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.

The Pythagoreans figured out what makes strings sound harmonious

*(Image: Nils Jorgensen/Rex Features)*
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.

**Graphic:**See 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 as earthquakes.
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 theories.

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 force.

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 waves.

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
Tesla, Guglielmo 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 particle.

### 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 and frequencies.

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 history books.

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 relativity improves on Newton. And even that theory may not be the last word.

**Ian Stewart**is a mathematician at the University of Warwick, UK. His latest book,*17 Equations That Changed the World*, is published by Profile*http://www.newscientist.com/*

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