- This is the most amazing formula of all and nearly all
mathematicians put it at the top of their list of favourite formulae.
It's known as Euler's Formula, although so is another
- The formula contains three constants,
all with different origins. e is used for logarithms and is
connected with exponential growth and radioactive decay, i is
the basis for complex numbers and
comes, of course, from circles. Put these three together and bingo!
you get -1. I have seen this formula so many times but I never fail
to be amazed by it.
- Rearranging giveswhich
now includes five of the most important mathematical constants.
0 and 1 are the foundation stones of arithmetic and their importance
is perhaps best appreciated by studying group
- Even more amazingly, this formula is a special case of which
links e with trigonometric functions, another surprise.
- Keith Devlin's book Mathematics: the Science of Patterns Internet
sums things up very well, and there's an excerpt at Euler's
- Read Keith
Devlin's speech to graduating students Making
the Invisible Visible.
2. Fermat's Little Theorem
- There are two theorems linked to Fermat's
name and this is the other one. It says that
- if p
is a prime number and a
is an integer then ap-1
- 1 is divisible by p
- or, equivalently,
- if p
is a prime number and a
is an integer then ap
- a is divisible by p
- For example, if p = 13 then 612 - 1 = 2 176 782
335 which is divisible by 13.
- This formula is very useful. For example, you can use it to show
a number is not prime. 63 - 1 = 215 isn't
divisible by 4, so 4 can't be prime. This is blindingly obvious in
this case, but is useful for larger numbers.
- If ap-1 - 1 is divisible by p
for all integers a with no factor in common with p,
then it is highly probable that p is prime. However, it
needn't be and then p is called a Carmichael
number (561 is the smallest).
- For a proof of the theorem try Fermat's
- For more on the use of this result see Probable
3. Euler's formula V - E + F
- This is an important formula in graph theory. Draw any
two-dimensional graph, that is, a set of a points called vertices,
and some line segments called edges which connect the vertices. Make
sure it is in one piece (connected in mathematical language).
Then count up the number of vertices V, the number of edges E,
and the number of faces (regions) F that it encloses.
- Then no matter how you do this you will find V - E
+ F = 1, For example,
- (vertices are numbered in black, edges
in blue and faces
- thus V = 14, E
= 21, F = 8
so V - E + F = 14 - 21 + 8 = 1.
- Draw your own graph and check it. You will sometimes see the
formula written as
- because the face outside the graph has been included.
- What happens to this formula if you draw it on the surface of a
sphere, a torus or just in three dimensions? Euler's
Formula in Higher Dimensions
- Try to use this formula to show there are only 5 regular solids
(known as Platonic solids) Euler's
Five Platonic Solids and to show there is no solution to an old riddle:
- There are three houses each of which must be
connected to the water, gas and electricity mains. Can this be done
in such a way that the cables and pipes do not cross over or under
each other and do not pass through the houses or the mains supply?
developed graph theory to solve the Königsberg bridge problem.
There were seven bridges over the River Pregel at Königsberg in
Prussia. Euler solved the problem of whether there is a way of
traversing each bridge once and only once starting and returning to
the same point in the town. Picture
of Königsberg bridges. For more information see Topology
- This little formula, called the difference of two squares formula,
crops up everywhere, often without warning. For example, the
following problem leads to an outrageous polynomial unless you use
- A geometric series has first term p,
and eight times the sum of the first six terms is equal to nine times
the sum of the first three terms. Find the common ratio of the series.
used it to factorise large numbers (no calculator or computer was
available!) into the product of two numbers roughly equal in size.
For example to factorize the number 119143 try the smallest number x
so that x2 - 119143 is positive and keep
increasing x by 1 until x2 - 119143
is a perfect square, y2 say. Then x2 - 119143 = y2
and so it follows that 119143 = x2 - y2 = (x - y)(x + y)
thus factorising 119143 (and what do you get?). This method is
known, not surprisingly, as Fermat's
method of factoring.
- You can use the difference of two squares formula to amaze your
friends with your mathematical ability:
Find 352 - 342 easily. The formula
tells you it's 69 in an instant.
You can find squares easily. To find 562 take the two
numbers 6 less and 6 more than 56 and multiply them together as
50 × 62 = 3100 (you do know how to multiply
by 50 easily don't you?), then add 62 to get 3136 which is 562.
Use the formula to see how this works.
5. The Prime Number Theorem
- This is a difficult to prove formula which gives the number of
prime numbers less than a given number. It says:
- What this means is that if x is any positive real number
which is the number of primes less than x, is approximately (where
log is to base e, sometimes called the natural log or ln).
The ~ means that the approximation is such that
gets closer and closer to 1 as x gets larger.
- Here are some values to see that this formula is reasonable:
1 000 000
1 000 000 000
50 847 478
48 254 942
- Source: An
Introduction To The Theory Of Numbers
- Perhaps the power of this formula can only be realised when you
discover that the primes don't occur in any pattern so that their
occurrence is quite hard to pin down.
- There's lots more on this formula at How
Many Primes Are There?
- See also Bertrand's
6. Number of divisors
- Most people are unable to calculate how many divisors that a
number has without writing them all down, and doubtless wandering
whether they've missed some.
- Yet it can be done very easily provided you can factorize the
number into powers of prime numbers.
- For example, 108 = 22 × 33.
- Then you can list all the divisors, including 1 and 108, as made
up by multiplying each number in 1, 2, 22 with a number in
1, 3, 32, 33 giving the 12 combinations
- 1 × 1, 1 × 3, 1 × 32, 1 × 33
- 2 × 1, 2 × 3, 2 × 32, 2 × 33
- 22 × 1, 22 × 3, 22
× 32, 22 × 33
- Thus there are 12 divisors which comes from adding one to each of
the powers in 22 × 33 and then
multiplying them together.
- You should be able to see how this generalizes to the formula:
then the number of divisors of n is
and Chikwama get out of jail
- This is a special case (s = 2) of the Zeta Function,
first proved by Euler
after defeating the best efforts of famous mathematicians like Jacob
Bernoulli and Daniel
Bernoulli (all members of the same family).
- It is yet another formula that involves the ubiquitous which
seems to turn up in all sorts of surprising places.
- You can find fourteen different proofs of the formula at Robin
Chapman's Home Page, though they are mostly non-elementary. The
proofs Evaluating zeta(2) are in DVI, postscript (PS) or PDF
- What is even stranger is that if you pick two positive integers
at random, the odds of them having no common divisor are 1 in which
is 1 in 1.644934... . This result, along with others, appears at Notable
Properties of Specific Numbers.
8. Wallis's Product
- Yet another formula that gives ,
and it was discovered by John
- You'll find a simple program that will run on Casio 9x50
calculators which uses Wallis's product at Sanoy's
Casio Programs and is repeated here:
- Locate 1,1,"TRIES:"
- Locate 1,3,"PI:"
- Lbl 1
- Locate 4,3,2C
- Locate 7,1,B
- Goto 1
- so that you may be able to convert it to run elsewhere.
- A much more well-known formula for
is to use the series for tan-1 and the fact that
to get Gregory's Formula
- This is discussed in How
Pi is calculated.
9. Euler (Riemann) Zeta Function
- which can be summarized more neatly as
the Greek letter zeta and the sum is over all natural numbers n
while the product is over all prime numbers p.
- It is sometimes known as the Euler
zeta function, but because Riemann extended it to complex
numbers it is more often known as the Riemann
zeta function, which is the subject of one of the most famous
unsolved problems in mathematics The
- This formula is fascinating because it converts a sum of powers
to a product of prime numbers, and although it isn't difficult to
prove, it is still a surprising result.
- We have already said above that
but it is also true that
- see Dave's
10. Stirling's Formula
- This formula is used in probability to help estimate the value of n!
As in Prime Number Theorem ~ means that
the ratio of the two sides of the formula tend to 1 as n tends
- What is remarkable is how
and e get involved in estimating n! when there's no
obvious connection with multiplying the integers 1, 2, 3, ... , n together.
- Stirling's formula was actually discovered by De
Moivre although Stirling
did improve it. The formula nowadays is so well-known and seems to
have lost its magic so that there are very few references to it on
the internet, though you may be able to prove otherwise.