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
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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?
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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*.
Good Math/Bad Math
The beauty of math; the humor of stupidity.
I’ve never understood: in what sense are there more transcendentals than non-transcendentals? I’m guessing that this would have been answered in that measure theory course I never took…
Corkscrew: maybe in measure theory, but definitely in set theory. There are only countably many algebraic numbers; that is, they can be put in one-to-one correspondence with the natural numbers. The set of real numbers is uncountable; it’s infinite, but not countable, so it’s strictly larger. What’s more, the union of two countable sets is countable, so the set of transcendentals also must be uncountable. It’s in that strong sense that there are more transcendentals than algebraic numbers.
It’s true that transcendental numbers can’t be described by finitely many algebraic operations; however, there are also algebraic (nontranscendental) numbers having the same property. By Galois theory, such algebraic numbers have degree at least five. Wikipedia has a decent exposition.
In the font my browser displays GMBM in, √2 shows up looking like “the integral of 2”. 🙁
Think I’m getting a grasp of why it’s impossible to “square the circle,” now.
However, in the end, math always proves to be far more interesting, deep, and rich that whatever our feeble minds had imagined.
Cantor’s diagonalization argument wasn’t his first proof of the uncountability of the reals. It was just the first one readily accessible to dilettantes such as myself.
Rationals are countable, and the leftovers are called irrational. A countable set of irrationals are algebraic, and the rest a transcendental. A countable set of transcendentals are computable, and the rest are… uncomputable, I guess. A countable set of uncomputable numbers are definable (e.g., Chaitin’s constant), so undefinable real numbers make up the bulk of all real numbers.
I would definitely recommend Edward Burger’s “Making Transcendence Transparent” for anyone with a reasonable bit of math background who is interested in these issues. http://www.amazon.com/exec/obidos/tg/detail/-/0387214445 It’s very well-written and motivated, though if you don’t work through the numerous “challenges” you probably won’t follow the actual proofs.
“Since e^iπ is not transcendental – it’s 1”
Shouldn’t that be -1?
re: Chris Nelson
I don’t know what your setup is, but I was having the same problem in FireFox on XP. I’ve got MathML installed for some normally good math browsing, but I had no idea what √2 was supposed to be by just looking at it. I had to actually paste it into Notepad to see it correctly. Perhaps someone knows how to fix this?
Amos, what character encoding do you have set? It seems to make a difference. Unicode (UTF-8) works for me.
Observation: Math and physics have a common point here. In math we don’t know how to write down most of the numbers, but we know they are there. In physics we know that most of the mass of the universe (>90% “black mass”) is something we have no clue what is, but we know it is there.
Good there is still some science to be done…
Davis, I was already using UTF-8 and my other choices aren’t doing anything to help. My fonts are default. But it looks fine in IE… a strange little problem.
Chris, Amos, Davis et al:
I’m not using MathML; I haven’t had good luck with it in experiments in the past. I try to stick to HTML character entities that are *supposed* to work in firefox. Sorry for the problem with the radic.
I may try experimenting with MathML again sometime soon; later this month, after I wrap up category theory, I’m going to do some stuff about topology, and I expect it’ll be a lot easier *if* I can get MathML to work acceptably. (It’s either MathML, or pasting image files for all of the equations.)
“Think I’m getting a grasp of why it’s impossible to “square the circle,” now.”
Agreed 🙂
nice exposition. it’s sad how many people don’t understand what “irrational” means. in the whole .999…=1 fiasco on my blog, many, many commenters said that it can’t be true because .999… isn’t rational!
e^(i*pi) is -1 not 1
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I have listed Infinitudes [!!] of transcendental numbers as convergent sequences. They can be proved to be transcendental using liouville’s & mahler’s criteria etc.
I have also generalized the idea of Polynomials to Complete Labyrinth Functions! This has profound repurcussions in all branches of Mathematics.
Cantor’s diagonalization argument is a moronic error !
Big Infinity & small infinity !! how moronic one can be!
Two raised to aleph naught = aleph naught ~