Why do mathematicians study transcendental numbers?

[Marm]

Numbers such as π and e are transcendental and they have loads of uses in science and economics. Is this why mathematicians are still studying them, so that they can discover more numbers like these ones?

[Xei]

Do you understand what transcendental means? pi was first conceptualised thousands of years before the concept of transcendental numbers was developed, indeed, far before the idea would have even made any sense; e is more modern, but still predates the concept by at least a century (I'm guesstimating). They emerged by themselves for the purposes for which they were needed, not by a conscious effort to find new, useful numbers; this doesn't really make any sense. Most of the 'study' mathematicians do about the transcendental numbers e and pi isn't actually into their property of being transcendental, a bit like how most of the study a zoologist does into mammals isn't about their property of being furry. pi and e occur literally all the time in applied mathematics; they are almost as ubiquitous as multiplication. You wouldn't be able to do most science without them, so they're clearly vital. Mathematicians do not always study things because they are applicable though; pure mathematics, usually by definition, has no practical application. It's an exercise of pure thought. The idea of transcendental numbers is a pure one, and as far as I can see will have no practical application, despite this area being intellectually interesting. One heuristic reason to help you see why is that measurements in science are always approximations, and the distinction between transcendentals and non-transcendentals is infinitely small and only really exists in the realm of thought. Whenever you assign something in nature a number, it is impossible to know if that number is really transcendental or not, because that would require infinite accuracy, which is impossible. Instead we just measure whichever decimal number (decimal numbers effectively being capable of expressing infinite accuracy) the number seems to be. It doesn't matter if a number in nature is transcendental (if that even makes sense) because nature is continuous; the difference between transcendentals and non-transcendentals is infinitely small, and hence the difference in any effects of these numbers will also be infinitely small.

[Marm]

Thanks for the response. That cleared up some misunderstanding I had about this topic. From what I understand, a transcendental number is a number that is not a solution of any algebraic solution. Or are their further definitions of transcendence? Also, can you please explain to me how Georg Cantor proved that the set of transcendental numbers is uncountable. I don't really understand the proof on wikipedia Cantor's diagonal argument - Wikipedia, the free encyclopedia and here Cantor Diagonal Method -- from Wolfram MathWorld

[DrunkenArse]

The definition of a transcendental number is that it's not algebraic, i.e. that it does not occur as the root of a polynomial with rational coefficients. If a number is the root of a polyynomial with rational coefficients, i.e. if f(x) = 0 then you can just multiply across by any common multiple of the denominators of the coefficients, call it c, and obtain the polynomial cf(x) = c*0 = 0 that has integer coefficients. For example f(x) = 1/2x^2 + 1/3x + 1/5 turns into 30*f(x) = 30*(1/2x^2 + 1/3x + 1/5) = 15x^2 + 10x + 6. So it's usual to define a transcendental number to be a number that doesn't occur as the root of any polynomial equation with integer coefficients. Cantor proved that the transcendental numbers are not countable by proving that the algebraic numbers are countable. What's left over after the algebraic numbers are removed is the transcendentals. Because the reals are not countable, the transcendentals must not be countable either. Essentially, all you have to do to carry out the proof is to prove that the reals are not countable and then count the algebraic reals.

[Xei]

Thanks for the response. That cleared up some misunderstanding I had about this topic. From what I understand, a transcendental number is a number that is not a solution of any algebraic solution. Or are their further definitions of transcendence? Yep that's totally right, although obviously to be complete we have to specify that the coefficients are integers; otherwise e wouldn't be transcendental because it's a root of x - e = 0, for example. In a sense it's an extension of irrational numbers. Irrational numbers cannot be written as a/b; this is equivalent to saying they are not the root of any first degree polynomial, specifically bx - a = 0. Transcendental numbers are the same but it's for any degree. Also, can you please explain to me how Georg Cantor proved that the set of transcendental numbers is uncountable. I don't really understand the proof on wikipedia Cantor's diagonal argument - Wikipedia, the free encyclopedia and here Cantor Diagonal Method -- from Wolfram MathWorld That's not so hard once you understand what countability is. Once you've got the basic idea, hopefully you can build on it with relative ease via reading. Two sets A and B have the same size if you can make a function that maps each element of A to a unique element in B, in other words a one to one mapping (the technical word you will see for this type of function is 'bijection'). A simple analogy: counting ten sheep. You map sheep one to the number 1, sheep two to the number 2, etcetera, and you end up with a one to one mapping between A = {the sheep} and B = {1, 2, ... , 10}. Then we say the size of the set of sheep is 10. It's the same with infinite sets. The most basic infinite set is the naturals, {1, 2, 3, ... }. We say a set is countable if it has the same size (by the above definition) as the naturals. For instance, the integers, {..., -2, -1, 0, 1, 2, ...}. If we zig zag back and forth, we can make a function that sends 0 to 1, 1 to 2, -1 to 3, 2 to 4, -2 to 5, 3 to 6, -3 to 7, and so on. The important thing is that this is one-to-one; each integer now has an associated natural number, and each natural number an associated integer. Thus the size of the set of integers is the same as the size of the naturals, in other words, the integers are countable. How about all natural coordinates, (n, m)? These are also countable. A neat way to do it is to send each pair to the natural number (2^n)*(3^m). The crucial thing is that it never maps two different pairs to the same natural number (because prime factorisations are unique). I brought up this example because it isn't actually one-to-one, but it's still okay. It's not one-to-one because some naturals don't get mapped to (for instance, 5) by any pair. However, heuristically, we know that the set of pairs of naturals is clearly infinite, so it must be at least as big as the naturals; and this mapping has managed to 'fit them' into the naturals, with extra room in fact, so it can't be any bigger than the naturals. Therefore it's the same size. This illustrates that to demonstrate countability, you just have to map your set to the naturals in a way that doesn't send two elements to the same natural (the technical word is 'injection'); it doesn't actually have to be one-to-one, as long as you fit your set inside. The rationals are also countable. The reals however are not, which is quite an astonishing fact (that some infinities are bigger than others). I've explained why before, you can see here. Clearly the reals are just the combination of two sets: the transcendentals and the non-transcendentals. As the non-transcendentals can be counted (a good challenge is to try to prove this roughly yourself; start by considering if the information about the coefficients can be counted), the transcendentals cannot; if they could be, then the non-transcendentals and transcendentals together could be counted, i.e. all the real numbers could be counted, which is not true. Intuitively, this means that almost all real numbers, if you pick one at random, are transcendental; algebraic numbers are, in a sense, 'rare'.

[sloth]

-gives wedgies to all of you, takes your lunch monies, and shoves you into your respective lockers.-

[Xei]

-fires you-

[Invader]

So why is it that pi's value came from the ratio of a circle's circumference to its diameter and not radius? Instead of 3.1415... we'd have 6.1830... And the equivalent angle for 360 degrees in radians would be pi, instead of 2pi. Seems cleaner that way. Is there a reason or is it ambiguous?

[Dannon Oneironaut]

How do mathematicians earn money? I can understand if you are a scientist or doing something useful with the math, but how do pure mathemeticians earn money? Do donors and investors say "Here is $10,000 if you can solve this equation, or create a cooler one?"

[Marm]

Are there any specific example where the research of transcendental numbers provided insight to other aspects of mathematics? (besides pi and e)

[sloth]

-fires you- Damn it.

[khh]

So why is it that pi's value came from the ratio of a circle's circumference to its diameter and not radius? Instead of 3.1415... we'd have 6.1830... And the equivalent angle for 360 degrees in radians would be pi, instead of 2pi. Seems cleaner that way. Is there a reason or is it ambiguous? π is used in more places than 2π. For instance the area of a circle is A = πr2. Thus it's simpler to have π mean π. edit: Also, 2π is easier to write than π/2

[Xei]

So why is it that pi's value came from the ratio of a circle's circumference to its diameter and not radius? Instead of 3.1415... we'd have 6.1830... And the equivalent angle for 360 degrees in radians would be pi, instead of 2pi. Seems cleaner that way. Is there a reason or is it ambiguous? Nah, it's just convention. Back in the time of the Greeks no doubt the more natural measure to take was all the way across the circle. When coordinates were developed and we started thinking of a circle as the path described by rotating the radius once around a point, 2pi would probably have been more natural, but it's just not worth the confusion. pi does turn up in aesthetically nice places without a 2 though. One of my favourite results is that the volume over the whole plane of a bell curve (e^-x^2) rotated around x = 0 is pi. It follows from this that the area over the whole x axis of e^-x^2 is sqrtpi. Also of course there is Euler's famous equation, e^i*pi + 1 = 0. How do mathematicians earn money? I can understand if you are a scientist or doing something useful with the math, but how do pure mathemeticians earn money? Do donors and investors say "Here is $10,000 if you can solve this equation, or create a cooler one?" Universities. They are paid to research into pure, and lecture both pure and applied. Are there any specific example where the research of transcendental numbers provided insight to other aspects of mathematics? (besides pi and e) I can't think of any other common numbers which are transcendental, but like I said, being transcendental is a fairly useless property and will probably not give insights into any other field except its own.

[Xei]

So why is it that pi's value came from the ratio of a circle's circumference to its diameter and not radius? Instead of 3.1415... we'd have 6.1830... And the equivalent angle for 360 degrees in radians would be pi, instead of 2pi. Seems cleaner that way. Is there a reason or is it ambiguous? Nah, it's just convention. Back in the time of the Greeks no doubt the more natural measure to take was all the way across the circle. When coordinates were developed and we started thinking of a circle as the path described by rotating the radius once around a point, 2pi would probably have been more natural, but it's just not worth the confusion. pi does turn up in aesthetically nice places without a 2 though. One of my favourite results is that the volume over the whole plane of a bell curve (e^-x^2) rotated around x = 0 is pi. It follows from this that the area over the whole x axis of e^-x^2 is sqrtpi. Also of course there is Euler's famous equation, e^i*pi + 1 = 0. How do mathematicians earn money? I can understand if you are a scientist or doing something useful with the math, but how do pure mathemeticians earn money? Do donors and investors say "Here is $10,000 if you can solve this equation, or create a cooler one?" Universities. They are paid to research into pure, and lecture both pure and applied. Are there any specific example where the research of transcendental numbers provided insight to other aspects of mathematics? (besides pi and e) I can't think of any other common numbers which are transcendental, but like I said, being transcendental is a fairly useless property and will probably not give insights into any other field except its own.

[Marvo]

Interestingly, some "pure math" that didn't seem to have any practical applications, has with the development of electronics actually seen use. I believe the modulo operation is an example of this. Primes too.

[Xei]

I hear that said a lot, but that stuff is so basic it doesn't really make sense to call it pure or applied. I guess you're talking about things like RSA encryption, but that stuff is so simple it's only one step up from multiplication... we'd covered all that stuff within the first month of my maths course. It's based on obvious things like, 'if you add two numbers with remainders x and y after division by q, you'll get a number with remainder x + y after division by q'. Hardly in the spirit of pure mathematics research.

[Marvo]

That is true, it's all very basic. I was thinking of RSA encryption, yes. Modulo is also good for a lot of other things in programming though. It's still stuff that people figured out though, without having any real use of it, until thousands of years later.