What is超越數?

In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. The most prominent examples of transcendental numbers are π and e.

Although transcendental numbers are never rational, some irrational numbers are not transcendental: the square root of 2 is irrational, but it is a solution of the polynomial x2 − 2 = 0, so it is algebraic.

The transcendental numbers are uncountable. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. But Cantor's diagonal argument proves that the reals (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable. In a very real sense, then, there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult.

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超越數 is Transcendental Numbers which is a subset (but not a sub-field )of complex numbers. To know what is transcendental number, we should know what is Algebraic Numbers.
Definition: If an number is algebraic, then it is a root of non-zero irreducible polynomial with rational (or integral ) coefficient(s).
Definition: A complex number is transcendental if it is not algebraic.

Note: Algebraic Numbers found a sub-field of complex numbers.

I just end with a list of references here.

http://books.google.com/books?vid=ISBN0521286549&id=hAFfb8BBC0IC&dq=Alan+Baker
http://books.google.com/books?vid=ISBN052139791X&id=SmsCqiQMvvgC&dq=Alan+Baker
http://books.google.com/books?vid=ISBN0486495264&id=eR4XSdzTECUC&dq=Transcendental+and+algebraic+numbers
http://mathworld.wolfram.com/TranscendentalNumber.html
http://www.math.ust.hk/~makryu/

超越數是不能滿足任何整系數代數方程的數。這即是超越數是代數數的相反，也即是說若 x 是一個超越數，那麼對於任何整數 都符合：


超越數的例子包括：

劉維爾 (Liouville) 常數：
它是第一個確認為超越數的數，是於 1844年劉維爾發現的。
e
ea，其中 a 是代數數。
π
eπ
。
更一般地，若 a 為零和一以外的任何代數數及 b 為無理代數數則 ab 必為超越數。希爾伯特第七問題便是問若 b 只是無理數那麼 ab 是否也是超越數。此問題到目前為止還未解決。
sin 1
ln a，其中 a 為非一正有理數。
Γ (1/3) 、 Γ(1/4) 及 Γ (1/6)（參見伽傌函數）。
所有超越數構成的集是一個不可數集。這暗示超越數遠多於代數數。可是，現今發現的超越數極少，因為要證明一個數是超越數或代數數是十分困難的。

超越數的發現令一些古代尺規作圖問題的不可能性得以證明。這包括著名的化圓為方問題，因 π 是超越數而被確定為不可能的了。


***sorry!!!no english!!!***