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 Euler’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}}.

Hilbert’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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