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Chapter 2
Numbers
2.1 Introduction
Numbers are at the heart of mathematics. Some, like prime numbers, are enigmatic,
and defy all attempts to predict their arrival; while others, like integers, are well
behaved, and can be counted on! This chapter revises some basic ideas about numbers and how we refer to them in the context of mathematics for computer graphics.
In particular, I include: natural, prime, integer, rational, irrational, real, imaginary
and complex numbers. I also provide a short description on set notation, and our
positional number system.
2.2 Background
Over the centuries, mathematicians have realised that in order to progress, they must
give precise definitions to their discoveries, ideas and concepts, so that they can
be built upon and referenced by new mathematical inventions. In the event of any
new discovery, these definitions have to be occasionally changed or extended. For
example, once upon a time, integers, rational and irrational numbers, satisfied all the
needs of mathematicians, until imaginary quantities were invented. Today, complex
numbers have helped shape the current number system hierarchy. Consequently,
there must be clear definitions for numbers, and the operators that act upon them.
Therefore, we need to identify the types of numbers that exist, what they are used
for, and any problems that arise when they are stored in a computer.
2.3 Set Notation
A set is a collection of related things identified by a unique name. For example,
I own a set of books stored in my library, and a set of photographs stored on my
computer. I could call them my-books and my-photos. Similarly, numbers are organised into sets represented by a single letter such as R for reals and Z for integers.
J. Vince, Mathematics for Computer Graphics,
Undergraduate Topics in Computer Science,
DOI 10.1007/978-1-4471-6290-2_2, © Springer-Verlag London 2014
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2 Numbers
Set theory provides a logical framework for manipulating and interrogating sets,
which does not concern us here, apart from the statement confirming that something
is a member of a set. For example, the symbol ∈ means “is a member of”, which
permits me to state
QFCG ∈ my-books
where QFCG stands for my book Quaternions for Computer Graphics.
It is also common practice to use a symbol such as λ (lambda), to define a number
as follows:
λ∈R
which means that λ is a member of the set of real numbers R, and is thus a real
quantity.
2.4 Positional Number System
All sorts of number system have been proposed by previous civilisations, but our
current system is a positional number system using a base ten. The number 1234
stands for the sum of one thousand, plus two hundreds, plus three tens, plus four
ones, which is expressed:
1234 = 1 × 1000 + 2 × 100 + 3 × 10 + 4 × 1.
It should be obvious that the base ten is nothing special; it just so happens that
human beings have evolved with ten digits, which we use for counting. This suggests
that any number can be used as a base: 2, 3, 4, 5, 6, 7, etc. In fact, the decimal number
system is not very convenient for computer technology, where digital circuits switch
on and off trillions of times a second using binary numbers—numbers to a base 2—
with great ease. In this text there is no real need to explore such numbers, and are
left to programmers who have to master number systems such as binary (base 2),
octal (base 8) and hexidecimal (base 16).
2.5 Natural Numbers
The natural numbers [1, 2, 3, 4, . . .] are used for counting, ordering and labelling
and represented by the set N. When zero is added to the set, N∗ is used. Note that
negative numbers are not included. We use natural numbers to subscript a quantity
to distinguish one element from another, e.g. x1 , x2 , x3 , x4 , . . . .
2.6 Prime Numbers
A prime number [2, 3, 5, 7, 11, 13, 17, . . .] is a natural number that can be divided
only by 1 and itself, without leaving a remainder. There are 25 primes less than 100,
2.7 Integer Numbers
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168 primes less than 1,000 and 455,052,512 primes less than 10,000,000,000. The
fundamental theory of arithmetic states: “Any positive integer (other than 1) can be
written as the product of two or more prime numbers in one and only one way.” For
example:
25 = 5 × 5
26 = 2 × 13
27 = 3 × 3 × 3
28 = 2 × 2 × 7
29 = 29
30 = 2 × 3 × 5
92365 = 5 × 7 × 7 × 13 × 29.
In 1742, the German mathematician Christian Goldbach (1690–1764) conjectured
that every even integer greater than 2 could be written as the sum of two primes:
4=2+2
14 = 11 + 3
18 = 11 + 7, etc.
No one has ever found an exception to this conjecture, and no one has ever confirmed
it. Although prime numbers are enigmatic and have taxed the brains of the greatest
mathematicians, they unfortunately, play no part in computer graphics!
2.7 Integer Numbers
Integers include positive and negative natural numbers: [. . . , −2, −1, 0, 1, 2, 3, . . . ]
and are represented by the set Z, or Z∗ without zero. Zero, in any form, is a useful
concept, but does not sit comfortably alongside other numbers, especially when it is
a divisor.
2.8 Rational Numbers
Rational or fractional numbers are numbers that can
√ be represented as a fraction
and are represented by the set Q. For example, 2, 16, 0.25 are rational numbers
because
2=
4
2
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2 Numbers
Fig. 2.1 The number line
√
8
16 = 4 =
2
1
0.25 = .
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Some rational numbers can be stored accurately inside a computer, but many others
can only be stored approximately. For example, 4/3 = 1.333333 . . . produces an
infinite sequence of threes and has to be truncated when stored as a binary number.
2.9 Irrational Numbers
An irrational number cannot be represented as a fraction. Examples are
√
2 = 1.414213562 . . .
π = 3.141592653 . . .
e = 2.718281828 . . .
Such numbers never terminate and are always subject to a small error when stored
within a computer.
2.10 Real Numbers
Rational and irrational numbers together comprise the set of real numbers which
are represented by the letter R, or R∗ without zero. Typical examples are 1.5, 0.004,
12.999 and 23.0.
2.11 The Number Line
It is useful to visualise numbers in the form of a number line, as shown in Fig. 2.1,
which helps us understand complex numbers.
2.12 Complex Numbers
The Swiss mathematician Leonhard Euler (1707–1783) (pronounced “Oiler”)
played a significant role in putting complex numbers on the map. His ideas on
2.12
Complex Numbers
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Fig. 2.2 Rotating numbers
through 180◦ reverses their
sign
rotations are also used in computer graphics to locate objects and virtual cameras in
space, as we shall see later on.
Complex numbers resolve some awkward problems that arise when attempting
to solve certain types of equations and are represented by the set C. For example,
x 2 − 4 = 0 has solutions x = ±2. But x 2 + 4 = 0 has no real-number solution. However, the number line provides a graphical interpretation for a new type of number:
the complex number. The name is rather misleading: it is not complex; it is rather
simple.
Consider the scenario depicted in Fig. 2.2. Any number on the number line is
related to the same number with the opposite sign via an anticlockwise rotation of
180◦ . For example, if 2 is rotated 180◦ about zero, it becomes −2, and if −3 is
rotated 180◦ about zero it becomes 3.
We can now write −2 = −1 × 2, or 3 = −1 × −3, where −1 is effectively a
rotation through 180◦ . But a rotation of 180◦ can be interpreted as two consecutive
rotations of 90◦ , and the question now arises: what represents a rotation of 90◦ ?
Well, let’s assume we don’t know the answer to this question—even though some
of you do—we can at least give a name to the operation, and what better name to
use than i.
So the letter i represents an anticlockwise rotation of 90◦ . Therefore 2i is equivalent to lifting 2 out of the number line, rotating it 90◦ and leaving it hanging in
limbo. But if we take this “imaginary” number and subject it to a further rotation
we can write 2ii = −2, which means
of 90◦ , i.e. 2ii, it becomes −2. Therefore,
√
that ii = −1. But if this is so, i = −1! Therefore, i is not really a number, but an
operator—an operator that effectively rotates a number anticlockwise through 90◦ .
So now we have two types of number: ordinary numbers and complex numbers.
Ordinary numbers are the everyday numbers we use for counting and so on, whereas
complex numbers have a mixture of real and imaginary components, and help resolve a wide range of mathematical problems. For example, a complex number z
has the following definition:
z = a + bi,
a, b ∈ R
where b is multiplied by i, and makes it imaginary.
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Fig. 2.3 The graphical representation of complex numbers
Complex numbers obey all the normal laws of algebra. For example, if we multiply (a + bi) by (c + di) we have
(a + bi)(c + di) = ac + adi + bci + bdi2 .
Collecting up like terms and substituting −1 for i2 we get
(a + bi)(c + di) = ac + (ad + bc)i − bd
and simplifies to
(a + bi)(c + di) = ac − bd + (ad + bc)i
which is another complex number.
Something interesting happens when we multiply a complex number by its conjugate, which is the same complex number but with the sign of the imaginary part
reversed:
(a + bi)(a − bi) = a 2 − abi + bai − b2 i2 .
Collecting up like terms and simplifying we obtain
(a + bi)(a − bi) = a 2 + b2
which is a real number because the imaginary part has been cancelled out by the
action of the conjugate.
Figure 2.3 shows how complex numbers are represented graphically: the horizontal number line represents the real part, and the vertical number line represents
the imaginary part.
2.13
Summary
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For example, the complex number P (1 + 2i) in Fig. 2.3 is rotated 90◦ to Q by
multiplying it by i. Let’s do this, and remember that i2 = −1:
i(1 + 2i) = i + 2i2
=i−2
= −2 + i.
The point Q(−2 + i) is rotated 90◦ to R by multiplying it by i:
i(−2 + i) = −2i + i2
= −2i − 1
= −1 − 2i.
The point R(−1 − 2i) is rotated 90◦ to S by multiplying it by i:
i(−1 − 2i) = −i − 2i2
= −i + 2
= 2 − i.
Finally, the point S(2 − i) is rotated 90◦ back to P by multiplying it by i:
i(2 − i) = 2i − i2
= 2i + 1
= 1 + 2i.
Historically, complex numbers have not played a big part in computer graphics, but imaginary quantities are very important in quaternions and geometric algebra, which are covered in later chapters. Complex numbers are intimately related to
Cartesian coordinates, in that the ordered pair (x, y) ≡ (x + yi), and that they led to
the discovery of vectors and quaternions.
Before concluding this chapter, I cannot fail to include the famous equation discovered by Euler:
eiπ + 1 = 0
which unifies 0, 1, e, π and i in a simple and beautiful arrangement, and is on a par
with Einstein’s e = mc2 .
2.13 Summary
Apart from the natural numbers, integers, rational, irrational, prime, real and complex numbers, there are also Fermat, Mersenne, amicable, chromic, cubic, Fibonacci, pentagonal, perfect, random, square and tetrahedral numbers, which although equally interesting, don’t concern us in this text.
Now that we know something about some important number sets, let’s revise
some ideas behind algebra.
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