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
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Mark A. Herkommer
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