Chapter 2 Numbers. 2.1 Introduction. 2.2 Background. 2.3 Set Notation

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1 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 / _2, Springer-Verlag London

2 4 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 = 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. x 1,x 2,x 3,x 4, 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,

3 2.7 Integer Numbers 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 = = = = = = = In 1742, the German mathematician Christian Goldbach ( ) conjectured that every even integer greater than 2 could be written as the sum of two primes: 4 = = = , 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, orz without zero. Zero, in any form, is a useful concept, but does not sit comfortably alongside other numbers, especially when it is adivisor. 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

4 6 2 Numbers Fig. 2.1 The number line 16 = 4 = = 1 4. Some rational numbers can be stored accurately inside a computer, but many others can only be stored approximately. For example, 4/3 = 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 = π = e = Such numbers never terminate and are always subject to a small error when stored within a computer Real Numbers Rational and irrational numbers together comprise the set of real numbers which are represented by the letter R,orR without zero. Typical examples are 1.5, 0.004, and 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 Complex Numbers The Swiss mathematician Leonhard Euler ( ) (pronounced Oiler ) played a significant role in putting complex numbers on the map. His ideas on

5 2.12 Complex Numbers 7 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 = 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 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 of 90, i.e. 2ii, it becomes 2. Therefore, we can write 2ii = 2, which means that ii = 1.Butifthisisso,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.

6 8 2 Numbers 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 + bdi 2. Collecting up like terms and substituting 1 fori 2 we get and simplifies to (a + bi)(c + di) = ac + (ad + bc)i bd (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 b 2 i 2. Collecting up like terms and simplifying we obtain (a + bi)(a bi) = a 2 + b 2 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.

7 2.13 Summary 9 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 i 2 = 1: i(1 + 2i) = i + 2i 2 = i 2 = 2 + i. The point Q( 2 + i) is rotated 90 to R by multiplying it by i: i( 2 + i) = 2i + i 2 = 2i 1 = 1 2i. The point R( 1 2i) is rotated 90 to S by multiplying it by i: i( 1 2i) = i 2i 2 = 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 i 2 = 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: e iπ + 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 = mc 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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