1.1.1 Rules of arithmetic: rational and irrational numbers

The first contact with mathematics is usually via counting, using the positive integers 1, 2, 3, 4, … (also called natural numbers). Later, fractional numbers such as , etc. and negative numbers −1, −3, , , etc. are introduced, together with the rules for combining positive and negative numbers and the basic laws of arithmetic. As we will build on these laws later in this chapter, it is worth reminding oneself of what they are by stating them in a somewhat formal way as follows.

1. Commutativity: The result of subtracting or dividing two integers is dependent on the order in which the operations are performed, but addition and multiplication are independent of the order. For example,

   (1.1a)

   but

   (1.1b)

   where ≠ means not equal to.

2. Associativity: The result of subtracting or dividing three or more integers is dependent on the way the integers are associated, but is independent of the association for addition and multiplication. Examples are

   (1.2a)

   but

   (1.2b)

3. Distributivity: Multiplication is distributed over addition and subtraction from both left and right, whereas division is only distributed over addition and subtraction from the right. For example, for multiplication:

   (1.3a)

   and

   (1.3b)

   but for division, from the right we have

   (1.3c)

   whereas division from the left gives

   (1.3d)

Positive and negative integers and fractions can all be expressed in the general form n/m, where n and m are integers (with m ≠ 0 because division by zero is undefined). A number of this form is called a rational number. The latter is said to be proper if its numerator is less than its denominator, otherwise it is said to be improper. The operations of addition, subtraction, multiplication and division, when applied to rational numbers, always result in another rational number. In the case of fractions, multiplication is applied to the numerators and denominators separately; for division, the fraction is inverted and then multiplied. Examples are

(1.4a)

For addition (and subtraction) all the terms must be taken over a common denominator. An example is:

(1.4b)

Not all numbers can be written in the form n/m. The exceptions are called irrational numbers. Examples are the square root of 2, that is, , and the ratio of the circumference of a circle to its diameter, that is, π = 3.1415926…, where the dots indicate a non-recurring sequence of numbers. Irrational numbers, when expressed in decimal form, always lead to such non-recurrence sequences, but even rational numbers when expressed in this form may not always terminate, for example . The proof that a given number is irrational can be very difficult, but is given for one particularly simple case in Section 1.2.2.

In practice, an irrational number may be represented by a rational number to any accuracy one wishes. Thus π is often represented as in rough calculations, or as in more accurate work. Rational and irrational numbers together make up the class of so-called real numbers that are themselves part of a larger class of numbers called complex numbers that we will meet in Chapter 6. It is worth remarking that infinity, denoted by the symbol ∞, is not itself a real number. It is used to indicate that a quantity may become arbitrarily large.

In the examples above of irrational numbers, the sequence of numbers after the decimal point is endless and so, in practice, one has to decide where to terminate the string. This is called rounding. There are two methods of doing this: quote either the number of significant figures or the number of decimal places. Consider the number 1234.567…. To two decimal places this is 1234.57; the last figure has been rounded up to 7 because the next number in the string after 6 is 7, which is greater than 5. Likewise, we would round down if the next number in the string were less than 5. If the next number in the string were 5, then the 5 and the next number following it are rounded up or down to the nearest even multiple of 10, and the zero dropped. For example, 1234.565 to two decimal places would be 1234.56, whereas 1234.575 would be rounded to 1234.58. If we were to quote 1234.567 to five significant figures, it would be 1234.6 and to three significant figures it would be 1230.

1.1.2 Factors, powers and rationalisation

Integer numbers may often be represented as the product of a number of smaller integers. This is an example of a process called factorisation, that is, decomposition into a product of smaller terms, or factors. For example, 24 is equal to 2 × 2 × 2 × 3. In this example, the integers in the product cannot themselves be factorised further. Such integers are called prime numbers. (By convention, unity is not considered a prime number.) By considering all products of the prime numbers in the factorisation, we arrive at the result that the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, that is, these are all the numbers that divide exactly into 24. If we have several numbers, the highest common factor (HCF) is the largest factor that can divide exactly into all the numbers. The lowest common multiple (LCM) is the smallest number into which all the given numbers will divide exactly. Thus the HCF of 24 and 36 is 12 and the LCM of all three numbers is 72.

In the example of the factorisation of the number 24 above, the factor 2 occurs three times. It is common to encounter situations where a number is multiplied by itself several times. A convenient notation for this is to introduce the idea of a power (or index) n, such that, for example, 5ⁿ ≡ 5 × 5 × 5… n times. To emphasise that this relation defines the index n, the usual two-line equality sign has been replaced by a three-line equality sign (≡). So, using powers, we could also write 24 in the compact prime-number factorised form 24 = 3 × 2³. Any real number p to power zero is by definition equal to unity, that is, p⁰ ≡ 1 for any p.

By writing out in full, it is easy to see that multiplying the same integers each raised to a power is equivalent to adding the powers. Thus

(1.5a)

and analogously for division,

(1.5b)

A power can also be a fraction or rational number, since, for example, the combination rule (1.5a) implies 5^1/2 × 5^1/2 = 5¹, so that . Similarly the expression 3^1/33⁰3^4/3/27^1/3, for example, can be simplified to give

(1.5c)

An example of the use of factors is to express numbers in so-called scientific notation (also called normal form). In this representation, any real number is written as the product of a number between −10 and +10 (excluding the numbers ± 10 themselves), with as many decimal places as required, and a power of 10. The number 1245.678 to four significant figures in scientific notation is therefore 1.246 × 10³.

It is conventional to write arithmetical forms in a compact form and to remove as far as possible fractional powers from the denominator of a fraction, a process called rationalisation. For example, consider the form

(1.6a)

By taking the terms over a common denominator and then multiplying numerator and denominator by , we have

(1.6b)

which is the rationalised form of (1.6a).

*1.1.3 Number systems¹

All the numbers in the previous sections are expressed in the decimal system, where the ‘basis’ set of integers is 0, 1, 2, …, 9. Real numbers in this system are said to be to ‘base 10’. In the number 234, for example, the integers 2, 3 and 4 indicate how many powers of 10 are present, reading from the right, i.e.

(1.7)

Any other base could equally well be used and in some circumstances other number systems are more appropriate. The most widely used number system other than base 10 is the binary system, based on the two integers 0 and 1, that is, base 2, so we will only discuss this case. Its importance stems from its use in computers, because the simplest electrical switch has just two states, ‘open’ and ‘closed’. To distinguish numbers in this system we will write them with a subscript 2.

As an example, consider the number 123. In the binary system this is 1111011₂. To check: in the decimal system,

(1.8)

Fractions are accommodated by using negative values for the indices. Thus the number 6.25 in the binary system is 110.01₂. To check: in the decimal system,

(1.9)

To convert a number in one basis to another is straightforward, if rather tedious. Consider, for example, the conversion of the number 51.78 to the binary system. We start with the integer 51 and find the largest value of an integer n such that 2ⁿ is less than or equal to 51 and then note the remainder R = 51 − 2ⁿ. This is then repeated by again finding the largest number n such that 2ⁿ is less than or equal to R, and continued in this way until the remainder is zero. We thus obtain:

so that in the binary system

(1.10a)

Similarly, we can convert the numbers after the decimal point using negative powers. This gives

so again in the binary system,

(1.11a)

and finally,

(1.11b)

in the binary system, which represents the decimal number to an accuracy of two decimal places.

All the normal arithmetic operations of addition, subtraction, multiplication and division can be carried out in any number system. For example, in the binary system, we have the basic result 1₂ + 1₂ = 10₂. So adding the numbers 101₂ and 1101₂ gives 101₂ + 1101₂ = 10010₂. To check, we can again use the decimal system. Thus,

(1.12a)

with

(1.12b)

As an example of multiplication, consider the numbers 5 and 7. In the binary system these are 101₂ and 111₂, respectively, and multiplying them together gives, using 1₂ + 1₂ = 10₂,

Once again, we can check the result using the decimal system:

(1.13)

As an example of division, consider the numbers 51 and 3. In the binary system these are 110011₂ and 11₂, respectively, and dividing them we have

So the quotient is 10001₂, which in the decimal system is 2⁴ + 2⁰ = 17, as required.

1.2.1 Rules of elementary algebra

Algebra enables us to consider general expressions like, for example, (x + y)², where x and y can be any real number. When manipulating real numbers as symbols, the fundamental rules of algebra apply. These are analogous to the basic rules of arithmetic given in Section 1.1 and can be summarised as follows.²

1. Commutativity: Addition and multiplication are commutative operations, i.e.

   (1.14a)

   and

   (1.14b)

   In contrast, subtraction and division are only commutative operations under special circumstances. Thus,

   

   and

   

2. Associativity: Addition and multiplication are associative operations, i.e.,

   (1.15a)

   and

   (1.15b)

   Subtraction and division are not associative operations except in very special circumstances. Thus,

   

   and

   

   as is easily verified by choosing any particular values for x, y and z.

3. Distributivity: The basic rule is

   (1.16a)

   Together with the commutative law of multiplication, this implies

   (1.16b)

   since

   

   In addition, by noting that (y − z) = (y + ( − z)) etc., one sees that these results imply that multiplication is distributed over addition and subtraction from both the left and the right, i.e.

   (1.16c)

   Finally, since (x + y)/z = (x + y)z^{− 1}, equation (1.16b) implies that division is distributed over addition and subtraction from the right, i.e.

   (1.16d)

   but not from the left, i.e.

   

4. The law of indices: This is

   (1.17)

   with xⁿ/x^m = x^{n − m}, and where, by definition, x⁰ ≡ 1.

The nine laws (1.14)–(1.17) are the fundamental laws of elementary algebra. To illustrate their use, consider the proof of the familiar result

We have,

It should be emphasised that although the above rules are obeyed by the real variables of elementary algebra, in later chapters we will encounter other mathematical quantities, such as vectors and matrices, that do not necessarily obey all these rules.

*1.2.2 Proof of the irrationality of

Now we have introduced algebraic symbols and the idea of powers, we can return to the discussion of Section 1.1.1 and prove that is an irrational number. The proof uses a general method called reductio ad absurdum, or proof by contradiction; that is, we assume the opposite, and prove it leads to a contradiction. This is a commonly used method of proof in mathematics. Suppose is rational. It then follows that

(1.18)

where p and q are integers, and we may, without loss of generality, assume that they are the smallest integers for which this is possible, that is, they have no common factors. Then from (1.18), we have

(1.19)

so that p² is even. Furthermore, since the square of an odd number is odd and the square of an even number is even, p itself must be even; and since p and q have no common factors, q must be odd, since otherwise both would be divisible by 2. On the other hand, since p is even, we can write p = 2r, where r is an integer. Substituting this in (1.19) now gives q² = 2r², so that q is even, in contradiction to our previous result. Hence the assumption that is rational must be false, and can only be an irrational number.

1.2.3 Formulas, identities and equations

The use of symbols enables general algebraic expressions to be constructed. An example is a formula, which is an algebraic expression relating two or more quantities. Thus the volume of a rectangular solid, given by volume = length × breadth × height, may be written V = lbh. Given numerical values for l, b and h, we can calculate a value for the volume V. Formulas may be manipulated to more convenient forms providing certain rules are respected. These include (1) taking terms from one side to the other reverses their sign; and (2) division (multiplication) on one side becomes multiplication (division) on the other. For example, if S = ab + c, then S − c = ab, a = (S − c)/b etc.

As with numerical forms, it is usual to rationalise algebraic expressions where possible. Thus,

(1.20)

Sometimes factorisation may be used to simplify the results. For example,

(1.21)

where we have used the results

(1.22a)

and

(1.22b)

Equations (1.21)–(1.22) are examples of identities, because they are true for all values of x, and the three-line equality symbol (mentioned earlier) is also sometimes used to emphasise this, although in this book we will reserve its use for definitions.

In contrast, the expression on the left-hand side of (1.22a) can also be written f(x) = x² − 3x + 2 and setting f(x) equal to a specific value gives an equation that will only have solutions (or roots) for specific values of x. In the case of (1.22a), setting f(x) = 0 yields the two solutions x = 1 and x = 2.

1.2.4 The binomial theorem

An important class of algebraic expressions consists of the binomials (x + y)ⁿ, where the integer n ≥ 0. These can be built up starting from (x + y)⁰ = 1 by successively multiplying by (x + y) to give

and so on. The coefficients of the terms in this sequence form the Pascal triangle are

in which the elements of each row sum to 2ⁿ, and those in the (n + 1)th row are given by the sum of the neighbouring elements in the row. Thus, the fourth element for n = 4 is given by the sum of the neighbouring elements 3 and 1 in the row with n = 3.

These results are generalised to arbitrary n by the binomial theorem. In this theorem, the binomial expansion is written in the form

(1.23)

where the summation symbol means that a sum is to be taken over all terms labelled by k = 0, 1, 2, …, n. Here k is called a dummy index because the sum, that is, the left-hand side of (1.23), does not depend on the index k. The binomial coefficients are defined by

(1.24a)

where n! indicates the factorial

(1.25)

with 0! ≡ 1 by definition, so that

(1.26)

An alternative notation that is used is

(1.24b)

The binomial coefficients (1.24), which occur frequently in, for example, probability theory and statistical physics, have a number of important properties which include

(1.27a)

(1.27b)

(1.27c)

and

(1.27d)

The first three of these follow trivially from the definition (1.24). The fourth, called Pascal's rule, is just the relation between the elements of the and rows of the Pascal triangle mentioned above. To prove (1.27d), we note that

as required.

It remains to prove the binomial theorem (1.23). This is done by another general method, that of induction: one proves that if a proposition is true for a value n, it is true for a value n + 1. Then provided it is true for n = 1, its validity for all positive integers n is established. We therefore assume that (1.23) is valid for a value n = m. Multiplying by (x + y) then gives

(1.28)

where we have substituted j = k + 1 in the second term. The value of this term is unchanged by relabeling the dummy index j → k, so that (1.28) becomes

(1.29)

where we have separated off the first and last terms in the first and second summations in (1.28), respectively. Since (1.27a) holds for arbitrary n, we may replace m by (m + 1) in the first and last terms in (1.29); and substituting (1.27d) in the middle term then gives

This is just the binomial theorem (1.23) for index n = m + 1, so that if the theorem holds for index n = m, it holds for n = m + 1. Since it is trivially true for n = 1, this implies it holds for all positive integers n, as required.

1.2.5 Absolute values and inequalities

We are often interested in the numerical values of real numbers and variables without regard to their signs. This is called the modulus (or absolute value), with the notation . We will also be using inequalities, with the symbols > meaning ‘greater than’ and < meaning ‘less than’. Thus 3 < 4 < 7 is the statement that 3 is less than 4 which in turn is less than 7. A related statement is 7 > 4 > 3, that is, 7 is greater than 4, which in turn is greater than 3.

Using algebraic quantities, the definition of the modulus is

(1.30)

Therefore,

(1.31)

where a is a real number and the symbol ⇒ means ‘implies’. Generalising further to include the possibility that |x| = a, that is |x| ≤ a, we have − a ≤ x ≤ a, where we have used the obvious notation ≤ to mean ‘less than or equal to’. In general, if a ≤ x ≤ b, where a and b are real numbers, then we say that x lies in a closed interval (or range) of length (b − a). Likewise, if a < x < b, the interval is said to be open. Using the definition of the modulus, gives

(1.32)

The manipulation of inequalities differs from the manipulation of equalities, so we will discuss it in some detail. Terms may be taken from one side of an inequality to the other if their sign is changed. Also, adding a constant (positive or negative) to the terms of an inequality, or multiplying it by a positive constant, does not alter its validity. Thus, by adding a to each part of the inequality (1.32), we have a − b < x < a + b. However, multiplying or dividing by a negative number will reverse the sense of the inequality. For example, multiplying both sides of the inequality x < 6 by − 1 does not imply − x < −6, which obviously contradicts the original inequality, but rather − x > −6, that is, the sense of the inequality is reversed. For this reason, particular care should be taken when simplifying an inequality involving algebraic quantities, such as

(1.33)

Cross-multiplying is not permitted, because the denominators may be negative. Rather, the inequality should be simplified by taking the terms over a common denominator. For (1.33),

(1.34a)

so that

(1.34b)

which implies that the inequality is true only for or x < −1.

To illustrate these results, consider the inequality

(1.35)

There are two possible cases:

(1.36a)

i.e. x > 1, and

(1.36b)

where we now have to reverse the direction of the inequalities, i.e. . Another example is

(1.37)

Thus either both brackets are positive, or both are negative. In the first case and in the second case x < −4.

Care must also be taken when manipulating pairs of inequalities. Thus for addition, while

(1.38a)

on adding the two inequalities, we cannot deduce by subtraction that . Likewise, if , are positive quantities, then

(1.38b)

but this conclusion does not follow if any of are negative numbers. For division, x > y and do not imply , even for positive numbers. The validity of these statements can be verified by some simple numerical examples. Thus if we take x = 3, y = 2 and , then , but , where the symbol means ‘not greater than’. The other statements can also be confirmed by using specific numbers.

1.3.1 Functions

Suppose two variables, x and y, are related in such a way that there is a single value of y corresponding to each value of x that lies within a given range a < x < b. Then the dependent variable y is said to be a single-valued function of the independent variable x, whose value may be varied at will within the allowed range. This is written

where f(x) specifies the particular function. In many cases, f(x) is an explicit function, for example,

(1.39)

Functions can also be defined implicitly, for example as the solutions of a given equation. A simple example would be to define y = f(x) as the solution of the equation y² = x − 1 for x > 1. In this case, there are two solutions, and , and such cases are referred to as multi-valued functions.³ Alternatively, one can impose a subsidiary condition, for example y > 0, to ensure that the solution is unique, in accord with our original definition.

It is often useful to represent functions by graphs, which summarise, and give considerable insight into their properties. Figure 1.1 shows a graph of the function (1.39) in the range − 2.5 < x < 4.5. The graph shows that the function has one maximum and one minimum in this range and that the solutions of the equation f(x) = 0 are x = −2, 1 and 4.

Functions, whether of algebraic form or not, may be characterised by a variety of general properties and below we list some of these for use in later chapters.

If f( − x) = f(x) for all values of x, the function is said to be even (or symmetric), whereas if f( − x) = −f(x) for all values of x, the function is said to be odd (or antisymmetric). The simple examples

are shown in Figure 1.2. Although most functions have no specific symmetry, any function can always be written as the sum of even and odd functions. To see this we can write

(1.40a)

where

(1.40b)

are symmetric and anti-symmetric functions by construction. As an example, consider the function

(1.41a)

from which we have

(1.41b)

and hence from (1.40),

(1.41c)

The usefulness of this decomposition is that exploitation of the symmetry of a function can often lead to simplifications in calculations. We will see examples of this in later chapters.

The function f(x) is a prescription for calculating f given the value of x. We often need to know the prescription for the inverse process, that is, to find what value (or values) of x corresponds to a given value of f. This is called the inverse function of f and is written f^{− 1}(x). The notation is not perfect, because there is a danger of confusion with 1/f(x). It is important to remember that they are not the same. The inverse function corresponding to y = f(x) is found by transposing the equation so that x is given as a function of y and then replacing y by x, and x by the inverse function y^{− 1}(x). Thus if y(x) = x³ + 3, then x = (y − 3)^1/3 and hence the inverse function is y^{− 1}(x) = (x − 3)^1/3. The inverse function may be multivalued. Thus, if f(x) = x², then ‘inverting’ gives the function , with two values.

Most functions we will discuss are continuous. We will define this term more precisely in Chapter 3, but roughly speaking it means that the values of y vary smoothly without sudden ‘jumps’ when the value of x is slowly varied. Functions that do not have this property are said to be discontinuous and we will see that frequently met functions are often of this type. A common situation is when a function is of the form 1/f(x), where f(x) is zero at some point x₀ and changes sign as x passes through the point; for example f(x) = (x − x₀). In this case the function will pass from + ∞ to − ∞ as x passes through the value x₀.

Finally, the argument of a function can itself be another function, in which case we speak of a ‘function of a function’. Thus, if

(1.42)

then p as a function of q is given by

(1.43a)

and likewise q as a function of p is

(1.43b)

1.3.2 Cartesian co-ordinates

Algebra and geometry are united by the use of co-ordinates, which enable geometrical forms to be described by algebraic equations. Here we illustrate this by considering Cartesian co-ordinates, mainly in two dimensions, leaving other co-ordinate systems to later chapters.

In two-dimensional Cartesian co-ordinates the position of a point P in a plane is specified relative to a chosen pair of horizontal and vertical axes, called the x- and y-axes respectively. The corresponding co-ordinates are written P = (x, y), where x and y are the projections of the point onto the x and y-axes respectively, as shown for two points A(x₁, y₁) and B(x₂, y₂) in Figure 1.3. The axes themselves intersect at the origin, that is, the point (x, y) = (0, 0).

Using Cartesian co-ordinates we can deduce a number of useful results. Thus the distance between any two points A(x₁, y₁) and B(x₂, y₂) is given by

(1.44)

which follows from using the Pythagoras Theorem⁴ for the triangle ABN in Figure 1.3. Likewise, the gradient, or slope, of the straight line AB joining A and B is given by

(1.45)

and the co-ordinates of the midpoint of AB are

(1.46)

Any line in the xy-plane implies an equation relating the x and y co-ordinates of any point which lies upon it. Consider for example a circle centre C(a, b) and radius r, as shown in Figure 1.4. If P(x, y) is any point on the circumference, then by using the Pythagoras Theorem in the triangle PCN, we have

(1.47)

which is therefore the equation of the circle in Cartesian co-ordinates.

An even simpler geometrical figure is a straight line. In this case, the co-ordinates P(x, y) of any point lying on a straight line satisfy a linear equation of the form

(1.48)

where m and c are constants. In Figure 1.5(a) the resulting lines are shown for m = 1 and different values of c, and in Figure 1.5(b) for c = 2 but with m varying. It can be seen from Figure 1.5(a) that c is the y co-ordinate of the point where the line cuts the vertical (i.e. y) axis (this is called the intercept) and m is the gradient. In Figure 1.5 the gradients are all positive, but m can also take negative values (or zero) in which case the line slopes downwards to the right (or is horizontal).

Equations like (1.47) and (1.48) enable many results to be derived very easily. For example, at the point of intersection of two straight lines y = m₁x + c₁ and y = m₂x + c₂, we have

so that x = (c₁ − c₂)/(m₂ − m₁) and the value of y follows from the equation of either straight line. In particular we see that for parallel lines, that is, lines which have the same slopes m₁ = m₂, but different intercepts, c₁ ≠ c₂, there is no solution, thus proving that ‘parallel lines never meet’.

In general, for any curve y = f(x), we define the tangent at a point as the straight line that just touches the curve at the point, so that the gradient of a curve at any point is equal to the gradient of the tangent at that point. This is illustrated in Figure 1.6(a), which shows the cubic polynomial x³ + 2x² − 5x, together with the tangent drawn at the point (2,6). Finding the gradient by graphical methods will only give an estimate, because the accuracy depends on how well one can draw the tangent. We will see in later chapters that there are better methods for finding gradients. Figure 1.6(b) shows the function 2/(x² − 1), together with tangents drawn at the points x = ±1. Notice that this function is discontinuous at x = ±1, and the gradients at these points are infinite.

Other results can be found by geometrical methods. One example is to prove that the product of the gradients of two perpendicular lines is −1. Let the gradients of the two perpendicular lines PA and PC in Figure 1.7 be m₁ and m₂, respectively. Since the two lines are perpendicular, the triangles PAB and PCD are similar, with AB/PB = DC/PD. Now m₁ = AB/PB and m₂ = −PD/DC and thus m₁m₂ = −1.

This result can used to find the equation of the perpendicular bisector of the line joining the points E(3, 0) and F(5, 6). The straight line connecting EF has an equation of the form y = mx + c. Since it passes through the points E(3, 0) and F(5, 6), then 6 = 5m + c and 0 = 3m + c, giving m = 3 and c = −9. Thus the perpendicular bisector has a gradient and is of the form . Finally, as it passes through the midpoint (4, 3), we have .

Inequalities in x and y define regions of the xy-plane, that can be combined to find areas of allowed values. For example, Figure 1.8 shows the xy-plane with a number of shaded areas. These indicate the areas satisfying the set of inequalities

and

where the last equation restricts P(x, y) to points outside the circle (x − 3)² + y² = 4. The coloured region thus represents the area occupied by all points that simultaneously satisfy the three inequalities x > 3.5, y > −1.0 and y² < 4 − (x − 3)².

All the above discussion has been in the context of two dimensions, but it can easily be generalised to three dimensions. In this case we construct three axes x, y and z, with the property that if the thumb and first two fingers of the right hand are arranged so that they are mutually perpendicular, then the first and second fingers point along the positive x- and y-axes, respectively, and the thumb points along the positive z-axis. This is called a right-handed Cartesian co-ordinate system and is shown in Figure 1.9. Alternatively, one can say that the rotations x → y, y → z and z → x are all in the sense of a right-handed screw. A point P in three-dimensional space is then described by co-ordinates (x, y, z), as shown in Figure 1.9.

As examples of the generalisation, the distance between any two points in two dimension (1.44) becomes the distance between any two points A(x₁, y₁, z₁) and B(x₂, y₂, z₁) in three dimensions, and is given by

(1.49)

Similarly, the equation

(1.50)

is the generalisation of the equation of a circle (1.47) and describes a sphere with centre at the point (x, y, z) = (a, b, c). Finally, if the equation of a straight line in two dimensions (1.48) is generalised to

(1.51)

where a, b, c and d are constants, it describes a plane in three dimensions. To describe a straight line in three dimensions requires two equations, for example,

(1.52)

which determine both the y and z co-ordinates for a given value of x.