Why are real numbers usually split into Rational/Algebraic/Transcendental?

[deluks917]

I think its fairly obvious to most people why a number being rational (or not) is extremely important. But I honestly do not see why being transcendental is as interesting of a property (though its clearly somewhat interesting). What interesting applications are there of knowing a number is transcendental, not just algebraic?

Talking to many "non-math" people they often seem to confuse a number being transcendental with it being normal. That is, in any basis b, the natural density of each of the b values has natural density 1/b. (informally each value is "equally likely"). Of course they should get their definitions right. But I am fairly sympathetic to their mistake. A number being normal gives a lot of intuition about the number.

Btw I am a phd student in math. Studying PDE. I have a reasonably good grasp of analysis but honestly not the best understanding of abstract algebra.

 

[SteamKing]

I think its fairly obvious to most people why a number being rational (or not) is extremely important. But I honestly do not see why being transcendental is as interesting of a property (though its clearly somewhat interesting). What interesting applications are there of knowing a number is transcendental, not just algebraic?

Ask your colleagues who are number theorists.

Talking to many "non-math" people they often seem to confuse a number being transcendental with it being normal. That is, in any basis b, the natural density of each of the b values has natural density 1/b. (informally each value is "equally likely"). Of course they should get their definitions right. But I am fairly sympathetic to their mistake. A number being normal gives a lot of intuition about the number.

Your definition of a 'non-math' person seems to be at variance with what would generally be expected. I don't think the average person walking down the street could give the definition of a 'normal' number as you have described it.

Btw I am a phd student in math. Studying PDE. I have a reasonably good grasp of analysis but honestly not the best understanding of abstract algebra.

Rational numbers and algebraic numbers exhibit certain properties, which can be used in different areas of math. This is not to say that a rational number cannot be algebraic, and vice versa. Proving that a certain number is irrational, for example, by initially hypothesizing that the number is rational, and then going thru a proof sequence which results in a contradiction, is a classic number theory example.

 

[Matterwave]

Rational numbers and algebraic numbers exhibit certain properties, which can be used in different areas of math. This is not to say that a rational number cannot be algebraic, and vice versa. Proving that a certain number is irrational, for example, by initially hypothesizing that the number is rational, and then going thru a proof sequence which results in a contradiction, is a classic number theory example.

Not only can rational numbers be algebraic, indeed all rational numbers are algebraic, they solve the algebraic (linear) equation $ax+b=0$ for some integers $a,b$. There are many algebraic numbers which are not rational; however, such as $\sqrt{2}$ which solves the algebraic equation $x^2-2=0$.

@OP: I'm not even a mathematician, so I don't think I can give you much more insight than you already have, but I was told that transcendental numbers are interesting because in some sense they can't be "constructed" out of integers with the normal - everyday operations that we are familiar with (addition subtraction multiplication and division) like the algebraic numbers can be. I don't know if that makes these numbers interesting to you but it does make it interesting to some people at least.

 

[SteamKing]

Not only can rational numbers be algebraic, indeed all rational numbers are algebraic, ...

You are correct, of course. I slipped that sentence about rationals and algebraic numbers in my post without reading it carefully.

One other famous relation which links four key numbers together is, of course, the famous Euler's identity, which links the numbers e, i, pi, one and zero.

http://en.wikipedia.org/wiki/Euler's_identity

Here, you have the imaginary unit, two transcendental numbers, and the integers zero and one all linked. Transcendental numbers would get plenty of scrutiny for this fact alone.

 

[homeomorphic]

Here's one motivation:

http://en.wikipedia.org/wiki/Squaring_the_circle

Another thing is just that it's somewhat surprising that algebraic numbers are so special. They are countable, so "almost every" real number is transcendental.

I'm not sure transcendental numbers are terribly important, overall, though.

 

[platetheduke]

The significance of a number being transcendental is specifically that it bears no algebraic relation to the integers (or rationals). While the transcendentals are not a big area of study in their own right, proving that a number is transcendental is significant. For instance, proving that $e$ is transcendental tells us that $e$ cannot be described in algebraic closed form. While that is disappointing (because it makes math harder), it also prevents people from wasting time on unsolvable problems.

Algebraic numbers are nice because they satisfy polynomial identities (by definition).

Rational numbers are nice because we can calculate with them easily without rounding and they are dense in the reals.