Transcendental
Numbers
(Some of the more
interesting)
Clear cut and paste job.
- pi = 3.1415 ...
- e = 2.718 ...
- Euler's constant,
gamma = 0.577215 ...
= (1 + 1/2 + 1/3
+ 1/4 + ... + 1/n - ln(n))
(Not proven to be transcendental, but generally believed to be
by mathematicians.)
- Catalan's constant,
G = sum (-1)**k / (2k + 1 )**2 =
1 - 1/9 + 1/25 - 1/49 + ...
(Not proven to be transcendental, but generally believed to be
by mathematicians.)
- Liouville's number
0.110001000000000000000001000 ...
which has a one in the 1st, 2nd, 6th, 24th, factorials, etc. places and
zeros
elsewhere.
- Chaitin's "constant", the probability that a random
algorithm halts. (Noam Elkies of Harvard notes that not only is this
number
transcendental but it is also incomputable.)
- Chapernowne's
number, 0.12345678910111213141516171819202122232425...
This is constructed by concatenating the digits of the positive
integers.
(Can you see the pattern?)
- Special values of the zeta function, such as
zeta (3).
(Transcendental functions can usually be expected to give
transcendental results at rational points.)
- ln(2).
- Hilbert's number,
2(sqrt 2 ).
(This is called Hilbert's number because the proof of whether or
not it is transcendental was one of Hilbert's
famous problems.
In fact, according to the Gelfond-Schneider theorem, any number
of the form
ab
is transcendental where
a
and
b
are algebraic
(a ne 0, a ne 1 )
and
b
is not a rational number.
Many
trigonometric or hyperbolic functions of non-zero algebraic numbers
are transcendental.)
- epi
- pie
(Not proven to be transcendental, but generally believed to be
by mathematicians.)
- Morse-Thue's number, 0.01101001 ...
- ii = 0.207879576... (Here
i
is the imaginary number
sqrt(-1). Isn't this a real beauty? How many people have actually
considered raising i to the i power?
If
a
is algebraic and
b
is algebraic but irrational
then
ab
is transcendental. Since
i
is algebraic but irrational, the theorem applies.
Note also:
ii
is equal to
e(- pi / 2 )
and several other values.
Consider
ii = e(i log i ) =
e( i times i pi / 2 ) .
Since log is multivalued, there are other possible values for
ii.
Here is how you can compute the value of ii = 0.207879576...
1. Since e^(ix) = Cos x + i Sin x, then let x =
Pi/2.
2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90
deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i.
3. Therefore e^(iPi/2) = i.
4. Take the ith power of both sides, the right side being i^i and the
left side = [e^(iPi/2)]^i = e^(-Pi/2).
5. Therefore i^i = e^(-Pi/2) = .207879576...
- Feigenbaum numbers, e.g.
4.669 ... .
(These are related to properties of dynamical systems with
period-doubling. The ratio of successive differences between
period-doubling bifurcation parameters approaches the number
4.669 ... ,
and it has been discovered in many physical systems before
they enter the chaotic regime. It has not been proven
to be transcendental, but is generally believed to be.)
Quiz 3: Are there more
transcendental numbers than integers, less, same?
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