
Mathematical
Constants
1.
 This is the most well known mathematical constant, and is the
ratio of the circumference of a circle to its diameter  the fact
that this is the same for all circles is amazing in itself.
 The value
has been estimated since biblical times, Pi
through the ages, and a (wrong) value was almost written into a law.
 You will know that
can be approximated by
which only gives the value correct to 2 decimal places.
is correct to 6 decimal places, andto
9 decimal places.
has been calculated to over 50
billion digits. See Table
of current records for the computation of constants for the
current record for
and other constants.

is not only irrational (it cannot be written
exactly as the ratio of two integers) it is also transcendental
(it is not the solution of any polynomial equation with rational
coefficients). You will find proofs in Transcendence
of Pi and in T_{E}X format.
Not many transcendental numbers are known (e
and Liouville's number are another two
examples) but in fact in 1874 Cantor
showed that almost all real numbers are transcendental. You can see
some of the proofs if you follow the links at Foundations.
 There are lots of formulae for .
A couple appear in formulae and
here are two more:
 This formula was discovered by François
Viète:

 and Isaac
Newton used 22 terms of this series to find
correct to 16 places:


appears everywhere and you will find it popping up in many of the
pages of this site.
 It is the subject of a film called, naturally, Pi. See Pi
The Movie and the excellent Internet
Movie Database.
 A good page to start your researches into
is The Pi Pages
though you may want to turn down the sound since it recites the
digits of
in French  don't ask why!
 How
Pi is calculated shows how to calculate .
 Useless
facts about Pi is what it says it is.
 Pi
Facts gives 71 facts about.
 The
Ridiculously Enhanced Pi Page is worth a visit
 There are lots of really daft sites devoted to
like Pi Day Songs.
 You'll find formulae involvingat
Expansions for PI.

attracts a lot of attention and that is reflected in the huge number
of sites on the internet. You can spend lots of happy days
researching these sites.
2. e
 It is harder to give a simple definition for e like that for;
e is usually introduced at A level. One way to define it is
to say that it is the number that makes the shaded area in the graph
of equal
to 1.

 The value of e is 2.718 281 828 459 045 ... and it has the
unusual attribute that the numbers 1828 repeat early on in the
decimal expansion. e is another example of a transcendental
number (see Liouville).
 Its series expansion

 is well known, and converges quickly, giving good estimates for e.
 That would be the end of the matter if it wasn't for the fact
that e like
keeps popping up in all sorts of unexpected places. You will find it
occurs in lots of places on this site, most famously along
with .
 One of the first properties that students meet is that the rate
of change of e^{x} is e^{x}, so it's
exceptionally easy to differentiate or integrate, and its use in
calculus is important.
 In addition, it occurs in formulae that deal with anything from
population increase to radioactive decay. Exponential
Growth: The Andromeda Strain, Growth
and Decay (do try Problem 1 on that page).
 It occurs in interest calculations: you should know that getting
interest at at 6% twice a year yields more than 12% per year.
Interest added at 1% per month is even better. If you keep increasing
the frequency of adding interest to daily, hourly, minutely (is there
such a word?) etc you will get a higher return but there's a limit
and that limit is e. This is because

 See Compound
Interest and Math
Buffs Find an Easier "e".
 It also pops up in probability. For example, if you have a number
of letters and the same number of envelopes, then the probability
that every letter is placed in the wrong envelope approaches
as the number of letters and envelopes increases. This is the same
problem as turning up pairs of cards from two packs, with the
probability that there is no match among the 52 pairs being approximately.
Paychecks
and Envelopes
 Finally, can you find the value of ?
This number is interesting because it appears to be a large integer
but it isn't. So how many decimal places are required before you can
see that it isn't an integer?
 Useful pages on e are:
 The Euler's Number Home Page
 The
Natural Logarithmic Base
 Some
History Surrounding e
 Why
is e so important?
3. i
 i stands for imaginary number because it seemed
like that when first used. Yet it is no more imaginary than negative
numbers (how often have you seen 2 pieces of cake?) or irrational
numbers like,
which Pythagoras had trouble
with. Indeed it was the use of i that revolutionised the use
of electricity enabling the design of dynamos and electric motors Applications
of Complex Numbers , Real
Life Applications of Imaginary Numbers. Mathematicians used i
long before they understood it, because they found it was so useful.
 i arose naturally when trying to solve equations like x^{2}
+ 1 = 0. Using i and complex numbers like 2 + 3i means
that one can say that all quadratic equations have two solutions
(though they may be equal). Indeed the Fundamental
Theorem of Algebra (you'll find proofs at Intuition
and Rigor) says that every polynomial of degree n has n
roots (root is another word for solution). The set of complex numbers is
then the top of the chain leading from natural numbers to integers
to rationals to reals and finally to the complex numbers (though see Number
Systems). is
then called algebraically closed in mathematical jargon Algebraic
structures.
 In the sixteenth century, Cardan
(also known as Cardano),
who was one of the first to give a formula for solving a cubic, was
able to solve the problem of dividing 10 into two parts such that the
product is 40, by giving the answer as
but he wasn't happy with these 'truly sophisticated' numbers.
 i is fun to play around with, so for example

 i reveals an unexpected
link between e and the trigonometric functions sin
and cos thus unifying two areas of mathematics. This leads to
simple proofs of trig identities and results like De
Moivre's theorem Expansion
of sin and cos using De Moivre's theorem

 which appears more naturally as

 using the unexpected link.
 Unlike many of the constants on this page there is no decimal
equivalent nor is it possible to compare the size of i with
these constants. Think about why this is so.
 One does have to be careful when using i because it
doesn't behave quite like a real number 1=2:
A Proof using Complex Numbers.
 It is a shame that complex numbers have dropped out of some A
level syllabuses because they open the door to a whole new world Basic
Definitions, Introduction
to complex numbers .
4. Euler's Constant
 It isn't difficult to show Testing
For Convergence that the series converges
if s > 1 and diverges if s < 1. The sums of this
series when s = 2, 4, 6 and 8 are given in Euler
Zeta Function.
 But what happens if s = 1? The answer is that it diverges
but very slowly. In fact as N gets larger the series ,
called the harmonic series, gets closer and closer to whereis
known as Euler's
or Mascheroni's
constant and is approximately 0.577 215 664 901 ... It is believed
that is
irrational but it hasn't been proved.
 So how slowly does the harmonic series diverge? If N = 10^{8}
then is
approximately 21.3. Thus the sum of one hundred million terms is
only just over twenty. That's slow divergence for you!
 EulerMascheroni
Constant has lots more on .
5. The Golden Ratio
 The golden ratiohas
the valuewhich
is 1.618 033 988 ... It is obviously irrational (why?) and algebraic
(the opposite of transcendental) being
one of the two solutions of x^{2}  x  1 = 0.
The other solution is which
is 0.618 033 988 ... (confusingly, this value is sometimes called
the Golden Ratio). Notice that .
 is
another one of those numbers that pops up in all sorts of surprising
places. If you draw a rectangle with sides whose lengths are in the
ratio :
1, then this rectangle is said to be aesthetically pleasing to the
eye and many Renaissance artists used it in their paintings The
Golden Mean, Golden
Figures.
 If you take a 'golden rectangle' Some
Golden Geometry then it can be divided into a square plus a
similar rectangle The
Golden Spiral

 The original rectangle gives rise to a new one in which the ratio
of the sides is now which
is the same as :
1 (why?). Now keep dividing in this way and you can join up the
vertices to get a logarithmic spiral, also known as an equiangular
spiral. The Golden
Bowls & the Logarithmic Spiral, Logarithmic
Spirals (beautiful pictures of shells), Equiangular
Spiral.
 This spiral occurs frequently in nature, for example in the
arrangement of sunflower heads and spiral shells Fibonacci
Numbers and Nature.
 can
be written as the continued fractionthough
this converges very slowly. An
Introduction to Continued Fractions.
 It is also intimately connected to the Fibonnaci
numbers Fibonacci
Numbers and the Golden Section
1, 1, 2, 3, 5, 8, 13, 21, 34
where each term is the sum of the previous two. The continued
fraction forgives
convergentswhose
numbers are the Fibonacci numbers Continued
fractions and the Fibonacci sequence. Furthermore, the n^{th}
Fibonacci number equals the nearest integer to ,
and it follows that the ratio of successive Fibonnaci numbers tends to.
 There are lots of sites aboutand
they include:
 Phi's
Fascinating Figures makesvery
interesting.
 Phi:
That Golden Number includes lots of geometric pictures with
Golden Triangles, Golden Ellipse, Whirling Triangles, Whirling
Rectangles amongst them.
 The
Golden Mean includes lots of references.


6. Countable cardinal
 Mathematicians looked to the Greek alphabet for many of their
symbols, with
being the most common. Then they ran out of Greek letters and turned
to the Hebrew alphabet for aleph ,
to Gothic letters like ,
to doublelined Latin letters (called blackboard bold) like
for the set of real numbers, to made up letters likeand,
occasionally, to the Cyrillic alphabet.
 So what is?
It is to do with infinite sets. Such sets can be difficult to deal
with, so that it is awkward to compare sizes of infinite sets. The
key is to use oneone correspondence. If you look in a
classroom you can easily see whether there are more seats than
students without counting. What you are doing is pairing off each
filled seat with the student sitting on them then seeing if there are
any seats (or students) left over. If there's no seat or student left
over then you have established a oneone correspondence between the
set of chairs and the set of students.
 Cantor applied this principle to
infinite sets and gave the setof
all integers a 'size' which is called cardinality, of.
Any set which could be put into oneone correspondence withalso
has cardinalityand
is called countable. This leads to interesting results. For
example, each even number 2n can be paired with the integer n
so the set of even numbers is also countable. Cantor
showed that the set of rational numbers
is also countable but that the set of real numbersis
not. The question whether there is a cardinal number betweenand
that ofis
called the Continuum Hypothesis and it has been answered in a most
surprising way.
 The Continuum Hypothesis
 Mudd
Math Fun Facts: Continuum Hypothesis
 A
Crash Course in the Mathematics Of Infinite Sets
 Infinite
Reflections is all about infinity.
 Hilbert's
Hotel will make you think about infinity.
7.
 This is the number that's used to introduce students to the idea
of an irrational number. A rational number is
expressible as the ratio of two integers. All finite decimals (like
1.23456) and repeating decimals (like 0.3333...) are rational. Any
real number that isn't rational is called irrational. (See Cantor.)
Look at Pythagoras for links to
proofs that is
irrational.
 You can approximate to
by the sequence

 and it can be shown that this is the best possible sequence of
approximations. Can you work out how each term is obtained from the
previous one?
8. Liouville's number
 In 1844 Liouville
showed that the number

 was transcendental, and this was
the first such number known. It was a struggle to proveand
e were also transcendental (Hermite
proved e was transcendental in 1873 and Lindemann provedwas
transcendental in 1882) and in 1874 Cantor
shocked the mathematical world by showing that almost all real
numbers are transcendental, so if you pick a real number at random it
is virtually certain to be transcendental. Common they may be, but
very few actual examples are known. The
15 Most Famous Transcendental Numbers.
9. Googol and Googolplex
 The mathematician Edward Kasner's 9yearold nephew coined the
names googol for the number that is 1 followed by a hundred
zeroes, 10^{100} and googolplex for 1 followed by a
googol of zeroes, 10^{googol}.
 These numbers are very large and impossible to visualise. A
googol is larger than any known estimate of the number of atoms in
the universe and thus it would be physically impossible to write out
the even larger googolplex since there are aren't the atoms available
to make up the zeroes.
 The human mind finds it difficult to comprehend such vast
numbers. To illustrate this, guess how long you think it would take
to count from one to a million, which is a mere 10^{6},
counting at a rate of one number per second without a break. How long
would it take to count to a billion (US version)
10^{9}? Check your guesses by using a calculator (think a
for minute and should be clear how to do this).
 Humour
: What's a GooGol?
 Large
Numbers  Notes
 The Googolplex Page
is an amusing look at googolplex.
10. Graham's number
 This is an outrageously large number which puts googol and
googolplex in the shade. It is so large that a special notation has
to be invented in order to describe it. It is actually an estimate
for a number that occurs in combinatorics and the joke is that the
number is believed to be 6, so Graham's
number must rank as one of the worst estimates ever.
 Graham's
Number gives a good description of the number.
 Large
Numbers
 Big Numbers.
You'll find more constants at Favorite
Mathematical Constants, a few
more at Some
Important Constants and at RJN's
More Digits of Irrational Numbers Page, which gives the decimal
expansions of many of them to millions of digits.
