The 17 Equations That Changed The
Course Of History
ANDY KIERSZ
MAR. 12, 2014, 11:38 AM
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):

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

6) Euler's Polyhedra Formula: Polyhedra are the threedimensional
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.

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.

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 — xt+1 , the level of some quantity x in the next time period — is given by the formula on
the right, and it depends on xt, 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.