At first numbers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers.

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1 1 Numers One thread in the history of mathematics has een the extension of what is meant y a numer. This has led to the invention of new symols and techniques of calculation. When you have completed this chapter, you should e ale to recognise various numer systems, and know the notation for them understand inequality relations, and the rules for calculating with them know what is meant y the modulus of a numer, and how it can e used e familiar with techniques for calculating with surds. 1.1 Different kinds of numer At first numers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers. Sometimes you also need the numer, or zero. For example, suppose you are recording the numer of sisters of every person in the class. Some will have one, two, three,... sisters, ut some will have none. The numers, 1, 2, 3,... are called natural numers. Then people found that numers could also e useful for measurement and in commerce. For these purposes they also needed fractions. Integers and fractions together make up the rational numers. These are numers which can e expressed in the form p where p and q q are integers, and q is not. One of the most remarkale discoveries of the ancient Greek mathematicians was that there are numers which cannot e expressed like this. These are called irrational numers. The first such numer to e found was 2, which is the length of the diagonal of a square with side 1 unit, y Pythagoras theorem (see Fig. 1.1). The argument that the Greeks used to prove this can e adapted to show that the square root, cue root,... of any positive integer is either an integer or an irrational numer. For example, the square root of 8 is an irrational numer, ut the cue root of 8 is the integer 2. Many other numers are now known to e irrational, of which the most famous is π Fig. 1.1 Rational and irrational numers together make up the real numers. When rational numers are written as decimals, they either come to a stop after a numer of places, or the sequence of decimal digits eventually starts repeating in a regular pattern. For example, 7 =.7, 7 = , 7 = , 7 = , =.5, 7 = , 7 =.4375, 7 =

2 2 Higher Level 1 The reverse is also true. If a decimal numer stops or repeats indefinitely then it is a rational numer. So if an irrational numer is written as a decimal, the pattern of the decimal digits never repeats however long you continue the calculation. Integers, rational and irrational numers and real numers can all e either positive or negative. It is helpful to have special symols to denote the different kinds of numer. The set of integers {..., 3, 2, 1,, 1, 2, 3,...} is written as Z; the set of (positive and negative) rational numers is Q; and the set of real numers is R. If you want just the positive numers, you use the symols Z +, Q +, R +. The set of natural numers, {, 1, 2, 3,...}, is written as N; there are no negative natural numers. You may wonder why these letters were chosen. The notation was first used y German mathematicians in the 19th century, and the German word for numer is Zahl. Hence the choice of Z for the integers. The letter Q proaly came from Quotient, which is the result of dividing one numer y another. You proaly know the symol, which stands for is an element of. The statement x Z + means x elongs to the set of positive integers ; that is, x is a positive integer. So x Z + and x is a positive integer are two ways of saying the same thing. You can also draw diagrams of these sets of numers on a numer line, as in Fig The arrows at the ends of the lines indicate the positive direction; the usual convention is for this to point to the right. The larger the numer, the further it is to the right on the line Fig. 1.2 The point which represents is called the origin on the numer line. It is usually denoted y the letter O. But you will notice a snag. There are more real numers than rational numers. But in the figure R and Q, and R + and Q +, look the same. This is ecause there are rational numers as close as you like to any real numer. For example, π R, ut π is not a memer of Q. However, (which is the 8 decimal place approximation to π) is a memer of Q, and you can t distinguish π from in the figure. The only numers that can e shown on a calculator are rational numers. Calculators can t handle irrational numers. So when you key in, 2 on your calculator, and it displays , this is only an approximation to 2. If you square the rational numer , you get = , ut 2 2 = 2 exactly.

3 1 Numers Notation for inequalities You often want to compare one numer with another and say which is the igger. This comparison is expressed y using the inequality symols >, <, and. The symol a > means that a is greater than. You can visualise this geometrically as in Fig. 1.3, which shows three numer lines, with a to the right of. Notice that it does not matter whether a and are positive or negative. The position of a and in relation to zero on the numer line is irrelevant. In all three lines, a >. As an example, in the ottom line, 4 > 7. a a Fig. 1.3 a Similarly, the symol a < means that a is less than. You can visualise this geometrically on a numer line, with a to the left of. These expressions are equivalent. a > < a a is greater than is less than a The symol a means either a > or a = ; that is, a is greater than or equal to, ut not less than,. Similarly, the symol a means either a < or a = ; that is, a is less than or equal to, ut not greater than,. These expressions are equivalent. a a a is greater than or equal to is less than or equal to a Some ooks use the symols and in place of and. The symols < and > are called strict inequalities, and the symols and are called weak inequalities. Example Write down the set of numers x such that x N and x < 6. N is the set of natural numers {, 1, 2, 3,...}. The largest numer in N less than 6 is 5. So the set of numers such that x N and x < 6is{, 1, 2, 3, 4, 5}. Example The points A and B on the numer line represent the numers 2 and 3. Use inequalities to descrie the numers represented y the line segment [AB] shown in Fig The notation [AB] is explained in Section 8.2.

4 4 Higher Level 1 A O B 2 3 Fig. 1.4 All the points of the line segment, except A itself, are to the right of A. So the numers they represent satisfy the inequality x 2. They are also, except B itself, to the left of B, so the numers satisfy x 3. You could also write x 2as 2 x. So 2 x and x 3. These two inequalities can e comined in a single statement as 2 x 3. When you write an inequality of the kind r < x and x < s in the form r < x < s, it is essential that r < s. It makes no sense to write 7 < x < 3; how can x e oth greater than 7 and less than 3? An inequality of the type r < x < s (or r < x s or r x < s or r x s) is called an interval. It consists of all the numers etween r and s (including r or s where the sign adjacent to them is ). The word interval is also used for an inequality such as x > r, which consists of all the numers greater than r (and similarly for x r, x < s and x s). 1.3 Solving linear inequalities When you solve an equation like 3x + 7 = 5, you use two rules: you can add (or sutract) the same numer on oth sides of the equation you can multiply (or divide) oth sides of the equation y the same numer. In this example, sutracting 7 from oth sides and then dividing y 3 leads to the solution x = 4. An inequality like 3x + 7 > 5 doesn t have a single solution for x, ut it can e replaced y a simpler statement aout the value of x. To find this, you need rules for working with inequalities. These are similar to those for equations, ut with one very important difference. Adding or sutracting the same numer on oth sides You can add or sutract the same numer on oth sides of an inequality. Justifying such a step involves showing that, for any numer c, if a > then a + c > + c. This is saying that if a is to the right of on the numer line, then a + c is to the right of + c. Figure 1.5 shows that this is true whether c is positive or negative. Since sutracting c is the same as adding c, you can also sutract the same numer from oth sides. c c c + c a a + c c is positive c + c a + c a c is negative Fig. 1.5

5 1 Numers 5 Example If x 3 < 4, what can you say aout the value of x? You can add 3 on oth sides of the inequality, which gives that is x < 4 + 3, x < 1. In this example, the inequality x < 1 is called the solution of the inequality x 3 < 4. It is the simplest statement you can make aout x which is equivalent to the given inequality. Multiplying oth sides y a positive numer You can multiply (or divide) oth sides of an inequality y a positive numer. Example Solve the inequality 1 3 x 2. Multiply oth sides of the inequality y 3. This gives the equivalent inequality that is 3 ( 1 3 x) 3 2, x 6. Here is a justification of the step, if a > and c >, then ca > c. As a >, a is to the right of on the numer line. As c >, ca and c are enlargements of the positions of a and relative to the numer. c a ca c a ca c ca a Fig. 1.6 Figure 1.6 shows that, whether a and are positive or negative, ca is to the right of c, so ca > c. Multiplying oth sides y a negative numer If a >, and you sutract a + from oth sides, then you get > a, which is the same as a <. This shows that if you multiply oth sides of an inequality y 1, then you change the direction of the inequality. Suppose that you wish to multiply the inequality a > y 2. This is the same as multiplying a < y 2, so 2a < 2.

6 6 Higher Level 1 You can also think of multiplying y 2 as reflecting the points corresponding to a and in the origin, and then multiplying y 2 as an enlargement (see Fig. 1.7). ca c Fig. 1.7 a You can summarise this y saying that if you multiply (or divide) oth sides of an inequality y a negative numer, you must change the direction of the inequality. Thus if a > and c <, then ca < c. Example Solve the inequality 3x < 21. In this example you need to divide oth sides y 3. Rememering to change the direction of the inequality, 3x < 21 ecomes x > 7. Summary of operations on inequalities You can add or sutract a numer on oth sides of an inequality. You can multiply or divide an inequality y a positive numer. You can multiply or divide an inequality y a negative numer, ut you must change the direction of the inequality. Solving inequalities is simply a matter of exploiting these three rules. You can link the inequality operation involving multiplication with + +=+. For if a > and c >, oth a and c are positive numers, so c(a ) is also positive. So ca c is positive, ca c > and ca > c. Example Solve the inequality x 5. Begin y sutracting 3 from oth sides, to get 1 2 x 2. Now multiply oth sides y 2, rememering to change the direction of the inequality. This gives x 4. Example Solve the inequality 4 2x > 3x. Begin y adding 2x on oth sides, to get 4 > 5x. Dividing oth sides y 5 then gives.8 > x. The answer is usually written with x on the left side, as x <.8.

7 1 Numers 7 Exercise 1A 1 Which of the numer systems N, Z, Q, R, Z +, Q + and R + contain the following numers? 3 (a) 6 () 6.6 (c) (continued indefinitely) (d) 125 (e) 125 (f) (g) π (h) 2 2 State in list form the sets of numers x which satisfy the following. (a) x Z + and x 3 () x N and 2x < 9 (c) x Z and 2 x < 2 (d) x N and 3 x > 1 (e) x Z + and 4x 7 < 9 (f) x N and 1 3x > 1 3 Solve the following inequalities, where x R. (a) x 3 > 11 () 2x (c) 5x (d) 3x 1 13 (e) 5x 2 (f) 3x 12 (g) 5x + 1 (i) 3 (m) 7 3x 2 (q) 3x 5 2(3 x) x (h) 3x > 3 (j) 4 3x 1 (k) 2 6x (l) 6 5x > 1 < 1 (n) x x (o) 2x + 5 < 4x 7 (p) 4x 3(2 x) Modulus notation Suppose that you want to find the difference etween the heights of two children. With numerical information, the answer is quite straightforward: if their heights are 9 cm and 1 cm, you would answer 1 cm; and if their heights were 1 cm and 9 cm, you would still answer 1 cm. But how would you answer the question if their heights were H cm and h cm? The answer is, it depends which is igger: if H > h, you would answer (H h) cm; if h > H you would answer (h H) cm; and if h = H you would answer cm, which is either (H h) cm or (h H) cm. Questions like this, in which you want an answer which is always positive or zero, lead to the idea of the modulus of a numer. This is a quantity which tells you the size of a numer regardless of its sign. For example, the modulus of 15 is 15, ut the modulus of 15 is also 15. The notation for modulus is to write the numer etween a pair of vertical lines. So you would write 15 =15 and 15 =15. The modulus is sometimes called the asolute value of the numer. Some calculators have a key marked as which produces the modulus. If yours has, try using it with a variety of inputs.

8 8 Higher Level 1 The modulus can e defined formally like this: The modulus of x, written x and pronounced mod x, is defined y x =x if x, x = x if x <. Using the modulus notation, you can now write the difference in heights as H h whether H > h, h > H or h = H. Another situation when the modulus is useful is when you talk aout numers which are large numerically, ut which are negative, such as 1 or 1. These are negative numers with large modulus. For example, for large positive values of x, the value of 1 is close to. The same is true for x negative values of x with large modulus. So you can say that, when x is large, 1 x is close to zero; or in a numerical example, when x > 1, 1 x < Modulus on the numer line In Fig. 1.8 A and B are points on a numer line with O A B coordinates a and. How can you express the distance AB in a terms of a and? Fig. 1.8 If B is to the right of A, then > a, so a > and the distance is a. If B is to the left of A, then < a, so a < and the distance is a = ( a). If B and A coincide, then = a, so a = and the distance is. You will recognise this as the definition of a. The distance etween points on the numer line representing numers a and is a. As a special case, if a point X has coordinate x, then x is the distance of X from the origin. Now suppose that x = 3. What can you say aout X? There are two possiilities: since the distance OX = 3, either X is 3 units to the right of O so that x = 3, or X is 3 units to the left of O so that x = 3. This is also true in reverse. If x = 3orx = 3, then X is 3 units from O, so x = 3. A convenient way of summarising this is to write x = 3 means that x = 3 or x = 3. Similarly you can write x 3 as 3 x 3, since these are two different ways of saying that X is within 3 units of O on the numer line.

9 1 Numers 9 In this example there is nothing special aout the numer 3. You could use the same argument with 3 replaced y any positive numer a to show that x a means that a x a. You can get a useful generalisation y replacing x in this statement y x k, where k is a constant: If a is a positive numer, x k a means that a x k a. Now add k to each side of the inequalities a x k and x k a. You then get k a x and x k + a. That is, k a x k + a. The equivalence then ecomes: If a is a positive numer, x k a means that k a x k + a. This result is used when you give a numer correct to a certain numer of decimal places. For example, to say that x = 3.87 correct to 2 decimal places is in effect saying that x Fig. 1.9 shows that x means x , or x x Fig. 1.9 Exercise 1B 1 Write down the values of (a) 6, () 13 15, (c) If x > 1, what can you say aout 1? Write your answer using inequality notation. x2 3 The mass of a prize pumpkin is 4.76 kg correct to 2 decimal places. Denoting the mass y m kg, write this statement in mathematical form (a) without modulus notation, () using modulus notation. 4 The length and width of a rectangle are measured as 4.8 cm and 3.6 cm, oth correct to 1 decimal place. Write a statement aout the perimeter, P cm, in mathematical form (a) without modulus notation, () using modulus notation. 5 Repeat Question 4 for the area, Acm 2. 6 Write the given inequalities in an equivalent form of the type a < x < or a x. (a) x 3 < 1 () x (c) 2x 3.1 (d) 4x 3 8

10 1 Higher Level 1 7 Rewrite the given inequalities using modulus notation. (a) 1 x 2 () 1 < x < 3 (c) 3.8 x 3.5 (d) 2.3 < x < What can you say aout x if x Z? 1.6 Surds and their properties When you have met expressions such as 2, 8 and 12 efore, it is likely that you have used a calculator to express them in decimal form. You might have written 2 = or 2 = correct to 3 decimal places or Why is the statement 2 = incorrect? Expressions like 2or 3 9 are called surds. However, expressions like 4or 3 27, which are whole numers, are not surds. This section is aout calculating with surds. You need to rememer that x always means the positive square root of x, orifx is. The main properties that you will use are: If x and y are positive numers, then x x xy = x y and y =. y You can use these properties to express square roots like 8 and 12 in terms of smaller surds: 8 = 4 2 = 4 2 = 2 2; 12 = 4 3 = 4 3 = 2 3. You can also simplify products and quotients of square roots: 18 2 = 18 2 = 36 = 6; 27 3 = Use a calculator to check the results of these calculations = 9 = 3. To show that the statements in the lue ox are true, rememer that the numer x has two properties: x x = x, and x is positive. Similarly y y = y, and y is positive. So, to prove that xy = x y, you have to show that ( x y) ( x y) = xy, and that x y is positive.