Top Ten Transcendental Numbers

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Here are the top ten transcendental numbers, as put together by Dave Richeson.

1.

${\displaystyle 0.1100010000000000000000010\ldots= \sum_{k=1}^{\infty}\frac{1}{10^{k!}}}$

(Liouville, 1851): the first known transcendental number not expressed as a continued fraction.

2.

${e}$

(Hermite, 1873): the first non-contrived example of a transcendental number.

3.

${\pi}$

(Lindeman, 1882): use the Lindemann-Weierstrass theorem (below) and Eulerâ€™s identity,

${e^{\pi i}=-1}$

This showed that it is impossible to square the circle.

Lindemann-Weierstrass Theorem (1882/1885). If ${a_{1},a_{2},\ldots,a_{m}}$ are distinct algebraic numbers and ${b_{1},\ldots,b_{m}}$ are nonzero algebraic numbers, then ${b_{1}e^{a_{1}}+b_{2}e^{a_{2}}+\cdots+b_{m}e^{a_{m}}\ne 0}$.

4.

${\sin(1)}$

use the Lindemann-Weierstrass theorem and the fact that

${\displaystyle \sin z=\frac{e^{iz}-e^{-iz}}{2i}}$

5.

${\ln(2)}$

use the Lindemann-Weierstrass theorem and the fact that ${\ln x}$ is the inverse function for ${e^{x}}$.

Hilbertâ€™s 7th problem (1900). If ${a}$ and ${b}$ are algebraic numbers with ${a\ne0,1}$ and ${b}$ not rational, then ${a^{b}}$ is transcendental.

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