Category Archives: Numbers

Categorical Numbers

Leave a reply

Sorry 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 demonstrate.
But then, I came across this lovely little article on wikipedia. It turns out that you can [build the natural numbers entirely in terms of categories][wikipedia] in a beautifully simple way.
Before we dive into categorical numbers, let’s take a moment to remember what we really mean by natural numbers.
Peano Arithmetic
——————
The meanings of natural numbers are normally defined in terms of what we call *Peano arithmetic*. Peano arithmetic defines the natural numbers, **N** in terms of five fundamental axioms:
1. 0 ∈ **N** (0 is a natural number.)
2. ∀ a ∈ **N** ∃ a’=s(n), a’ ∈ **N**. (Every natural number a has a successor, s(a))
3. ¬ ∃ a ∈ **N** : s(a)=0. (No natural number has 0 as its successor.)
4. ∀ a,b ∈ **N**: s(a)=s(b) ⇒ a=b. (Each number has a unique successor.)
5. (P(0) ∧ ∀ a : P(a) ⇒ P(s(a))) ⇒ ∀ b ∈ **N** : P(b). (If a property P is true for 0; and it can be shown that for any number a, if it’s true for a, then it’s true for b, then that means that it’s true for all natural numbers. This is the *axiom of induction*.)
To get to arithmetic, we need to add two operations: addition and multiplication.
We can define addition recursively quite easily:
1. ∀ x ∈ **N**: x+0 = x
2. ∀ x,y ∈ **N**: s(x)+y=x+s(y)
And once we have addition, we can do multiplication:
1. ∀ x ∈ **N**: x * 0 = 0 ∧ 0 * x = 0
2. ∀ x ∈ **N**: 1 * x = x
3. ∀ x,y ∈ **N**: s(x)*y = (x*y) + y
To sum up: Peano arithmetic is based on a definition of the natural numbers in terms of five fundamental properties, called the Peano axioms, built on “0”, a successor function, and induction. If we have those five properties, we can define addition and multiplication; and given those, we have the ability to build up arbitrary functions, sets, and the other fundamentals of mathematics in terms of the natural numbers.
Peano Naturals and Categories
———————————
The peano axioms can be captured in a category diagram for a relatively simple object called a *natural number object (NNO)*. A natural number object can exist in many categories: basically any category with a terminal object *can* have NNOs. Any closed cartesian category *does* have an NNO.
There are two alternate constructions; a general construct that works for all NNO-containing categories; and a more specific one for closed cartesian categories. (You can show that they are ultimately the same in a CCC.)
I actually think that the general one is easier to understand, so that’s what I’ll describe. This is basically a construction that captures the ideas of successor and recursion.
Given a category C with a terminal object **1**. A *natural number object* is an object *N* with two arrows z : 1 → *N*, and s : *N* → *N*, where z and s have the following property.
(∀ a ∈ Obj(C), (q : 1 → a, f : a → a) ∈ Arrows(C)) : u º z = q ∧ u º s = f º u.
That statement means the same thing as saying that the following diagram commutes:

cat-numbers.jpg

Let’s just walk through that a bit. Think of the object **N** as a set. “s” is successor. So S is an arrow from *N* to *N* that maps each number to its successor. *N* is “rooted” by the terminal object; that gives it its basic structure. The “z” arrow from the terminal object maps to the zero value in *N*. So we’ve got an arrow that selects zero; and we’ve got an arrow that defines the successor.
Think of “q” as a predicate, and “u” as a statement that a value in *N* satisfies the predicate.
So u º z, that is the path from 1 to a by stepping through z and u is a statement that the predicate “q” is true for z (zero); because composing u (the number I’m composed with satisfies q) and z (the number zero) gives us “q” (zero satifies q).
Now the induction rule part. *N* maps to *N* by s – that’s saying that s is an arrow from a number in *N* to its successor. If you have an arrow composition that says “q is true for n”, then f can be composed onto that giving you “q is true for n + 1”.
If for all q, u is unique – that is, there is only *one* mapping where induction works; then you’ve got a categorical structure that satisfies the Peano axioms.
[wikipedia]: http://en.wikipedia.org/wiki/Natural_number_object

φ, the Golden Ratio

Leave a reply

Lots 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 cool is representational: if you write φ as a continued fraction, it’s [1;1,1,1,1…]; and if you write it as a continued square root, it’s 1 + sqrt(1+sqrt(1+sqrt(1+sqrt(1…)))).
What is φ?
—————–
φ is the value so-called “golden ratio”: it’s the number that is a solution for the equation (a+b)/a = (a/b). (1+sqrt(5))/2. It’s the ratio where if you take a rectangle where the ratio of the length of the sides is 1:φ, then if you remove the largest possible square from it, you’ll get another rectangle whose sides have the ration φ:1. If you take the largest square from that, you’ll get a rectangle whose sides have the ratio 1:φ. And so on:

phi.jpg

Allegedly, it’s the proportion of sides of a rectangle that produce the most aesthetically beautiful appearance. I’m not enough of a visual artist to judge that, so I’ve always just taken that on faith. But it *does* shows up in many places in geometry. For example, if you draw a five-pointed star, the ratio of the length of one of the point-to-point lines of the star to the length of the sides of the pentagon inside the star are φ:1:

pentaphi.jpg

φ is also related to the fibonacci series. In case you don’t remember, the fibonacci series is the set of numbers where each number in the series is the sum of the two previous: 1,1,2,3,5,8,13,… If Fib(n) is the *n*th number in the series, you can compute it as:
phi-fix-eq.gif
History

Quaternions: upping the dimensions of complex numbers

34 Replies

Quaternions
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 distinct unit for each dimension. In complex, those were 1 and i. For quaternions, you need for units: we’ll call them “1”, “i”, “j”, and “k”. In quaternion math, the units have the following properties:
1. i2 = j2 = k2 = ijk = -1
2. ij = k, jk = i, and ki=j
3. ji = -k, kj = -i, and ik=-j
No, that last one is *not* an error. Quaternion multiplication is *not* commutative: ab &neq; ba. It *is* associative over multiplication, meaning (a*b)*c = a*(b*c). And fortunately, it’s both commutative and associative over addition.
Anyway, we write a quaternion as “a + bi + cj + dk”; as in “2 + 4i -3j + 7k”.
Aside from the commutativity thing, quaternions work well as numbers; they’re a non-abelian group over multiplication (assuming you omit 0); they’re an abelian group over addition; they form a division algebra *over* the reals (meaning that you can define an algebra with operations up to multiplication and division over the quaternions, and operations in this algebra will generate the same results as normal real algebra if the i,j, and k components are 0.)
History
———
Quaternions are a relatively recent creation, so we know the history quite precisely. Quaternions were invented by a gentleman named William Hamilton on October 16th, 1843, while walking over the Broom bridge with his wife. He had been working on trying to work out something like the complex numbers with more dimensions for a while, and the crucial identity “ijk=-1” struck him while he was walking over the bridge. He immediately *carved it into the bridge* before he forgot it. Fortunately, this act of vandalism wasn’t taken seriously: the Broom bridge now has a plaque commemorating it!
Quaternion_Plague_on_Broom_Bridge.jpg
Quaternions were controversial from the start. The fundamental step that Hamilton had taken in making quaternions work was abandoning commutativity, and many people believed that this was simply *wrong*. For example, Lord Kelvin said “Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.”
Why are they non-commutative?
——————————-
Quaternions are decidedly strange things. As I mentioned above, they’re non-commutative. Doing math with things where ab &neq; ba was a *huge* step, and it can be strange. But it is necessary for quaternions. Here’s why.
Let’s take the fundamental identity of quaternions: ijk = -1
Now, let’s multiply both sides by k: ij(k2) = -1k
So, ij(-1)=-k, and ij=k.
Now, multiply by *i* on the left on both sides: i2j = ik
And so, -1j = ik, and ik=-j.
Keep playing with multiplication in the identities, and you find that the identities simple *do not work* unless it’s non-commutative.
What are they good for?
————————-
For the most part, quaternions aren’t used much anymore. Initially, they were used in many places where we now use complex vectors, but that’s mostly faded away. What they *are* used for is all sorts of math that in any way involves *rotation*. Quaternions capture the essential properties of rotation.
To describe a rotation well, you actually need to have *extra* dimensions. To describe an *n*-dimensional rotation, you need *n+2* dimensions. Since in our three dimensional world, we rotate things around a *line* which is two dimensional, the description of the rotation needs *four* dimensions. And further, rotation itself is non-commutative.
Take a book and put it flat on the table, with the binding to the left. Flip it so that the binding stays on the same side, but now you’ve got the back cover on top, upside-down. Then rotate it clockwise 90 degrees on the table. You now have the book with its back cover up, and the binding facing away from you.
Now, start over: book right side up, binding facing left. Rotate it clockwise 90 degrees. Now flip it the same way you did before (over the x axis if you think of left as -x, right as +x, away from you as +y, up as +z, etc.)
rotate.jpg
What’s the result? You get the book with its back cover up, and the binding factor *towards* you. It’s 180 degrees different depending on which order you do it in. And 180 degrees is a reverse equivalent to the sign difference that you get in quaternion multiplication! Quaternions have exactly the right properties for describing rotations – and in particular, the *composition* of rotations.
The use of quaternions for rotations is used in computer graphics (when something rotates on the screen of your PS2, it’s doing quaternion math!); in satellite navigation; and in relativity equations (for symmetry wrt rotation).

Ω: my favorite strange number

Leave a reply

Ω 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 probability*. The idea of it is that it encodes the *probability* that a given infinitely long random bit string contains a prefix that represents a halting program.
It’s a strange notion, and I’m going to take a few paragraphs to try to elaborate on what that means, before I go into detail about how the number is generated, and what sorts of bizzare properties it has.
Remember that in the theory of computation, one of the most fundamental results is the non-computability of *the halting problem*. The halting problem is the question “Given a program P and input I, if I run P on I, will it ever stop?” You cannot write a program that reads P and I and gives you the answer to the halting problem. It’s impossible. And what’s more, the statement that the halting problem is not computable is actually *equivalent* to the fundamental statement of Gödel’s incompleteness theorem.
Ω is something related to the halting problem, but stranger. The fundamental question of Ω is: if you hand me a string of 0s and 1s, and I’m allowed to look at it one bit at a time, what’s the probability that *eventually* the part that I’ve seen will be a program that eventually stops?
When you look at this definition, your reaction *should* be “Huh? Who cares?”
The catch is that this number – this probability – is a number which is easy to define; it’s not computable; it’s completely *uncompressible*; it’s *normal*.
Let’s take a moment and look at those properties:
1. Non-computable: no program can compute Ω. In fact, beyond a certain value N (which is non-computable!), you cannot compute the value of *any bit* of Ω.
2. Uncompressible: there is no way to represent Ω in a non-infinite way; in fact, there is no way to represent *any substring* of Ω in less bits than there are in that substring.
3. Normal: the digits of Ω are completely random and unpatterned; the value of and digit in Ω is equally likely to be a zero or a one; any selected *pair* of digits is equally likely to be any of the 4 possibilities 00, 01, 10, 100; and so on.
So, now we know a little bit about why Ω is so strange, but we still haven’t really defined it precisely. What is Ω? How does it get these bizzare properties?
Well, as I said at the beginning, Ω isn’t a single number; it’s a family of numbers. The value of *an* Ω is based on two things: an effective (that is, turing equivalent) computing device; and a prefix-free encoding of programs for that computing device as strings of bits.
(The prefix-free bit is important, and it’s also probably not familiar to most people, so let’s take a moment to define it. A prefix-free encoding is an encoding where for any given string which is valid in the encoding, *no prefix* of that string is a valid string in the encoding. If you’re familiar with data compression, Huffman codes are a common example of a prefix-free encoding.)
So let’s assume we have a computing device, which we’ll call φ. We’ll write the result of running φ on a program encoding as the binary number p as &phi(p). And let’s assume that we’ve set up φ so that it only accepts programs in a prefix-free encoding, which we’ll call ε; and the set of programs coded in ε, we’ll call Ε; and we’ll write φ(p)↓ to mean φ(p) halts. Then we can define Ω as:

Ωφ,ε = Σp ∈ Ε|p↓ 2-len(p)

So: for each program in our prefix-free encoding, if it halts, we *add* 2-length(p) to Ω.
Ω is an incredibly odd number. As I said before, it’s random, uncompressible, and has a fully normal distribution of digits.
But where it gets fascinatingly bizzare is when you start considering its computability properties.
Ω is *definable*. We can (and have) provided a specific, precise definition of it. We’ve even described a *procedure* by which you can conceptually generate it. But despite that, it’s *deeply* uncomputable. There are procedures where we can compute a finite prefix of it. But there’s a limit: there’s a point beyond which we cannot compute *any* digits of it. *And* there is no way to compute that point. So, for example, there’s a very interesting paper where the authors [computed the value of Ω for a Minsky machine][omega]; they were able to compute 84 bits of it; but the last 20 are *unreliable*, because it’s uncertain whether or not they’re actually beyond the limit, and so they may be wrong. But we can’t tell!
What does Ω mean? It’s actually something quite meaningful. It’s a number that encodes some of the very deepest information about *what* it’s possible to compute. It gives a way to measure the probability of computability. In a very real sense, it represents the overall nature and structure of computability in terms of a discrete probability. It’s an amazingly dense container of information – as a n infinitely long number and so thoroughly non-compressible, it contains an unmeasurable quantity of information. And we can’t get near most of it!
[omega]: http://www.expmath.org/expmath/volumes/11/11.3/Calude361_370.pdf

Irrational and Transcendental Numbers

Leave a reply

If 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 fraction, then they go on forever. If you write them in decimal form, they go on forever without repeating. π, √2, *e* – they’re all irrational. (Incidentally, the reason that they’re called “irrational” isn’t because they don’t make sense, or because their decimal representation is crazy, but because they can’t be written *as ratios*. My high school algebra teacher who first talked about irrational numbers said that they were irrational because numbers that never repeated in decimal form weren’t sensible.)
The transcendentals are even worse. Transcendental numbers are irrational; but not only can transcendental numbers not be written as a ratio of integers; not only do their decimal forms go on forever without repeating; transcendental numbers are numbers that *can’t* be described by algebraic operations: there’s no finite sequence of multiplications, divisions, additions, subtractions, exponents, and roots that will give you the value of a transcendental number. √2 is not transcendental; *e* is.
History
———–
The first disappointment involving the irrational numbers happened in Greece, around 500 bce. A rather brilliant man by the name of Hippasus, who was part of the school of Pythagoras, was studying roots. He worked out a geometric proof of the fact that √2 could not be written as a ratio of integers. He showed it to his teacher, Pythagoras. Pythagoras was convinced that numbers were clean and perfect, and could not accept the idea of irrational numbers. After analyzing Hippasus’s proof, and being unable to find any error in it, he became so enraged that he *drowned* poor Hippasus.
A few hundred years later, Eudoxus worked out the basic theory of irrationals; and it was published as a part of Euclid’s mathematical texts.
From that point, the study of irrationals pretty much disappeared for nearly 2000 years. It wasn’t until the 17th century that people started really looking at them again. And once again, it led to disappointment; but at least no one got killed this time.
Mathematicians had come up with yet another idea of what the perfection of math meant – this time using algebra. They decided that it made sense that algebra could describe all numbers; so you could write an equation to define any number using a polynomial with rational coefficients; that is, an equation using addition, subtraction, multiple, division, exponents, and roots.
Leibniz was studying algebra and numbers, and he’s the one who made the unfortunate discovery: lots of irrational numbers are algebraic; but lots of them *aren’t*. He discovered it indirectly, by way of the sin function. You see, sin(x) *can’t* be computed from x using algebra. There’s no algebraic function that can compute it. Leibniz called sin a transcendental function, since it went beyond algebra.
Building on the work of Leibniz, Liouville worked out that you could easily construct numbers that couldn’t be computed using algebra. For example, the constant named after Liouville consists of a string of 0s and 1s where 10-i is 1 if/f there is some integer n such that n!=i.
Not too long after, it was discovered that *e* was transcendental. The main reason for this being interesting is because up until that point, every transcendental number known was *constructed*; *e* is a natural, unavoidable constant. Once *e* was shown irrational, others followed; in one neat side-note, π was shown to be transcendental *using e*. One of the properties that they discovered after the transcendence of *e* was that any transcendental number raised to a non-transcendental power was transcendental. Since eiπ is *not* transcendental – it’s 1 – then π must be transcendental.
The final disappointment in this area came soon after; Cantor, studying the irrationals, came up with the infamous “Cantor’s diagonalization” argument, which shows that there are *more* transcendental numbers than there are algebraic ones. *Most* numbers are not only irrational; they’re transcendental.
What does it mean, and why does it matter?
———————————————-
Irrational and transcendental numbers are everywhere. Most numbers aren’t rational. Most numbers aren’t even algebraic. That’s a very strange notion: *we can’t write most numbers down*.
Even stranger, even though we know, per Cantor, that most numbers are transcendental, it’s *incredibly* difficult to prove that any particular number is transcendental. Most of them are; but we can’t even figure out *which ones*.
What does that mean? That our math-fu isn’t nearly as strong as we like to believe. Most numbers are beyond us.
Some interesting numbers that we know are either irrational or transcendental:
1. *e*: transcendental.
2. π: transcendental.
3. √2 : irrational, but algebraic,
4. √x, for all x that are not perfect squares are irrational.
5. log23 is irrational.
6. Ω, Chaitin’s constant, is transcendental.
What’s interesting is that we really don’t know very much about how transcendentals interact; and given the difficulty of proving that something is transcendental, even for the most well-known transcendentals, we don’t know much of what happens when you put them together. π+*e*; π×*e*; ππ; *e**e*, πe are all numbers that we *don’t know* if they;’re transcendental. In fact, for π+*e* (and π-*e*) we don’t even know if it’s *irrational*.
That’s the thing about these number. We have *such* a weak grasp of them that even things that seem like they should be easy and fundamental, *we don’t know how to do*.

Something Nifty: A Taste of Simple Continued Fractions

Leave a reply

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. 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 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?
One of the answers is something called *continued fractions*. A continued fraction is a very neat thing. Here’s the idea: take a number where you don’t know it’s fractional form. Pick the nearest simple fraction 1/n that’s just a *little bit too large*. If you were looking at, say, 0.4, you’d take 1/2 – it’s a bit bigger. Now – you’ve got an approximation, but it’s too large. So that means that the demoninator is *too small*. So you add a correction to the denominator to make it a little bit bigger. And you just keep doing that – you approximate the correction to the denominator by adding a fraction to the denominator that’s just a little too big, and then you add a correction to *that* correction.
Let’s look at an example: 2.3456
1. It’s close to 2. So we start with 2 + (0.3456)
2. Now, we start approximating the fraction. The way we do that is we take the *reciprocal* of 0.3456 and take the integer part of it: 1/0.3456 rounded down is 2 . So we make it 2 + 1/2; and we know that the denominator is off by .3088.
3. We take the reciprocal again, and get 3, and it’s off by .736
4. We take the reciprocal again, and get 1, and it’s off by 0.264
5. Next we get 3, but it’s off by 208/1000
6. Then 4, off by 0.168
7. Then 5, off by .16
8. Then 6, off by .25
9. Then 4, off by 0; so now we have an exact result.
So as a continued fraction, 2.3456 looks like:
continued.jpg
As a shorthand, continued fractions are normally written using a list notation inside of square brackets: the integer part, following by a semicolon, followed by a comma-separated list of the denominators of each of the fractions. So our continued fraction for 2.3456 would be written [2; 2, 3, 1, 3, 4, 5, 6, 4].
There’s a very cool visual way of understanding that algorithm. I’m not going to show it for 2.3456, because it’s a bit too much… But let’s look at something simpler: let’s try to write 9/16ths as a continued fraction. Basically, we make a grid consisting of 16 squares across by 9 squares up and down. We draw the *largest* square we can on that grid. The number of squares of that size that we can draw is the first digit of the continued fraction. Now there’s a rectangle left over: we draw the largest squares we can, there. And so on:

square-continued.jpg

So the continued fraction for 9/16ths is [0; 1, 1, 3, 2].
Using continued fractions, we can represent *any* rational number in a finite-length continued fraction.
One incredibly nifty thing about this way of writing numbers is: what’s the reciprocal of 2.3456, aka [2; 2, 3, 1, 3, 4, 5, 6, 4]? It’s [0; 2, 2, 3, 1, 3, 4, 5, 6, 4]. We just add a zero to the front as the integer part, and push everything else one place to the right. If it was a zero in front, then we would have removed the zero and pulled everything else one place to the left.
Irrational numbers are represented as *infinite* continued fractions. So there’s an infinite series of correction fractions. You can understand it as a series of every-improving approximations of the value of the number. And you can define it using a recurrence relation (that is, a recursive formula) for how to get to the next digit.
For example, π = [3; 7, 15, 1, 292, 1, …]. If we work that out, the first six places of the continued fraction for pi work out in decimal form to 3.14159265392. That’s correct to the first *11* places in decimal. Not bad, eh?
A very cool property of the continued fractions is: square roots written as continued fractions *always repeat*. Even cooler? What’s the square root of two as a continued fraction? [1; 2, 2, 2, 2, …. ].

e – the Unnatural Natural Number

Leave a reply

Looks 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?
————
*e* is a transcendental irrational number. It’s roughly 2.718281828459045. It’s also the base of the natural logarithm. That means that by definition, if ln(x)=y, then *e*y=x. Given my highly warped sense of humor, and my love of bad puns (especially bad *geek* puns) , I like to call *e* the *unnatural natural number*. (It’s natural in the sense that it’s the base of the natural logarithm; but it’s not a natural number according to the usual definition of natural numbers. Hey, I warned you that it was a bad geek pun.)
But that’s not a sufficient answer. We call it the *natural* logarithm. Why is that bizzare irrational number just a bit smaller than 2 3/4 *natural*?
Take the curve y=1/x. The area under the curve from 1 to n is the natural log of n. *e* is the point on the x axis where the area under the curve from 1 is equal to one:
ln.jpg
It’s also what you get if you you add up the reciprocal of the factorials of every natural number: (1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …)
It’s also what you get if you take the limit: *lim*n → ∞ (1 + 1/n)n.
It’s also what you get if you work out this very strange looking series:

2 + 1/(1+1/(2+2/(3+3/(4+..))))

It’s also the base of a very strange equation: the derivative of *e*x is… *e*x.
And of course, as I mentioned yesterday, it’s the number that makes the most amazing equation in mathematics work: *e*iπ=-1.
Why does it come up so often? It’s really deeply fundamental. It’s tied to the fundamental structure of numbers. It really is a deeply *natural* number; it’s tied into the shape of a circle, to the basic 1/x curve. There are dozens of different ways of defining it, because it’s so deeply embedded in the structure of *everything*.
Wikipedia even points out that if you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly *e* dollars. (That’s not too suprising; it’s just another way of stating the integral definition of *e*, but it’s got a nice intuitiveness to it.)
History