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 = 10^{0}; log(10) = 1, since 10 =

10^{1}; and log(100) = 2, since 100 =

10^{2}.

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,

m_{1} and m_{2}, 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 : x^{2} + 4 = 0 has no real number solution,

but it does have a complex solution: the square root of -4, or

2*i*. 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 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.

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 = mc^{2} 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.

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