfuck to many people today. If you haven't taken a lot of math, think of i in the same way you think of 0. It's a little harder to conceive 0 or i of than it is to conceive of 1 or 2, but they're no less legitimate as numbers. I'd say that imaginary numbers are not in fact imaginary (trivia: René Decartes actually coined the term "imaginary number" solely to discredit the idea; he didn't believe they were mathematically valid), but that's a little misleading, so here's a better way of putting it: "imaginary" numbers are just no less imaginary than all other numbers. 1-2 might sound goofy until you decided to create a number -1 such that 1-2=-1. Then, once you've done that, negative numbers have all sorts of useful real-world applications: debt is the most obvious, but vectors are another big one, etc. Similarly, the square root of -1, (-1)^(1/2), seems crazy until you create a number i such that i^2=-1. And once you do, it similarly has many useful applications that unfortunately are not as obvious as the debt example. Trust me, though, they do exist.

I highly recommend reading Godel, Escher, Bach if you ever get a chance. It includes a discussion of numbers, and one of the proposed hitches to Godel's theorem is the idea that there may be a whole separate set of numbers that can be introduced that do not follow the rules perscribed for any numbers we currently use. But THAT is a discussion for another list. A much more boring list that probably does not belong on this site. Wow, this list is really boring, isn't it? Sorry, guys, that's what you get with me. Either vitriolic anger or turbo-nerd. NO COMPROMISES \m/

1. Euler's number, e

And now to number one, e. By far the number on this list with the most astounding amount of properties, e is arguably the most important mathematical constant. Here is the simplest and most representative definition of e that I can give: e is the unique constant such that e^x is the derivative of e^x. That makes a little sense to people who sorta remember calculus, so here's a simpler way of putting it: if you draw a graph of y=e^x, its slope is equal to its y-value. That doesn't make much more sense to people who still don't remember calculus well, so here's a pretty picture:
The point in the picture where the red and blue lines cross is (x,y)=(0,1). At that point, y=e^x=e^0=1. The red line is tangent to the blue line (the blue line is the graph of y=e^x), and at that point, they have the exact same slope. The slope of the red line is equal to 1. Imagine sliding the red line along the blue line to another point where they barely touch. At that point as well, the slope (steepness) of the red line will be exactly equal to the y-value where the red and blue lines touch.

Seem a little convoluted, maybe? It surprisingly isn't. This is the only nontrivial function in existence for which this is true. It makes the constant e linked to calculus in a way that no other constant I know is so linked to a mathematical field. Pi is linked to geometry, sure, but in an almost incidental way. e (2.712828128246, call me on that one too) is so tied into the fabric of all mathematical fields that are based on calculus (that is, the majority of them) that it ends up having almost supernatural properties.

For instance--and this one has as much to do with i as it does with e, but e provides the really cool part of it, I think: e^(ix)=cos(x)+i*sin(x). That's Euler's formula. Basically everything in math is Euler's, get used to it. I'm not going to take the time to fully explain this, but it has a very interesting case: e^(i*pi)=cos(pi)+i*sin(pi)=-1+0=-1. Or, put more simply,


That's one equation that uses each of the five most important numbers in math once. You know why? It's because e is fucking nuts. (Note that 17 and 18 are not included because those were jokes, see, and phi is not included because it's kind of a piece of shit.)



Chris said...

"This is the only function in existence for which this is true."

That's not true: y=0 , the most boring equation in the world, has a slope of 0 at every point.

Rob said...

Duly noted, thank you! I will edit to say that it is the only nontrivial function for which this is true.

Connor Mooneyhan said...

Great post, but dude you've gotta work on that "e" estimation. It's 2.718281828459...
Other than that I totally agree with the listing!