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Fourteen.
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50
mathematical ideas
you really need to know
01 Zero
02 Number systems
03 Fractions
04 Squares and square roots
05 π
06 e
07 Infinity
08 Imaginary numbers
09 Primes
10 Perfect numbers
11 Fibonacci numbers
12 Golden rectangles
13 Pascal’s triangle
14 Algebra
15 Euclid’s algorithm
16 Logic
17 Proof
3
18 Sets
19 Calculus
20 Constructions
21 Triangles
22 Curves
23 Topology
24 Dimension
25 Fractals
26 Chaos
27 The parallel postulate
28 Discrete geometry
29 Graphs
30 The four-colour problem
31 Probability
32 Bayes’s theory
33 The birthday problem
34 Distributions
35 The normal curve
36 Connecting data
37 Genetics
38 Groups
39 Matrices
40 Codes
41 Advanced counting
42 Magic squares
43 Latin squares
44 Money mathematics
45 The diet problem
46 The travelling salesperson
47 Game theory
48 Relativity
49 Fermat’s last theorem
50 The Riemann hypothesis
Glossary
Index
Introduction
Mathematics is a vast subject and no one can possibly know it all. What one can do is explore and
find an individual pathway. The possibilities open to us here will lead to other times and different
cultures and to ideas that have intrigued mathematicians for centuries.
Mathematics is both ancient and modern and is built up from widespread cultural and political
influences. From India and Arabia we derive our modern numbering system but it is one tempered
with historical barnacles. The ‘base 60’ of the Babylonians of two or three millennia BC shows up in
our own culture – we have 60 seconds in a minute and 60 minutes in an hour; a right angle is still
90 degrees and not 100 grads as revolutionary France adopted in a first move towards
decimalization.
The technological triumphs of the modern age depend on mathematics and surely there is no
longer any pride left in announcing to have been no good at it when at school. Of course school
mathematics is a different thing, often taught with an eye to examinations. The time pressure of
school does not help either, for mathematics is a subject where there is no merit in being fast.
People need time to allow the ideas to sink in. Some of the greatest mathematicians have been
painfully slow as they strove to understand the deep concepts of their subject.
There is no hurry with this book. It can be dipped into at leisure. Take your time and discover
what these ideas you may have heard of really mean. Beginning with Zero, or elsewhere if you wish,
you can move on a trip between islands of mathematical ideas. For instance, you can become
knowledgeable about Game theory and next read about Magic squares. Alternatively you can move
from Golden rectangles to the famous Fermat’s last theorem, or any other path.
This is an exciting time for mathematics. Some of its major problems have been solved in recent
times. Modern computing developments have helped with some but been helpless against others.
The Four-colour problem was solved with the aid of a computer, but the Riemann hypothesis, the
final chapter of the book, remains unsolved – by computer or any other means.
Mathematics is for all. The popularity of Sudoku is evidence that people can do mathematics
(without knowing it) and enjoy it too. In mathematics, like art or music, there have been the
geniuses but theirs is not the whole story. You will see several leaders making entrances and exits in
some chapters only to reappear in others. Leonhard Euler, whose tercentenary occurs in 2007, is a
frequent visitor to these pages. But, real progress in mathematics is the work of ‘the many’
accumulated over centuries. The choice of 50 topics is a personal one but I have tried to keep a
balance. There are everyday and advanced items, pure and applied mathematics, abstract and
concrete, the old and the new. Mathematics though is one united subject and the difficulty in writing
has not been in choosing topics, but in leaving some out. There could have been 500 ideas but 50
are enough for a good beginning to your mathematical career.
01 Zero
At a young age we make an unsteady entrance into numberland. We learn that 1 is first
in the ‘number alphabet’, and that it introduces the counting numbers 1, 2, 3, 4, 5,. . .
Counting numbers are just that: they count real things – apples, oranges, bananas,
pears. It is only later that we can count the number of apples in a box when there are
none.
Even the early Greeks, who advanced science and mathematics by quantum
leaps, and the Romans, renowned for their feats of engineering, lacked an
effective way of dealing with the number of apples in an empty box. They failed
to give ‘nothing’ a name. The Romans had their ways of combining I, V, X, L, C,
D and M but where was 0? They did not count ‘nothing’.
How did zero become accepted?
The use of a symbol designating ‘nothingness’ is thought to have originated
thousands of years ago. The Maya civilization in what is now Mexico used zero in
various forms. A little later, the astronomer Claudius Ptolemy, influenced by the
Babylonians, used a symbol akin to our modern 0 as a placeholder in his number
system. As a placeholder, zero could be used to distinguish between examples
(in modern notation) such as 75 and 705, instead of relying on context as the
Babylonians had done. This might be compared with the introduction of the
‘comma’ into language – both help with reading the right meaning. But, just as
the comma comes with a set of rules for its use – there have to be rules for using
zero.
The seventh-century Indian mathematician Brahmagupta treated zero as a
‘number’, not merely as a placeholder, and set out rules for dealing with it. These
included ‘the sum of a positive number and zero is positive’ and ‘the sum of zero
and zero is zero’. In thinking of zero as a number rather than a placeholder, he
was quite advanced. The Hindu-Arabic numbering system which included zero in
this way was promulgated in the West by Leonardo of Pisa – Fibonacci – in his
Liber Abaci (The Book of Counting) first published in 1202. Brought up in North
Africa and schooled in the Hindu-Arabian arithmetic, he recognized the power of
using the extra sign 0 combined with the Hindu symbols 1, 2, 3, 4, 5, 6, 7, 8 and
9.
The launch of zero into the number system posed a problem which
Brahmagupta had briefly addressed: how was this ‘interloper’ to be treated? He
had made a start but his nostrums were vague. How could zero be integrated
into the existing system of arithmetic in a more precise way? Some adjustments
were straightforward. When it came to addition and multiplication, 0 fitted in
neatly, but the operations of subtraction and division did not sit easily with the
‘foreigner’. Meanings were needed to ensure that 0 harmonized with the rest of
accepted arithmetic.
How does zero work?
Adding and multiplying with zero is straightforward and uncontentious – you
can add 0 to 10 to get a hundred – but we shall amean ‘add’ in the less
imaginative way of the numerical operation. Adding 0 to a number leaves that
number unchanged while multiplying 0 by any number always gives 0 as the
answer. For example, we have 7 + 0 = 7 and 7 × 0 = 0. Subtraction is a simple
operation but can lead to negatives, 7 0 = 7 and 0 7 = 7, while division
involving zero raises difficulties.
Let’s imagine a length to be measured with a measuring rod. Suppose the
measuring rod is actually 7 units in length. We are interested in how many
measuring rods we can lie along our given length. If the length to be measured
is actually 28 units the answer is 28 divided by 7 or in symbols 2 8 ÷ 7 = 4. A
better notation to express this division is
and then we can ‘cross-multiply’ to write this in terms of multiplication, as 2 8
= 7 × 4. What now can be made of 0 divided by 7? To help suggest an answer
in this case let us call the answer a so that
By cross-multiplication this is equivalent to 0 = 7 × a. If this is the case, the
only possible value for a is 0 itself because if the multiplication of two numbers
gives 0, one of them must be 0. Clearly it is not 7 so a must be a zero.
This is not the main difficulty with zero. The danger point is division by 0. If
we attempt to treat 7/0 in the same way as we did with 0/7, we would have the
equation
By cross-multiplication, 0 × b = 7 and we wind up with the nonsense that 0 =
7. By admitting the possibility of 7/0 being a number we have the potential for
numerical mayhem on a grand scale. The way out of this is to say that 7/0 is
undefined. It is not permissible to get any sense from the operation of dividing 7
(or any other nonzero number) by 0 and so we simply do not allow this
operation to take place. In a similar way it is not permissible to place a comma in
the mid,dle of a word without descending into nonsense.
The 12th-century Indian mathematician Bhaskara, following in the footsteps of
Brahmagupta, considered division by 0 and suggested that a number divided by
0 was infinite. This is reasonable because if we divide a number by a very small
number the answer is very large. For example, 7 divided by a tenth is 70, and by
a hundredth is 700. By making the denominator number smaller and smaller the
answer we get is larger and larger. In the ultimate smallness, 0 itself, the answer
should be infinity. By adopting this form of reasoning, we are put in the position
of explaining an even more bizarre concept – that is, infinity. Wrestling with
infinity does not help; infinity (with its standard notation ∞) does not conform to
the usual rules of arithmetic and is not a number in the usual sense.
If 7/0 presented a problem, what can be done with the even more bizarre
0/0? If 0/0 = c, by cross-multiplication, we arrive at the equation 0 = 0 ×c and
the fact that 0 = 0. This is not particularly illuminating but it is not nonsense
either. In fact, c can be any number and we do not arrive at an impossibility. We
reach the conclusion that 0/0 can be anything; in polite mathematical circles it is
called ‘indeterminate’.
All in all, when we consider dividing by zero we arrive at the conclusion that it
is best to exclude the operation from the way we do calculations. Arithmetic can
be conducted quite happily without it.
What use is zero?
9
We simply could not do without 0. The progress of science has depended on
it. We talk about zero degrees longitude, zero degrees on the temperature scale,
and likewise zero energy, and zero gravity. It has entered the non-scientific
language with such ideas as the zero-hour and zero-tolerance.CITAZIONEAll about nothing
The sum of zero and a positive number is positive
The sum of zero and a negative number is negative
The sum of a positive and a negative is their difference; or, if they are equal, zero
Zero divided by a negative or positive number is either zero or is expressed as a
fraction with zero as numerator and the finite quantity as denominator
Brahmagupta, AD628
Greater use could be made of it though. If you step off the 5th Ave sidewalk
in New York City and into the Empire State Building, you are in the magnificent
entrance lobby on Floor Number 1. This makes use of the ability of numbers to
order, 1 for ‘first’, 2 for ‘second’ and so on, up to 102 for ‘a hundred and second.’
In Europe they do have a Floor 0 but there is a reluctance to call it that.
Mathematics could not function without zero. It is in the kernel of
mathematical concepts which make the number system, algebra, and geometry
go round. On the number line 0 is the number that separates the positive
numbers from the negatives and thus occupies a privileged position. In the
decimal system, zero serves as a place holder which enables us to use both huge
numbers and microscopic figures.
Over the course of hundreds of years zero has become accepted and utilized,
becoming one of the greatest inventions of man. The 19th-century American
mathematician G.B. Halsted adapted Shakespeare’s Midsummer Night’s Dream to
write of it as the engine of progress that gives ‘to airy nothing, not merely a local
habitation and a name, a picture, a symbol, but helpful power, is the
characteristic of the Hindu race from whence it sprang’.
When 0 was introduced it must have been thought odd, but mathematicians
have a habit of fastening onto strange concepts which are proved useful much
later. The modern day equivalent occurs in set theory where the concept of a set
is a collection of elements. In this theory Φ designates the set without any
elements at all, the so-called ‘empty set’. Now that is an odd idea, but like 0 it is
indispensible.
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50 mathematical ideas you really need to know |
