The 17 equations that changed the course of history

Mathematics is all around us, and it has shaped our understanding
of the world in countless ways.

In 2013, mathematician and science author Ian Stewart published a
book on
17 Equations That Changed The World. We recently came across
this convenient table on Dr.
Paul Coxon’s twitter account by mathematics tutor and blogger
Larry Phillips that summarizes the equations. (Our
explanation of each is below):

Larry
Phillips, via @paulcoxon on Twitter

Here is a little bit more about these wonderful equations that
have shaped mathematics and human history:

Shutterstock/ igor.stevanovic

1) The Pythagorean Theorem: This theorem is
foundational to our understanding of geometry. It describes the
relationship between the sides of a right triangle on a flat
plane: square the lengths of the short sides, a and b, add those
together, and you get the square of the length of the long side,
c.

This relationship, in some ways, actually distinguishes our
normal, flat, Euclidean geometry from curved, non-Euclidean
geometry. For example, a right triangle drawn on the surface of a
sphere need not follow the Pythagorean theorem.

2) Logarithms: Logarithms are the
inverses, or opposites, of exponential functions. A logarithm for
a particular base tells you what power you need to raise that
base to to get a number. For example, the base 10 logarithm of 1
is log(1) = 0, since 1 = 10⁰; log(10) = 1, since 10 =
10¹; and log(100) = 2, since 100 =
10².

The equation in the graphic, log(ab) = log(a) + log(b),
shows one of the most useful applications of logarithms: they
turn multiplication into addition.

Until the development of the digital computer, this was the
most common way to quickly multiply together large numbers,
greatly speeding up calculations in physics, astronomy, and
engineering. 

3) Calculus: The formula given here is the
definition of the derivative in calculus. The derivative measures
the rate at which a quantity is changing. For example, we can
think of velocity, or speed, as being the derivative of position
— if you are walking at 3 miles per hour, then every hour, you
have changed your position by 3 miles.

Naturally, much of science is interested in understanding
how things change, and the derivative and the integral — the
other foundation of calculus — sit at the heart of how
mathematicians and scientists understand change.

Isaac
Newton

Wikimedia
Commons

4) Law of Gravity: Newton’s law of gravitation
describes the force of gravity between two objects, F, in terms
of a universal constant, G, the masses of the two objects,
m₁ and m₂, and the distance between the
objects, r. Newton’s law is a remarkable piece of scientific
history — it explains, almost perfectly, why the planets move in
the way they do. Also remarkable is its universal nature — this
is not just how gravity works on Earth, or in our solar system,
but anywhere in the universe.

Newton’s gravity held up very well for two hundred years,
and it was not until Einstein’s theory of general relativity that
it would be replaced.

5) The square root of -1: Mathematicians
have
always been expanding the idea of what numbers actually are,
going from natural numbers, to negative numbers, to fractions, to
the real numbers. The square root of -1, usually written
i, completes this process, giving rise to the complex
numbers.

Mathematically, the complex numbers are supremely elegant.
Algebra works perfectly the way we want it to — any equation has
a complex number solution, a situation that is not true for the
real numbers : x² + 4 = 0 has no real number solution,
but it does have a complex solution: the square root of -4, or
2i. Calculus can be extended to the complex numbers, and
by doing so, we find some amazing symmetries and properties of
these numbers. Those properties make the complex numbers
essential in electronics and signal processing.

A cube.

Wikimedia
Commons

6) Euler’s Polyhedra Formula: Polyhedra
are the three-dimensional versions of polygons, like the cube to
the right. The corners of a polyhedron are called its vertices,
the lines connecting the vertices are its edges, and the polygons
covering it are its faces.

A cube has 8 vertices, 12 edges, and 6 faces. If I add the
vertices and faces together, and subtract the edges, I get 8 + 6
– 12 = 2.

Euler’s formula states that, as long as your polyhedron is
somewhat well behaved, if you add the vertices and faces
together, and subtract the edges, you will always get 2. This
will be true whether your polyhedron has 4, 8, 12, 20, or any
number of faces.

Euler’s observation was one of the first examples of what is now
called a topological
invariant — some number or property shared by a class of
shapes that are similar to each other. The entire class of
"well-behaved" polyhedra will have V + F – E =
2. This observation, along
with with Euler’s solution to
the Bridges of Konigsburg problem, paved the way to the development of
topology, a branch of math essential to modern physics.

The normal
distribution.

economicshelp.org

7) Normal distribution: The normal probability
distribution, which has the familiar bell curve graph to the
left, is ubiquitous in statistics.

The normal curve is used in physics, biology, and the social
sciences to model various properties. One of the reasons the normal curve shows
up so often is that it describes
the behavior of large groups of independent
processes.

8) Wave Equation: This is a differential
equation, or an equation that describes how a property is
changing through time in terms of that property’s derivative, as
above. The wave equation
describes the behavior of waves — a vibrating guitar string,
ripples in a pond after a stone is thrown, or light coming out of
an incandescent bulb. The wave equation was an early differential
equation, and the techniques developed to solve the equation
opened the door to understanding other differential equations as
well.

9) Fourier Transform: The Fourier transform is
essential to understanding more complex wave structures, like
human speech. Given a complicated, messy wave function like a
recording of a person talking, the Fourier transform allows us to
break the messy function into a combination of a number of simple
waves, greatly simplifying analysis.

 The Fourier transform is
at the heart of modern signal processing and analysis, and data
compression. 

10) Navier-Stokes
Equations: Like the wave equation, this is a
differential equation. The Navier-Stokes equations describes the
behavior of flowing fluids — water moving through a pipe, air
flow over an airplane wing, or smoke rising from a cigarette.
While we have approximate solutions of the Navier-Stokes
equations that allow computers to simulate fluid motion fairly
well, it is still an open question (with
a million dollar prize) whether it is possible to construct
mathematically exact solutions to the equations.

11) Maxwell’s
Equations: This set of four differential equations
describes the behavior of and relationship between electricity
(E) and magnetism (H).

Maxwell’s equations are to
classical electromagnetism as Newton’s laws of motion and law of
universal gravitation are to classical mechanics — they are the
foundation of our explanation of how electromagnetism works on a
day to day scale. As we will see, however, modern physics relies
on a quantum mechanical explanation of electromagnetism, and it
is now clear that these elegant equations are just an
approximation that works well on human scales.

12) Second Law of
Thermodynamics: This states that, in a closed system,
entropy (S) is always steady or increasing. Thermodynamic entropy
is, roughly speaking, a measure of how disordered a system is. A
system that starts out in an ordered, uneven state — say, a hot
region next to a cold region — will always tend to even out, with
heat flowing from the hot area to the cold area until evenly
distributed.

The second law of
thermodynamics is one of the few cases in physics where time
matters in this way. Most physical processes are reversible — we
can run the equations backwards without messing things up. The
second law, however, only runs in this direction. If we put an
ice cube in a cup of hot coffee, we always see the ice cube melt,
and never see the coffee freeze.

Albert
Einstein

Associated
Press

13) Relativity: Einstein radically altered the
course of physics with his theories of special and general
relativity. The classic equation E = mc² states that
matter and energy are equivalent to each other. Special
relativity brought in ideas like the speed of light being a
universal speed limit and the passage of time being different for
people moving at different speeds.

General
relativity describes gravity as a curving and folding
of space and time themselves, and was the first major change to
our understanding of gravity since Newton’s law. General
relativity is essential to our understanding of the origins,
structure, and ultimate fate of the universe.

14) Schrodinger’s
Equation: This is the main equation in quantum
mechanics. As general relativity explains our universe at its
largest scales, this equation governs the behavior of atoms and
subatomic particles.

Modern quantum mechanics and
general relativity are the two most successful scientific
theories in history — all of the experimental observations we
have made to date are entirely consistent with their predictions.
Quantum mechanics is also necessary for most modern technology —
nuclear power, semiconductor-based computers, and lasers are all
built around quantum phenomena.

15) Information
Theory: The equation given here is for Shannon
information entropy. As with the thermodynamic entropy given
above, this is a measure of disorder. In this case, it measures
the information content of a message — a book, a JPEG picture
sent on the internet, or anything that can be represented
symbolically. The Shannon entropy of a message represents a lower
bound on how much that message can be compressed without losing
some of its content.

Shannon’s entropy measure
launched the mathematical study of information, and his results
are central to how we communicate over networks today.

16) Chaos
Theory: This equation is May’s logistic
map. It describes a process evolving through time —
x_t+1, the level of some quantity x in the next time
period — is given by the formula on the right, and it depends on
x_t, the level of x
right now. k is a chosen constant. For certain values of k, the
map shows chaotic behavior: if we start at some particular
initial value of x, the process will evolve one way, but if we
start at another initial value, even one very very close to the
first value, the process will evolve a completely different
way.

We see chaotic behavior —
behavior sensitive to initial conditions — like this in many
areas. Weather is a classic example — a small change in
atmospheric conditions on one day can lead to completely
different weather systems a few days later, most commonly
captured in the idea of a butterfly
flapping its wings on one continent causing a hurricane on
another continent. 

17) Black-Scholes
Equation: Another differential equation,
Black-Scholes 
describes how finance experts and traders
find prices for derivatives. Derivatives — financial products
based on some underlying asset, like a stock — are a major part
of the modern financial system.

The Black-Scholes equation
allows financial professionals to calculate the value of these
financial products, based on the properties of the derivative and
the underlying asset.

Here are some traders in
the S&P 500 options pit at the Chicago Board Options
Exchange. You won’t find a single person here that hasn’t heard
about the Black-Scholes equation.
REUTERS/Frank Polich