From 0 to Infinity in 26 Centuries (12 page)
Read From 0 to Infinity in 26 Centuries Online
Authors: Chris Waring
In action
The slide rule works because of the law of logarithms:
log (a × b) = log (a) + log (b)
Let’s test this out using some easy numbers:
log (10 × 1000) = log (10) + log (1000)
= 1 + 3
= 4
From this we know that 10 × 1000 = 10
4
= 10000.
Therefore, if you want to multiply two numbers together you can ‘take the log’ of each number, add them together and then raise ten by this number. Before the slide
rule, you would have had to look up this answer in a big, expensive book of tables – so the slide rule was a remarkably convenient development for mathematicians and scientists.
Jumping ahead
Rather than spacing out successive markings along an axis by adding on the same value each time, a logarithmic scale spaces them out by a multiple (normally of 10) each time.
For example, a graph with a normal
linear scale
would go 0, 1, 2, 3; a graph with a logarithmic scale would go 0, 1, 10, 100. The graphs of y=10
x
shown below are with a linear and
logarithmic scale respectively.
R
ENÉ
D
ESCARTES
(1596–1650)
Born in France, René Descartes was an important philosopher, perhaps best known for coining the statement ‘I think, therefore I am’.
A different equation
Descartes was a sickly child and was allowed to sleep in every morning to recuperate, which became a lifelong habit. Apparently, during one such lie-in Descartes was watching a
fly walk across the ceiling and wondered how he might be able to describe accurately the position of the fly at any given time. He realized that if he mapped out the ceiling with a square grid he
could use what we now call
coordinates
to record exactly where the fly was positioned.
These
Cartesian graphs
proved to be incredibly useful in mathematics because they linked the fields of geometry and algebra. Equations could now be
drawn on a set of numbered axes, which allowed them to be investigated more easily by sight, rather than solving them algebraically.
Cartesian geometry encouraged mathematicians to think about the graphical properties of equations, such as whether or not the lines of equations are parallel. To find out whether lines
are
parallel you need to work out their
gradient
. If two lines have the same gradient they must be parallel. Only the simplest linear equations are a straight line, which makes their
gradients straightforward to work out. Curves, however, have a changing gradient, but thanks to Descartes’ breakthrough the way was paved for Isaac Newton and Gottfried Wilhelm Leibniz to
discover calculus (see
here
).
The Proof of the Matter
Descartes’ philosophy also assumed a mathematical bent because he believed the universe was set out according to mathematical rules. For example, in
his book
Meditations on First Philosophy
, Descartes discards any beliefs he holds that are unproven, and then builds his philosophy of the way in which reality works without recourse to
anything other than proven fact. This sceptical viewpoint is one of the ideas that underpins modern science and makes mathematics the scientist’s most powerful and productive
tool.
P
IERRE DE
F
ERMAT
(1601–65)
A French lawyer, Pierre de Fermat spent his spare time pursuing his love of mathematics. Although he did not publish any of his ideas when he was alive, he did share them in
letter form with his mathematical contemporaries. Frustratingly, however, Fermat seldom found it necessary to provide proof for his work.
Fermat worked in many areas of mathematics, but the area for which he is famed is
number theory
: the study of
integers
(whole numbers) and the attempt to find integer solutions to
equations.
Fermat is famous for his ‘last theorem’, which he wrote in his copy of Diophantus’
Arithmetica
, and which was discovered by his son after his death (see
here
). Fermat
was also interested in
perfect numbers
(see box
here
) and primes.
In their prime
Fermat devised a method of testing whether or not a number is prime that relies on an algebraic trick known as the
difference of two squares
.
This says that:
a
2
- b
2
= (a + b)(a - b)
Not So Perfect
A perfect number is a number whose factors (not including the number itself) add up to make the number.
For example, 6 is a perfect number (the first one, in fact) because the factors of 6 are 1, 2 and 3 and 1 + 2 + 3 = 6
Perfect numbers are rare – the next one is 28, followed by 496 and then 8128. The fifth perfect number is 33,550,336.
Numbers whose factors add up to less than the number are called
deficient numbers
. For example, 8’s factors are 1, 2 and 4, which have a sum of 7, so 8 is
deficient.
Numbers whose factors add up to more than the number are called
abundant numbers
, e.g. the sum of 12’s factors are: 1 + 2 + 3 + 4 + 6 = 16.
In the case of some abundant numbers, no combination of its factors will make up the number. These numbers are called
weird numbers
. For example, 24 is not a weird
number because we can add its factors together (2, 4, 6 and 12) to make 24. The first weird number is 70 – we cannot add any combination of its factors 1, 2, 5, 7, 10, 14 and 35 to get a
total of 70.
For example, if a = 8 and b = 5:
8
2
- 5
2
= (8 + 5)(8 - 5)
64 - 25 = 13 × 3
39 = 39
Fermat needed to test odd numbers (because 2 is the only even prime) to see if they were prime. He made the number he was testing, n, equal to the difference of two squares:
n = a
2
- b
2
which means that:
n = (a + b)(a - b)
This shows that n is two numbers multiplied together, in which case n cannot be a prime number unless (a + b) = n and (a - b) = 1.
Fermat took the first statement and rearranged it:
a
2
- n = b
2
This meant that he could pick a starting value for a, square it and subtract n and see if he was left with a perfect square which is easily identifiable. If b
2
was
not a perfect square he would increase his starting value by one and try again until he either
found numbers for a and b that worked or got to a point where a × b was
larger than n.
B
LAISE
P
ASCAL
(1623–62)
Educated by his father, Blaise Pascal was a French prodigy who worked in the fields of mathematics, physics and religion. His precocious talent saw his first mathematical paper
published at the tender age of sixteen.
Speeding things up
Pascal’s father was a tax collector during a time of war in Europe, which made his job a somewhat onerous endeavour. Pascal sought to help his father by developing the
first mechanical calculator – a machine known as a ‘Pascaline’, designed to add and subtract numbers. After he’d created a number of prototypes, Pascal’s finished
product comprised a box with a series of numbered dials on its front and with a digit displayed above each dial. Numbers to be added were ‘dialled’ into the machine and the result would
be displayed.
Unfortunately, the Pascaline was very expensive to make and was therefore seen as more of a deluxe executive toy than a useful mathematical device. But Pascal’s contribution to mathematics
should not be underestimated – he paved the way for Leibniz and others to develop more effective mechanical calculators and, eventually, modern computing.
In all probability
Pascal was also interested in games of chance and gambling. His work with his acquaintance Pierre de Fermat (see
here
) led to the field of mathematics we now call
probability
. In probability, we talk about an event (e.g. rolling a die) having a certain number of outcomes (rolling a 1, 2, 3, 4, 5 or 6 has six outcomes). Each outcome has a probability
– for our die, each outcome is equally likely – which is expressed as a fraction (1/6), and the sum of the probabilities of all the events must add up to one. Probability is part of the
branch of mathematics called
statistics
, which has a wide variety of applications in science and economics.
Under pressure
As a scientist, Pascal was fascinated with the notion of a vacuum. At the time, many scientists conformed to the view expounded by Aristotle: vacuums cannot exist; you cannot
have emptiness because ‘nature abhors a vacuum’. However, Pascal noticed that if you place a glass beaker upside down in liquid (he used mercury) and then pull it out, there is a small
gap at the top of the up-ended beaker that somehow holds up the column of liquid below it. He reasoned that this could only be a vacuum and that it must provide some sort of suction force to hold
up the liquid.
Pascal went on to conduct more experiments on pressure within liquids and as a result the unit of pressure is called the Pascal (Pa) in his honour.
Absolute Proof
In 1654 Pascal had a profound religious experience and it changed the course of his life. He subsequently devoted himself to an ascetic existence and
focused on writing theological commentaries. He used his knowledge of probability to expound a reason for assuming God exists, now known as
Pascal’s wager
:
You cannot tell whether God does or does not exist through logic.
If you believe that he exists and he does not, you lose nothing.
If you believe that he does not exist and he does, you lose an eternal afterlife.
Therefore, there is nothing to lose and possibly infinite reward to be had from believing in God and nothing to gain from not believing in him.
So, on balance, you may as well believe God exists.
I
SAAC
N
EWTON
(1642–1727)
One of the greatest scientists of his era, Isaac Newton hailed from Lincolnshire and wrote one of the most important books ever to be written:
The Mathematical Principles of
Natural Philosophy,
often known by a shortened version of its Latin name
Principia Mathematica
. In the book Newton essentially rewrites the laws of physics that govern the way objects
move and react to forces exerted on them. With his laws, Newton was able to explain the motion of the planets and prove conclusively that the sun sits at the centre of the solar system.