This page contains an idiosyncratic and personal, and hopefully growing, selection
of mathematical and physical equations that I think are particularly
important or particularly intriguing. Not all of these
equations are complicated. Look and and see what you think.
Send me comments, or nominations of other equations or
inequalities.
This beautiful equation connects three major constants of
mathematics, Euler's Number e, the ratio of the
circumference of a circle to its diameter,
pi, and
the square root of -1, i.e., i.
Pi is defined to be the ratio of the circumference
c of any
circle divided by its diameter, d. Most people
when asked will
tell you that pi=3.14..., but that's just an
accident. You can see
pi to 10,000 digits right here
if you like, but that expression completely obscures the definition.
One way of defining Euler's number e is by this
formula. It can be interpreted as saying that if you
collect 100 percent interest annually and compound
continuously then you multiply your capital with e
every year.
A more mathematical definition of e is obtained
by asking which function f equals its own
derivative. The
answer to that question is f(x) = ex.
If that approach is chosen the
statement in the previous equation becomes a Theorem.
Again, you can see
e to 10,000 digits.
If a and b are the lengths of the two
short sides of a right triangle and c is its long
side then this formula holds. Conversely, if the formula
holds then a triangle whose sides have length a, b
and c is a right triangle. This formula is about
2,350 years old and due to Pythagoras of Samos. It
is used all over mathematics. The Greek thought of the
Theorem not algebraically as it is presented here, but
geometrically with the square numbers being represented as
squares attached to the edges of the triangle.
This formula expresses the fact that differentiation and
integration are inverse operations of each other.
This formula shows how to express an analytic function in
terms of its derivatives.
In this equation, A is a square matrix (often a
very large one), x is an unknown vector, and
lambda is an unknown real or complex number. Many
physical problems lead to equations like this. Usually the
numbers lambda that satisfy the equation are
significant to the dynamic behavior of the physical system,
i.e., the behavior as time goes on.
In this equation A and x are as before and
b is a known vector. The equation also describes
many physical systems and the solution x often
describes a physical situation either at one point in time
or for all time.
Nature likes to minimize things (like energy) and this
equation describes one particular minimization problem.
Given a function F one wants to find a function
u=u(x) such that the integral is as small as
possible.
The Mandelbrot Set is an extremly complex object that shows new structure
at all magnifications. It is the set of complex numbers c
for which the iteration indicated nearby remains bounded. Click on the equation
to see a page that has lots of pictures and that offers and documents
a Java applet you can use to explore the Mandelbrot set yourself.
This is arguably the most important equation of the bunch.
If you borrow an amount L dollars and pay it back
over N months at an annual interest rate of p
percent your monthly payment will be m dollars.
Let x and y be vectors that form two
sides of a triangle whose third side is x+y.
The expression ||x|| denotes the length of a
vector x. (It's more
generally called a norm in mathematics.)
The triangle inequality expresses the fact that the sum of
the lengths of any two sides of a triangle cannot be less than
the length of the third side. It is used ubiquitously
throughout mathematics. As an exercise you may want to prove the
You can do it! The main
use of the reverse triangle inequality is to provide a
challenging exercise to students. The argument is very
short and simple but you have to think of it. I once had
a graduate student who said he spent a total of 20 hours
finding a four line proof, but, he said, "it made me feel
really good!".
Mail me your proof if you like. It may spoil your fun,
but if you can't resist the temptation
a proof is just a click away.
Let S be a set, and let |S| denote its
cardinality. If S is a finite set then its
cardinality is the number of elements in it, and things are
not very interesting. But the concept of cardinality
makes sense also for infinite sets. That story makes
a fascinating webpage. The power set of a set is the
set of its subsets. It is easy to see that for finite
sets S the cardinality of the power set equals
2|S|. Thus we denote by
2|S| the cardinality of the power set
even for infinite sets S. Cantor's
Theorem states that the cardinality of the power set of a
set S always exceeds the cardinality of
S itself. That's obvious for finite sets but
far from trivial for infinite sets. You are invited to
look at
a proof of this remarkable fact.
Einstein's famous equations says that mass m is
equivalent to energy E, and the amount of energy
contained in a piece of mass is equal to the mass multiplied
with the square of the speed of light, c. Without
the fact described by this equation we wouldn't be around
since the energy we obtain from
the Sun is generated by converting mass to energy in the
process of nuclear fusion.
If you have two objects of mass m 1 and
m 2 at a distance d, then these
two objects will attract each other with a force F
given in this formula. G is the gravitational
constant. It equals approximately
6.67*10-11Nm2kg-2.
This formula determines the destiny of our Universe (i.e.,
whether it will expand forever or whether it will
ultimately collapse in a Big Crunch after having
originated in the Big Bang).
Fine print, your comments, more links, Peter Alfeld, PA1UM.
[15-Mar-1998]