The Pythagoreans figured out what makes strings sound harmonious (Image:
Nils Jorgensen/Rex Features)
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THE alarm rings. You glance at the clock. The time is 6.30 am. You haven't
even got out of bed, and already at least six mathematical equations have
influenced your life. The memory chip that stores the time in your clock
couldn't have been devised without a key equation in quantum mechanics.
Its time was set by a radio signal that we would never have dreamed of
inventing were it not for James Clerk Maxwell's four equations of
electromagnetism. And the signal itself travels according to what is known
as the wave equation.
We are afloat on a hidden ocean of equations. They are at work in
transport, the financial system, health and crime prevention and
detection, communications, food, water, heating and lighting. Step into
the shower and you benefit from equations used to regulate the water
supply. Your breakfast cereal comes from crops that were bred with the
help of statistical equations. Drive to work and your car's aerodynamic
design is in part down to the Navier-Stokes equations that describe how
air flows over and around it. Switching on its satnav involves quantum
physics again, plus Newton's laws of motion and gravity, which helped
launch the geopositioning satellites and set their orbits. It also uses
random number generator equations for timing signals, trigonometric
equations to compute location, and special and general relativity for
precise tracking of the satellites' motion under the Earth's gravity.
Without equations, most of our technology would never have been invented.
Of course, important inventions such as fire and the wheel came about
without any mathematical knowledge. Yet without equations we would be
stuck in a medieval world.
Equations reach far beyond technology too. Without them, we would have no
understanding of the physics that governs the tides, waves breaking on the
beach, the ever-changing weather, the movements of the planets, the
nuclear furnaces of the stars, the spirals of galaxies - the vastness of
the universe and our place within it.
There are thousands of important equations. The seven I focus on here -
the wave equation, Maxwell's four equations, the Fourier transform and
Schrödinger's equation - illustrate how empirical observations have led to
equations that we use both in science and in everyday life.
the seven equations
First, the wave equation. We live in a world of waves. Our ears detect
waves of compression in the air as sound, and our eyes detect light waves.
When an earthquake hits a town, the destruction is caused by seismic waves
moving through the Earth.
Mathematicians and scientists could hardly fail to think about waves, but
their starting point came from the arts: how does a violin string create
sound? The question goes back to the ancient Greek cult of the
Pythagoreans, who found that if two strings of the same type and tension
have lengths in a simple ratio, such as 2:1 or 3:2, they produce notes
that, together, sound unusually harmonious. More complex ratios are
discordant and unpleasant to the ear. It was Swiss mathematician Johann
Bernoulli who began to make
sense of these observations. In 1727 he modelled a violin string as a
large number of closely spaced point masses, linked together by springs.
He used Newton's laws to write down the system's equations of motion, and
solved them. From the solutions, he concluded that the simplest shape for
a vibrating string is a sine curve. There are other modes of vibration as
well - sine curves in which more than one wave fits into the length of the
string, known to musicians as harmonics.
From waves to wireless
Almost 20 years later, Jean
Le Rond d'Alembert followed
a similar procedure, but he focused on simplifying the equations of motion
rather than their solutions. What emerged was an elegant equation
describing how the shape of the string changes over time. This is the wave
equation, and it states that the acceleration of any small segment of the
string is proportional to the tension acting on it. It implies that waves
whose frequencies are not in simple ratios produce an unpleasant buzzing
noise known as "beats". This is one reason why simple numerical ratios
give notes that sound harmonious.
The wave equation can be modified to deal with more complex, messy
phenomena, such asearthquakes.
Sophisticated versions of the wave equation let seismologists detect what
is happening hundreds of miles beneath our feet. They can map the Earth's
tectonic plates as one slides beneath another, causing earthquakes and
volcanoes. The biggest prize in this area would be a reliable way to
predict earthquakes and volcanic eruptions, and many of the methods being
explored are underpinned by the wave equation.
But the most influential insight from the wave equation emerged from the
study of Maxwell's equations of electromagnetism. In 1820, most people lit
their houses using candles and lanterns. If you wanted to send a message,
you wrote a letter and put it on a horse-drawn carriage; for urgent
messages, you omitted the carriage. Within 100 years, homes and streets
had electric lighting, telegraphy meant messages could be transmitted
across continents, and people even began to talk to each other by
telephone. Radio communication had been demonstrated in laboratories, and
one entrepreneur had set up a factory selling "wirelesses" to the public.
This social and technological revolution was triggered by the discoveries
of two scientists. In about 1830, Michael
Faraday established the
basic physics of electromagnetism. Thirty years later,James
Clerk Maxwell embarked on a
quest to formulate a mathematical basis for Faraday's experiments and
At the time, most physicists working on electricity and magnetism were
looking for analogies with gravity, which they viewed as a force acting
between bodies at a distance. Faraday had a different idea: to explain the
series of experiments he conducted on electricity and magnetism, he
postulated that both phenomena are fields which pervade space, change over
time and can be detected by the forces they produce. Faraday posed his
theories in terms of geometric structures, such as lines of magnetic
Maxwell reformulated these ideas by analogy with the mathematics of fluid
flow. He reasoned that lines of force were analogous to the paths followed
by the molecules of a fluid and that the strength of the electric or
magnetic field was analogous to the velocity of the fluid. By 1864 Maxwell
had written down four equations for the basic interactions between the
electrical and magnetic fields. Two tell us that electricity and magnetism
cannot leak away. The other two tell us that when a region of electric
field spins in a small circle, it creates a magnetic field, and a spinning
region of magnetic field creates an electric field.
But it was what Maxwell did next that is so astonishing. By performing a
few simple manipulations on his equations, he succeeded in deriving the
wave equation and deduced that light must be an electromagnetic wave. This
alone was stupendous news, as no one had imagined such a fundamental link
between light, electricity and magnetism. And there was more. Light comes
in different colours, corresponding to different wavelengths. The
wavelengths we see are restricted by the chemistry of the eye's
light-detecting pigments. Maxwell's equations led to a dramatic prediction
- that electromagnetic waves of all wavelengths should exist. Some, with
much longer wavelengths than we can see, would transform the world: radio
In 1887, Heinrich Hertz demonstrated radio waves experimentally, but he
failed to appreciate their most revolutionary application. If you could
impress a signal on such a wave, you could talk to the world. Nikola
Marconi and others turned
the dream into reality, and the whole panoply of modern communications,
from radio and television to radar and microwave links for cellphones,
followed naturally. And it all stemmed from four equations and a couple of
short calculations. Maxwell's equations didn't just change the world. They
opened up a new one.
Just as important as what Maxwell's equations do describe is what they
don't. Although the equations revealed that light was a wave, physicists
soon found that its behaviour was sometimes at odds with this view. Shine
light on a metal and it creates electricity, a phenomenon called the
photoelectric effect. It made sense only if light behaved like a particle.
So was light a wave or a particle? Actually, a bit of both. Matter was
made from quantum waves, and a tightly knit bunch of waves acted like a
Dead or alive
In 1927 Erwin Schrödinger wrote down an equation
for quantum waves. It fitted experiments beautifully while painting a
picture of a very strange world, in which fundamental particles like the
electron are not well-defined objects, but probability clouds. An
electron's spin is like a coin that can be half heads and half tails until
it hits a table. Soon theorists were worrying about all manner of quantum
weirdness, such as cats that are simultaneously dead and alive, and
parallel universes in which Adolf Hitler won the second world war.
Quantum mechanics isn't confined to such philosophical enigmas. Almost all
modern gadgets - computers, cellphones, games consoles, cars,
refrigerators, ovens - contain memory chips based on the transistor, whose
operation relies on the quantum mechanics of semiconductors. New uses for
quantum mechanics arrive almost weekly. Quantum dots - tiny lumps of a
semiconductor - can emit light of any colour and are used for biological
imaging, where they replace traditional, often toxic, dyes. Engineers and
physicists are trying to invent a quantum
computer, one which can perform many different calculations in
parallel, just like the cat that is both alive and dead.
Lasers are another application of quantum mechanics. We use them to read
information from tiny pits or marks on CDs, DVDs and Blu-ray discs.
Astronomers use lasers
to measure the distance from the Earth to the moon. It might even be
possible to launch space vehicles from Earth on
the back of a powerful laser beam.
The final chapter in this story comes from an equation that helps us make
sense of waves. It starts in 1807, when Joseph Fourier devised an equation
for heat flow. He submitted a paper on it to the French Academy of
Sciences, but it was rejected. In 1812, the academy made heat the topic of
its annual prize. Fourier submitted a longer, revised paper - and won.
The most intriguing aspect of Fourier's prize-winning paper was not the
equation, but how he solved it. A typical problem was to find how the
temperature along a thin rod changes as time passes, given the initial
temperature profile. Fourier could solve this equation with ease if the
temperature varied like a sine wave along its length. So he represented a
more complicated profile as a combination of sine curves with different
wavelengths, solved the equation for each component sine curve, and added
these solutions together. Fourier claimed that this method worked for any
profile whatsoever, even a one where the temperature suddenly jumps in
value. All you had to do was add up an infinite number of contributions
from sine curves with more and more wiggles.
Even so, Fourier's new paper was criticised for not being rigorous enough,
and once more the French academy refused to publish it. In 1822 Fourier
ignored the objections and published his theory as a book. Two years
later, he got himself appointed secretary of the academy, thumbed his nose
at his critics, and published his original paper in the academy's journal.
However, the critics did have a point. Mathematicians were starting to
realise that infinite series were dangerous beasts; they didn't always
behave like nice, finite sums. Resolving these issues turned out to be
distinctly difficult, but the final verdict was that Fourier's idea could
be made rigorous by excluding highly irregular profiles. The result is the
Fourier transform, an equation that treats a time-varying signal as the
sum of a series of component sine curves and calculates their amplitudes
Today the Fourier transform affects our lives in myriad ways. For example,
we can use it to analyse the vibrational signal produced by an earthquake
and to calculate the frequencies at which the energy imparted by the
shaking ground is greatest. A sensible step towards earthquake-proofing a
building is to make sure that the building's preferred frequencies are
different from the earthquake's.
Other applications include removing noise from old sound recordings,
finding the structure of DNA using X-ray images, improving radio reception
and preventing unwanted vibrations in cars. Plus there is one that most of
us unwittingly take advantage of every time we take a digital photograph.
If you work out how much information is required to represent the colour
and brightness of each pixel in a digital image, you will discover that a
digital camera seems to cram into its memory card about 10 times as much
data as the card can possibly hold. Cameras do this using JPEG data
compression, which combines five different compression steps. One of them
is a digital version of the Fourier transform, which works with a signal
that changes not over time but across the image. The mathematics is
virtually identical. The other four steps reduce the data even further, to
about one-tenth of the original amount.
These are just seven of the many equations that we encounter every day,
not realising they are there. But the impact of equations on history goes
much further. A truly revolutionary equation can have a greater impact on
human existence than all the kings and queens whose machinations fill our
There is (or may be) one equation, above all, that physicists and
cosmologists would dearly love to lay their hands on: a theory of
everything that unifies quantum mechanics and relativity. The best known
of the many candidates is the theory of superstrings. But for all we know,
our equations for the physical world may just be oversimplified models
that fail to capture the deep structure of reality. Even if nature obeys
universal laws, they might not be expressible as equations.
Some scientists think that it is time we abandoned traditional equations
altogether in favour of algorithms - more general recipes for calculating
things that involve decision-making. But until that day dawns, if ever,
our greatest insights into nature's laws will continue to take the form of
equations, and we should learn to understand them and appreciate them.
Equations have a track record. They really have changed the world and they
will change it again.
The origin of equations
The ancient Babylonians and Greeks knew about equations, though they
wrote them using words and pictures. For the past 500 years,
mathematicians and scientists have used symbols, the crucial one being
the equals sign. Unusually, we know who invented it, and why. It was
Robert Recorde, who in 1557 wrote in his treatise The
Whetstone of Witte: "To avoide the tediouse repetition of these
woordes: is equalle to: I will sette as I doe often in woorke use, a
paire of paralleles, or gemowe lines of one lengthe: bicause noe .2.
thynges, can be moare equalle."
Theorems and theories
Some equations present logical relations between mathematical
quantities, and the task of mathematicians is to prove they are valid.
Others provide information about an unknown quantity; here the task is
to solve the equation and make the unknown known. Equations in pure
mathematics are generally of the first kind: they reveal patterns and
regularities in mathematics itself. Pythagoras's theorem, an equation
expressed in the language of geometry, is an example. Given Euclid's
basic geometric assumptions, Pythagoras's theorem is true.
Equations in applied mathematics and mathematical physics are usually of
the second kind. They express properties of the universe that could, in
principle, have been otherwise. For example, Newton's law of gravity
tells us how to calculate the attractive force between two bodies.
Solving the resulting equations tells us how planets orbit the sun or
how to plot a trajectory for a space probe. But Newton's law isn't a
mathematical theorem; the law of gravity might have been different.
Indeed, it is different: Einstein's general
relativityimproves on Newton. And even that theory may not be the
Our choice of Maxwell's equations
In this story, and in his latest book, 17
Equations that Changed the World, Ian Stewart uses Maxwell's
equation for electromagnetic waves propagating in a vacuum. In the
accompanying video, we use the longer version of the equation that
doesn't depend on the waves moving through a vacuum.
Ian Stewart is
a mathematician at the University of Warwick, UK. His latest book, In
Pursuit of the Unknown: 17 Equations That Changed the World, is
published by Basic Books/Profile