The 15 Most Famous Transcendental Numbers
Cliff Pickover.
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I am in love with the mysterious transcendental numbers.
Did you know that there
are "more" transcendental numbers than the more familiar
algebraic ones? Even so, only a few classes of transcendental numbers are widley known
to humans, and it's very difficult to prove that a particular
number is transcendental.
In 1844, math genius Joseph Liouville (18091882)
was the first to prove the existence of transcendental
numbers.
(More precisely, he was the first to prove that a specific number was
transcendental.)
Hermite proved that the number e
was transcendental in 1873. Lindeman proved that pi was transcendental in 1882.
For more information, see my book
Wonders
of Numbers
from
which this is excerpted.
The mathematical constant pi represents
the ratio of the circumference of a circle to its
diameter. It is the most famous ratio in mathematics both on Earth and
probably for any advanced civilization in the universe. The number pi,
like other fundamental constants of mathematics such as e = 2.718...,
is a transcendental number. The digits of pi and e never end, nor has
anyone detected an orderly pattern in their arrangement. Humans know
the value of pi to over a trillion digits.
Transcendental numbers cannot be expressed as the root of any
algebraic equation with rational coefficients. This means that pi
could not exactly satisfy equations of the type: pi^{2} = 10, or
9pi^{4} 
240pi^{2} + 1492 = 0. These are equations involving simple integers with
powers of pi. The numbers pi and e can be expressed as an endless
continued fraction or as the limit of an infinite series. The
remarkable fraction 355/113 expresses pi accurately to six decimal
places.
In 1882, German mathematician F. Lindemann proved that
pi
is transcendental, finally putting an end to 2,500 years
of speculation.
In effect, he proved that
pi
transcends the power of algebra to display it in its totality.
It can't be expressed in any finite series of arithmetical or
algebraic operations. Using a fixedsize font, it can't be written on a piece of paper
as big as the universe.
I also talk about all the mysteries of pi in my
book Keys to Infinity.
Many of you have probably heard of pi and e. But are there other
famous transcendental numbers? After conducting a brief survey of
readers, I made a list of the fifteen most famous
transcendental numbers. Can you list these in order of relative fame
and/or usage?

pi = 3.1415 ...

e = 2.718 ...
 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.)
 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, 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 GelfondSchneider theorem, any number
of the form
a^{b}
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 nonzero algebraic numbers
are transcendental.)

e^{pi}

pi^{e}
(Not proven to be transcendental, but generally believed to be
by mathematicians.)
 MorseThue's number, 0.01101001 ...

i^{i} = 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
a^{b}
is transcendental. Since
i
is algebraic but irrational, the theorem applies.
Note also:
i^{i}
is equal to
e^{( pi / 2 )}
and several other values.
Consider
i^{i} = e^{(i log i )} =
e^{( i times i pi / 2 )} .
Since log is multivalued, there are other possible values for
i^{i}.
Here is how you can compute the value of i^{i} = 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
perioddoubling. The ratio of successive differences between
perioddoubling 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.)
Long ago, Keith Briggs from the Mathematics Department of the
University of Melbourne in Australia
computed what he believed to be the worldrecord
for the number of digits for the Feigenbaum number:
4.
669201609102990671853203820466201617258185577475768632745651
343004134330211314737138689744023948013817165984855189815134
408627142027932522312442988890890859944935463236713411532481
714219947455644365823793202009561058330575458617652222070385
410646749494284981453391726200568755665952339875603825637225
Briggs carried out the computation using specialpurpose
software designed by David Bailey of NASA Ames running
on an IBM RISC System/6000. The computation required a few
hours of computation time.
Today, we know far more digits for the Feigenbaum constant. See this page
for more than 10,000 digits.
Ants and Transcendental Numbers
Imagine a race of talking ants.
The ants can compress
the infinite digits of
pi
in an interesting way. For example, let us
imagine that the ants can speak by manipulating their crude jaws.
The first ant in the long parade of ants screams out the first
digit,
"3". The next yells the number on its back, a
"1". The
next yells a "4", and so on. Further imagine that each ant
speaks its digit in
only half the time of the preceding ant. Each ant has a turn to
speak. Only the most recent digit is spoken at any instant. If
the first digit of
pi
requires 30
seconds to speak (due to the ant's cumbersome jaws and little
brain), might the entire ant colony will speak all the
digits of
pi
in
a minute?
(Again, this
is because
the infinite
sum
1/2 minute + 1/4 minute + 1/8 minute + ...
is equal to 1 minute.)
Astoundingly, at the end of the minute, might there be a
quicktalking ant that will actually say the "last" digit of
pi?
The geometer God, upon
hearing this last digit, may cry, "That's impossible,
because
pi
has no last digit!"
(See below for an email from Brian B. on the talking ants.)
Here are some nice web pages on transcendental numbers:
1,
2, and
3.
Here is a
book on transcendental numbers.
Dottie Number
Dottie number is the unique real root of cosx = x (namely, the unique real fixed point of the cosine function), which is 0.739085...
Dottie noticed that whenever she typed a number into her calculator and pressed the cosine button repeatedly, the result always converged to this value.
Wow. The number is wellknown, having appeared in numerous elementary works on algebra already by the late 1880s.
The Dottie number is transcendental as a consequence of the the LindemannWeierstrass theorem.
This text about Dottie is adpoted from MathWorld.
Comments from Colleagues on
Their Passion for Transcendental Numbers
RM Mentock comments:
Although they are not often recognized as such even by mathematicians,
there are a lot of commonlyused numbers that are also transcendental,
which can easily be shown by the GelfondSchneider theorem mentioned
in the tenth item on the list. If a is algebraic, and c is algebraic,
and b = logarithm (base a) of c is not rational, then b must be
transcendental or else the theorem would imply that c must be
transcendentala contradiction. Then, with a=10 and c=2, the log of
two, base ten, is transcendental, and so is any base ten logarithm of
any rational number other than rational powers of ten. The same holds
for any other rational logarithm baseso there are a lot of transcendental
numbers that are in common use. Or they were forty years ago, before
handheld calculators!
I'd also like to point out that any number can be used to produce a
transcendental by using Liouville's algorithm (see item number five). If
the number is terminating, convert it to nonterminating by subtracting one
from the last digit, and adding an infinite string of 9's to the end. Then
just put each of its digits where Liouville puts a one, even if the digit is
zero. The result will be a transcendental number.
Brian B. comments on the talking ants above:
On your transcendental number page (https://sprott.physics.wisc.edu/Pickover/trans.html), you say: "Astoundingly, at the end of the minute, there will be a quicktalking ant that will actually say the 'last' digit of pi!"
This is not correct. There is no last ant in an infinite line of ants, just as there's no last digit of pi.
This can be seen easily by having all the ants announce their digit at exactly the same time.
It takes zero seconds to say all the digits, but there is still no last digit, or last ant.
Or forget about the ants entirely and just have each digit announce itself at the same moment.
Still no last digit, though, just like there's no end to the natural numbers.
I hope you'll fix this on your page. I think it would probably be best just to remove the entire "ant" section. Thanks.
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