*Mark Herkommer
February 18, 2014
*

Continued fractions, an apparatus for representing numbers, has been used since the 16th century, and quite possibly earlier. Many types of common numbers can be represented by "simple" continued fractions. All rational numbers can be represented by a finite simple continued fraction. For example

13,020 / 5,797 = [2, 4, 15, 3]

Algebraic numbers, such as the square root of 3 or the golden ratio
(*phi*) are representated by perodic infinite simple continued fractions.

sqrt(3) = [ 1, 1, 2, 1, 2, 1, 2, ... ]

*phi* = [ 1, 1, 1, 1, 1, 1, 1, 1, ... ]

Transcendental numbers are a more complex class of numbers. Numbers such as
*e*, the natural logarithm base, can be expressed as non-periodic
infinite simple continued fractions. Although it is non-periodic, it can be
expressed in a closed form.

*e* = [ 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ... ] .

*pi*, another transcendental number, can be expressed as non-periodic
infinite simple continued fraction. Unlike *e*, it cannot be
representated in a closed form as a simple continued fraction.

*pi* = [ 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, ... ]

However, *pi* **can** be expressed in closed form using a more general form
of continued fraction. Thus, in "some" way,
*pi* is different, more complex, more transcendental, than *e*.

Using the continued fraction apparatus we can see that all real numbers can
be classified according to their demands upon the apparatus. Rationals are the
simplest, followed by algebraic numbers, next are transcendental numbers of
the first kind (e.g., *e*), then transcendental numbers of the second
kind (e.g., *pi*).

Constructed from a sequence of prime numbers, the Herkommer Number is more
transcendental than *pi* ; thus making it a transcendental number of the
third kind. Why? By its definition, the Herkommer Number *cannot* be
expressed in any closed form and therefore has a higher order of
transcendentalism than *pi*.

Incidently, *h* = 2.53602 70816 89339 (approximately).

Behold, the majesty and mystery of

*© Copyright, 1999, 2003, 2014. All Rights Reserved.*
*Mark A. Herkommer*