What is a transcendental number, examples and properties?


Happy new year to all my readers and followers.

Today, i will be celebrating the new year with you by writing on the transcendental numbers, their examples and properties. Although i have written on one of transcendentals in the past e.g Why does pi have a constant value of 3.14159265359... 

Transcendental numbers are simply non algebraic numbers, meaning, numbers that are not the root of any non-zero polynomial with rational coefficient.
The best known transcendental numbers are $\pi$ and $e$.
The history of transcendentals can be traced to Gottfried Leibniz in his paper in 1682 where he proved that $\sin{x}$ is not an algebraic function of $x$. Johann Lambert also proposed that $e$ and $\pi$ are transcendental numbers. John Liouville in 1851 also proved the existence of transcendental in his Liouville constant $\sum^\infty_{n=1}10^{-n}$.
In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers.
In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of $\pi$. He first proved that $e^a$ is transcendental when $a$ is any non-zero algebraic number. Then, since $e^{i\pi}=−1$ is algebraic then $i\pi$ must be transcendental. But since $i$ is algebraic, $\pi$ therefore must be transcendental.
In as much as only a few cases of transcendental numbers are known, almost all the real and complex numbers are transcendental since the algebraic numbers compose a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. We must also note that all real transcendental numbers are irrational numbers, since all rational numbers are algebraic while on the other hand not all irrational numbers are transcendental. e.g the root of the polynomial equation $x^2-2=0$ is a square root of $2$ i.e $\sqrt{2}$ which is an irrational number but not a transcendental number.

GENERAL PROPERTIES OF THE TRANSCENDENTAL NUMBERS.

  1. The set of transcendental numbers is uncountably infinite. e.g $\pi$ and $e$.
  2. No rational number is transcendental and all real transcendental numbers are irrational.
  3. Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. e. g $5\pi,\frac{\pi-3}{\sqrt{2}},(\sqrt{\pi}-\sqrt{3})^8$.
  4. All Liouville numbers are transcendental, but not all all transcendental numbers are Liouville numbers.
  5. The non-computable numbers are a strict subset of the transcendental numbers.

SOME NUMBERS THAT HAVE BEEN PROVEN TO BE TRANSCENDENTAL.

  1.  Lindemann–Weierstrass theorem: for any non-zero and algebraic $a$, then $e^a$ is a transcendental.
  2. Lindemann–Weierstrass theorem: shows that $\pi$ is a transcendental.
  3. Gelfond–Schneider theorem: if $a$ is algebraic and $a\neq{0,1}$ and $b$ is irrationally algebraic then $a^b$ is transcendental. e.g $2^\sqrt{2}$.
  4.  Gelfond–Schneider theorem: $e^\pi$ is transcendental as well as $e^\frac{\pi}{2}=i^i$.
  5.  Lindemann–Weierstrass theorem: for any nonzero algebraic number $a$, expressed in radians, $\sin{a}, \cos{a},\tan{a},\csc{a},\sec{a},\cot{a}$ and their hyperbolic counterparts are all transcendental.
  6. Gelfond-Schneider theorem: if $a$ and $b$ are positive integers not both powers of the same integer then $\log_b{a}$ is a transcendental.
  7. Gelfond-Schneider theorem: the square super-root of any natural number is either an integer or transcendental, $\sqrt{X_s}$ is transcendental.
  8.  the gamma functions $\Gamma(1/3),\Gamma(1/4),\Gamma(1/6)$ are all transcendental.
  9.  The Cahen's constant $0.64341054629...,$ is a transcendental.
  10.  The Chaitin's constant $\Omega$ (since it is a non-computable number).
  11.  The Gauss constant is a transcendental. 
  12. The two lemniscate constants $L_1$ and and $L_2$ are transcendental.
  13.  The Prouhet–Thue–Morse constant is a transcendental. 
  14.  The Komornik-Loreti constant is a transcendental.