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Vol. 1, No. 1, June 2012 | At Right Angles 61
To talk about a book on mathematics as ‘entertaining’ or
a ‘page-turner’ may look out of place; but that is exactly
how one would describe Simon Singh’s book, Fermat’s
Enigma. The book starts in a dramatic manner: “This was the
most important mathematics lecture of the century”. Singh is
writing about a lecture to be delivered by Andrew Wiles on 23
June, 1993; he was going to sketch a proof of Fermat’s last theo- rem in this lecture. It was known as the ‘last’ theorem because
it was the only remaining ‘theorem’ stated by the 17th century
mathematician Pierre de Fermat which had neither been proved
nor disproved, despite close attention given to it over the course
of three and a half centuries by some of the greatest mathemati- cians. (Technically it ought to have been called a ‘conjecture’
as no proof had been found as yet). One can imagine an atmo- sphere of tension and excitement in the lecture hall at the pros- pect of the theorem finally being proved.
Book Review:
Three centuries of brain racking discovery
xn
+yn
=zn ?
Fermat’s Enigma – The Epic Quest to Solve
The World's Greatest Mathematical Problem
by Simon Singh
Reviewed by Tanuj Shah
review
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62 At Right Angles | Vol. 1, No. 1, June 2012
What Singh manages to do in the book is weave a
story with several strands into a colourful tapestry.
The story navigates between biographical, histori- cal and mathematical topics in a fluid and intrigu- ing manner. It captures the spirit that drives and
inspires mathematicians to take on intellectual
challenges. The protagonist is Andrew Wiles, who
as a ten year old dreamed of solving one of the
most enduring problems of mathematics – that of
finding a proof of Fermat’s last theorem, or FLT as
it is called – and ultimately went on to solve it.
The FLT states that for the equation xn + yn
= zn
there are no solutions in positive integers when
n is an integer greater than 2.
Singh starts by looking at the origins of the equa- tion in ancient Greece in what we call the Theorem
of Pythagoras.1 This is the case n=2 of the equa- tion, that is, x2 + y2
= z2
. There is a short biogra- phy of Pythagoras, describing how he starts the
‘Pythagorean Brotherhood’ dedicated to discover- ing the meaning and purpose of life. He believed
that numbers held a special key to unlocking the
secrets of the universe. The Brotherhood was
fascinated by notions such as perfect numbers,
i.e., numbers whose proper divisors add up to
the number itself (for example, 6). Their world of
numbers consisted of the counting numbers and
rational numbers, which are ratios of counting
numbers. They found a surprising number of con- nections between these and nature, including the
ratios responsible for harmony in music. However
their strong belief in the importance of rational
numbers proved to undermine further progress
by the Brotherhood in the field of Mathematics.
There is an apocryphal story where one of the
disciples proved that 2 is irrational; the Broth- erhood felt that this threatened their worldview,
which was based on rational numbers, and sup- posedly the disciple was put to death. However
Pythagoras can be credited with laying the foun- dation of modern mathematics by introducing the
notion of proof: starting with a statement that is
self evident (an axiom) and arriving at a conclu- sion through a step-by-step logical argument. The
theorem that goes by his name is true for all right
angled triangles and it is not necessary to test it
on all right angled triangles, as it rests on logic
that cannot be refuted. (One of the few hundreds
of proofs that exist is given in the appendix of the
book). Teaching the Pythagorean Theorem by
giving the historical back ground would surely
broaden students’ horizons and deepen their
interest in the topic.
Singh makes a detour at this point, making a dis- tinction between mathematical proof and scientific
proof. The demands made by mathematical proof
are absolute; it has to be true for all cases whereas
a scientific theory is only a model or an approxi- mation to the truth. This is one reason why the
Pythagorean Theorem remains accepted 2500
years after it was first proved, while many scien- tific theories have been supplanted over the years.
These are ideas that a teacher can incorporate
while teaching the theorem. Further along, differ- ent types of proofs like proof by contradiction and
proof by induction are explained in a lucid manner
with examples given in the appendix, which an
average 14 or 15 year old would easily be able to
follow.
Chapter 2 and 3 focuses on some prominent
mathematicians who tried to tackle the problem,
starting with Fermat, the person who posed the
problem. Fermat was a civil servant – indeed, a
judge – who devoted all his leisure time to the
study of Mathematics. He was a very private man
and hardly met any other mathematician; the only
one with whom he collaborated with was Blaise
Pascal, on formulating the laws of probability.
With Father Marin Mersenne he would share his
findings, and Mersenne in turn would pass on the
news to other mathematicians. Fermat also had
a hand in developing calculus; he was one of the
first mathematicians to develop a way of finding
tangents to curves. However, the reason he has
become a household name is for his ‘last’ theorem,
which he jotted in the margins of the Arithmetica,
adding: “I have a truly marvelous demonstration of
this proposition which this margin is too narrow to
contain.” This statement spurred a large number
of mathematicians to try and prove it, while others
contested the claims.
Following this, the author gives a brief biography
of Leonhard Euler, one of the greatest mathemati- cians of the 18th Century. There is enough material
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Vol. 1, No. 1, June 2012 | At Right Angles 63
here that can be shared with students to give them
a glimpse of Euler, and they will certainly enjoy
tackling the problem of the Königsberg bridges.
There is also a small diversion into the structure
of numbers to introduce the idea of imaginary
or complex numbers. Euler, who made the first
breakthrough on the problem by giving a proof of
the case when n=3, had to make use of imaginary
numbers. Teachers can test students’ understand- ing of exponents by asking what other cases of
the theorem have been proved once you prove it
for n=3. (The theorem in this case states that the
equation x3
+y3 =z3
has no solutions in positive
integers.)
The next mathematician to make a breakthrough
on this problem was Sophie Germain. The author
engagingly brings through the difficulties women
had to undergo to establish themselves in this field,
which was regarded as a domain for men only. The
story of how her father tried to dissuade her from
pursuing mathematics by taking away the candles
will bring a tear to most. The determination with
which she continued her studies including taking
up an identity of a man will inspire girls; even now,
mathematics tends to be thought of as a subject
for boys. It will definitely help in puncturing some
stereotypes that people hold.
The second part of the book focuses on the discov- eries made in the 20th Century that finally helped in
cracking the theorem. Though one may not un- derstand the mathematics behind ‘modular forms’
or ‘elliptic curves’, Singh provides good analogies
to help the reader keep up with the story without
making any excessive technical demands. How two
Japanese mathematicians linked the above two
areas of mathematics with the Taniyama-Shimura
conjecture which led to new approaches in tackling
Fermat’s last theorem is described in a riveting
manner, with a touching account of the tragedy
that befell one of them. All along Singh keeps the
story moving by giving details of Wiles’ career and
his attempts at solving the problem, which finally
culminates in his lecture in Cambridge in June
1993. However, this is not the end of the story; at
the beginning there had been a hint that there was
more to come by saying “While a general mood of
euphoria filled the Newton Institute, everybody re- alised that the proof had to be rigorously checked
by a team of independent referees. However, as
Wiles enjoyed the moment, nobody could have
predicted the controversy that would evolve in the
months ahead.”
Simon Singh has managed to show that mathemati- cians are people with a great passion to discover
the highest truth, and he has certainly succeeded in
portraying mathematics as a subject of beauty. The
book will inspire teachers and students alike, and
is recommended to all classes of readers.
Tanuj Shah teaches Mathematics in Rishi Valley School. He has a deep passion for making mathematics
accessible and interesting for all and has developed hands-on self learning modules for the Junior School.
Tanuj Shah did his teacher training at Nottingham University and taught in various Schools in England
before joining Rishi Valley School. He may be contacted at tanuj@rishivalley.org
Reference
1 Editor's note: The article by J Shashidhar, elsewhere in this issue, gives more information on the history of the Pythagorean
Theorem.