2.
Another Frenchman of the 17th
Century, Pierre de Fermat, effectively
invented modern number theory virtually
single-handedly, despite being a small-town
amateur mathematician. Stimulated and
inspired by the “Arithmetica” of the
Hellenistic mathematician Diophantus, he
went on to discover several new patterns in
numbers which had defeated mathematicians
for centuries, and throughout his life he
devised a wide range of conjectures and
theorems. He is also given credit for early
developments that led to modern
calculus, and for early progress in probability
theory.
Pierre De Fermat (1601-1665)
3.
Fermat's mathematical work was
communicated mainly in letters
to friends, often with little or no
proof of his theorems. Although
he himself claimed to have
proved all his arithmetic
theorems, few records of his
proofs have survived, and many
mathematicians have doubted
some of his claims, especially
given the difficulty of some of
the problems and the limited
mathematical tools available to
Fermat.
One example of his many
theorems is the Two Square
Theorem, which shows that any
prime number which, when
divided by 4, leaves a remainder
of 1 (i.e. can be written in the
form 4n + 1), can always be rewritten as the sum of two
square numbers (see image at
right for examples).
Pierre De Fermat (1601-1665)
4.
His so-called Little Theorem is often used in the testing of large
prime numbers, and is the basis of the codes which protect our
credit cards in Internet transactions today. In simple (sic)
terms, it says that if we have two numbers a and p, where p is a
prime number and not a factor of a, then a multiplied by itself p1 times and then divided by p, will always leave a remainder of 1.
In mathematical terms, this is written: ap-1 = 1(mod p). For
example, if a = 7 and p = 3, then 72 ÷ 3 should leave a
remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.
Fermat identified a subset of numbers, now known as Fermat
numbers, which are of the form of one less than 2 to the power of
a power of 2, or, written mathematically, 22n + 1. The first five
such numbers are: 21 + 3 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1
= 257; and 216 + 1 = 65,537. Interestingly, these are all prime
numbers (and are known as Fermat primes), but all the higher
Fermat numbers which have been painstakingly identified over
the years are NOT prime numbers, which just goes to show the
value of inductive proof in mathematics.
Pierre De Fermat (1601-1665)
5.
Fermat’s Last Theorem
Fermat's pièce de
résistance, though, was his
famous Last Theorem, a
conjecture left unproven at his
death, and which puzzled
mathematicians for over 350
years. The theorem, originally
described in a scribbled note in
the margin of his copy of
Diophantus' “Arithmetica”, states
that no three positive integers
a, b and c can satisfy the
equation an + bn = cn for any
integer value of n greater than
two (i.e. squared). This
seemingly simple conjecture has
proved to be one of the world’s
hardest mathematical problems
to prove.
Pierre De Fermat (1601-1665)
6.
There are clearly many solutions - indeed, an
infinite number - when n = 2 (namely, all the
Pythagorean triples), but no solution could be
found for cubes or higher powers.
Tantalizingly, Fermat himself claimed to have a
proof, but wrote that “this margin is too small to
contain it”. As far as we know from the papers
which have come down to us, however, Fermat
only managed to partially prove the theorem for
the special case of n = 4, as did several other
mathematicians who applied themselves to it
(and indeed as had earlier mathematicians
dating back to Fibonacci, albeit not with the
same intent).
Pierre De Fermat (1601-1665)
7.
Over the centuries, several mathematical and
scientific academies offered substantial prizes for a
proof of the theorem, and to some extent it singlehandedly stimulated the development of algebraic
number theory in the 19th and 20th Centuries. It was
finally proved for ALL numbers only in 1995 (a proof
usually attributed to British mathematician Andrew
Wiles, although in reality it was a joint effort of
several steps involving many mathematicians over
several years). The final proof made use of complex
modern mathematics, such as the modularity
theorem for semi-stable elliptic curves, Galois
representations and Ribet’s epsilon theorem, all of
which were unavailable in Fermat’s time, so it seems
clear that Fermat's claim to have solved his last
theorem was almost certainly an exaggeration (or at
least a misunderstanding).
Pierre De Fermat (1601-1665)
8.
In addition to his work in number theory, Fermat
anticipated the development of calculus to some
extent, and his work in this field was invaluable later to
Newton and Leibniz. While investigating a technique for
finding the centers of gravity of various plane and solid
figures, he developed a method for determining
maxima, minima and tangents to various curves that was
essentially equivalent to differentiation. Also, using an
ingenious trick, he was able to reduce the integral of
general power functions to the sums of geometric series.
Fermat’s correspondence with his friend Pascal also helped
mathematicians grasp a very important concept in basic
probability which, although perhaps intuitive to us
now, was revolutionary in 1654, namely the idea of equally
probable outcomes and expected values.
Pierre De Fermat (1601-1665)
9.
The Frenchman Blaise Pascal was a prominent 17th
Century scientist, philosopher and mathematician.
Like so many great mathematicians, he was a child
prodigy and pursued many different avenues of
intellectual endeavor throughout his life. Much of his
early work was in the area of natural and applied
sciences, and he has a physical law named after him
(that “pressure exerted anywhere in a confined liquid
is transmitted equally and undiminished in all
directions throughout the liquid”), as well as the
international unit for the measurement of pressure.
In philosophy, Pascals’ Wager is his pragmatic
approach to believing in God on the grounds that is it
is a better “bet” than not to.
Blaise Pascal (1623-1662)
10.
The table of binomial coefficients
known as Pascal’s Triangle
But Pascal was also a
mathematician of the first order.
At the age of sixteen, he wrote a
significant treatise on the
subject of projective
geometry, known as Pascal's
Theorem, which states that, if a
hexagon is inscribed in a
circle, then the three
intersection points of opposite
sides lie on a single line, called
the Pascal line. As a young
man, he built a functional
calculating machine, able to
perform additions and
subtractions, to help his father
with his tax calculations.
Blaise Pascal (1623-1662)
11.
He is best known, however, for Pascal’s Triangle, a convenient tabular
presentation of binomial coefficients, where each number is the sum of the two
numbers directly above it. A binomial is a simple type of algebraic expression
which has just two terms operated on only by
addition, subtraction, multiplication and positive whole-number
exponents, such as (x + y)2. The coefficients produced when a binomial is
expanded form a symmetrical triangle (see image at right).
Pascal was far from the first to study this triangle. The Persian mathematician
Al-Karaji had produced something very similar as early as the 10th
Century, and the Triangle is called Yang Hui's Triangle in China after the 13th
Century Chinese mathematician, and Tartaglia’s Triangle in Italy after the
eponymous 16th Century Italian. But Pascal did contribute an elegant proof by
defining the numbers by recursion, and he also discovered many useful and
interesting patterns among the rows, columns and diagonals of the array of
numbers. For instance, looking at the diagonals alone, after the outside "skin"
of 1's, the next diagonal (1, 2, 3, 4, 5,...) is the natural numbers in order. The
next diagonal within that (1, 3, 6, 10, 15,...) is the triangular numbers in
order. The next (1, 4, 10, 20, 35,...) is the pyramidal triangular
numbers, etc, etc. It is also possible to find prime numbers, Fibonacci
numbers, Catalan numbers, and many other series, and even to find fractal
patterns within it.
Blaise Pascal (1623-1662)
12.
Pascal also made the conceptual leap to use the
Triangle to help solve problems in probability theory.
In fact, it was through his collaboration and
correspondence with his French contemporary Pierre
de Fermat and the Dutchman Christian Huygens on
the subject that the mathematical theory of
probability was born. Before Pascal, there was no
actual theory of probability - notwithstanding
Gerolamo Cardano’s early exposition in the 16th
Century - merely an understanding (of sorts) of how
to compute “chances” in dice and card games by
counting equally probable outcomes. Some
apparently quite elementary problems in probability
had eluded some of the best mathematicians, or
given rise to incorrect solutions.
Blaise Pascal (1623-1662)
13.
It fell to Pascal (with Fermat's help) to bring together
the separate threads of prior knowledge (including
Cardano's early work) and to introduce entirely new
mathematical techniques for the solution of problems
that had hitherto resisted solution. Two such
intransigent problems which Pascal and Fermat
applied themselves to were the Gambler’s Ruin
(determining the chances of winning for each of two
men playing a particular dice game with very specific
rules) and the Problem of Points (determining how a
game's winnings should be divided between two
equally skilled players if the game was ended
prematurely). His work on the Problem of Points in
particular, although unpublished at the time, was
highly influential in the unfolding new field.
Blaise Pascal (1623-1662)
14.
The Problem of Points at its simplest can be illustrated by a
simple game of “winner take all” involving the tossing of a
coin. The first of the two players (say, Fermat and Pascal)
to achieve ten points or wins is to receive a pot of 100
francs. But, if the game is interrupted at the point where
Fermat, say, is winning 8 points to 7, how is the 100 franc
pot to divide? Fermat claimed that, as he needed only two
more points to win the game, and Pascal needed three, the
game would have been over after four more tosses of the
coin (because, if Pascal did not get the necessary 3 points
for your victory over the four tosses, then Fermat must
have gained the necessary 2 points for his victory, and vice
versa. Fermat then exhaustively listed the possible
outcomes of the four tosses, and concluded that he would
win in 11 out of the 16 possible outcomes, so he suggested
that the 100 francs be split 11⁄16 (0.6875) to him and 5⁄16
(0.3125) to Pascal.
Blaise Pascal (1623-1662)
15.
Pascal then looked for a way of generalizing the problem
that would avoid the tedious listing of possibilities, and
realized that he could use rows from his triangle of
coefficients to generate the numbers, no matter how many
tosses of the coin remained. As Fermat needed 2 more
points to win the game and Pascal needed 3, he went to
the fifth (2 + 3) row of the triangle, i.e. 1, 4, 6, 4, 1. The
first 3 terms added together (1 + 4 + 6 = 11) represented
the outcomes where Fermat would win, and the last two
terms (4 + 1 = 5) the outcomes where Pascal would
win, out of the total number of outcomes represented by
the sum of the whole row (1 + 4 + 6 +4 +1 = 16).
Blaise Pascal (1623-1662)
16.
Pascal and Fermat had grasped through
their correspondence a very important
concept that, though perhaps intuitive to
us today, was all but revolutionary in
1654. This was the idea of equally
probable outcomes, that the probability
of something occurring could be
computed by enumerating the number of
equally likely ways it could occur, and
dividing this by the total number of
possible outcomes of the given situation.
This allowed the use of fractions and
ratios in the calculation of the likelihood
of events, and the operation of
multiplication and addition on these
fractional probabilities. For example, the
probability of throwing a 6 on a die twice
is 1⁄6 x 1⁄6 = 1⁄36 ("and" works like
multiplication); the probability of
throwing either a 3 or a 6 is 1⁄6 + 1⁄6 =
1⁄ ("or" works like addition).
3
Blaise Pascal (1623-1662)
17.
Later in life, Pascal and his sister Jacqueline strongly
identified with the extreme Catholic religious
movement of Jansenism. Following the death of his
father and a "mystical experience" in late 1654, he
had his "second conversion" and abandoned his
scientific work completely, devoting himself to
philosophy and theology. His two most famous
works, the "Lettres provinciales" and the
"Pensées", date from this period, the latter left
incomplete at his death in 1662. They remain Pascal’s
best known legacy, and he is usually remembered
today as one of the most important authors of the
French Classical Period and one of the greatest
masters of French prose, much more than for his
contributions to mathematics.
Blaise Pascal (1623-1662)
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