The 15 Most Famous
Transcendental Numbers
In 1844, math genius Joseph Liouville (1809-1882) was the first to
prove the existence of transcendental numbers.
Hermite proved that the number e was transcendental in 1873.Lindeman
proved that pi was transcendental in 1882. In 1882, German mathematician F.
Lindemann proved that pi
is transcendental. But are there other famous transcendental numbers. I made a
list of the fifteen most famous transcendental numbers.
1.
pi = 3.1415 ...
2.
e = 2.718 ...
3.
Euler's constant, gamma
= 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n -
ln(n)) (Not proven to be transcendental, but generally believed to be by
mathematicians.)
4.
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.)
5.
Liouville's number
0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th,
etc. places and zeros elsewhere.
6.
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.)
7.
Chapernowne's number,
0.12345678910111213141516171819202122232425... This is constructed by
concatenating the digits of the positive integers. (Can you see the pattern?)
8.
Special values of the
zeta function, such as zeta (3). (Transcendental functions can usually be
expected to give transcendental results at rational points.)
9.
ln(2).
10.
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.)
11.
epi
12.
pie (Not proven to be transcendental, but generally
believed to be by mathematicians.)
13.
Morse-Thue's number,
0.01101001 ...
14.
ii = 0.207879576... (Here i is the imaginary number
sqrt(-1). Isn't this a real beauty? How many people have actually considered
rasing 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: iiis
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...
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...
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...
15.
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.)
