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First Chapter
Cambridge University Press
9780521699280  Mathematics for the IB Diploma  Standard Level  by Hugh Neill and Douglas Quadling, Series Editor Hugh Neill
Excerpt
1 Numbers
One thread in the history of mathematics has been the extension of what is meant by a number. This has led to the invention of new symbols and techniques of calculation. When you have completed this chapter, you should
• be able to recognise various number systems, and know the notation for them
• understand inequality relations, and the rules for calculating with them
• know what is meant by the modulus of a number, and how it can be used
• be familiar with techniques for calculating with surds.
1.1 Different kinds of number
At first numbers were used only for counting, and 1, 2, 3, ... were all that was needed. These are called positive integers.
Sometimes you also need the number 0, or ‘zero’. For example, suppose you are recording the number of sisters of every person in the class. Some will have one, two, three, ... sisters, but some will have none. The numbers 0, 1, 2, 3, ... are called natural numbers.
Then people found that numbers could also beuseful for measurement and in commerce. For these purposes they also needed fractions. Integers and fractions together make up the rational numbers. These are numbers which can be expressed in the form p/q where p and q are intedgers, and q is not 0.
Image not available in HTML version
Figure 1.1
One of the most remarkable discoveries of the ancient Greek mathematicians was that there are numbers which cannot be expressed like this. These are called irrational numbers. The first such number to be found was √2, which is the length of the diagonal of a square with side 1 unit, by Pythagoras’ theorem (see Fig. 1.1).
The argument that the Greeks used to prove this can be adapted to show that the square root, cube root, ... of any positive integer is either an integer or an irrational number. For example, the square root of 8 is an irrational number, but the cube root of 8 is the integer 2. Many other numbers are now known to be irrational, of which the most famous is π.
Rational and irrational numbers together make up the real numbers.
When rational numbers are written as decimals, they either come to a stop after a number of places, or the sequence of decimal digits eventually starts repeating in a regular pattern. For example,
Display matter not available in HTML version
The reverse is also true. If a decimal number stops or repeats indefinitely then it is a rational number. So if an irrational number is written as a decimal, the pattern of the decimal digits never repeats however long you continue the calculation.
Integers, rational and irrational numbers and real numbers can all be either positive or negative.
It is helpful to have special symbols to denote the different kinds of number. The set of integers {...,  3,  2,  1,0,1,2,3,...} is written as ℤ the set of (positive and negative) rational numbers is ℚ and the set of real numbers is ℝ. If you want just the positive numbers, you use the symbols ℤ^{+}, ℚ^{+}, ℝ^{+}. The set of natural numbers, {0,1,2,3, ...}, is written as ℕ; there are no negative natural numbers.
You may wonder why these letters were chosen. The notation was first used by German mathematicians in the 19th century, and the German word for number is Zahl. Hence the choice of ℤ for the integers. The letter ℚ probably came from Quotient, which is the result of dividing one number by another.
You probably know the symbol Ɛ, which stands for ‘is an element of’. The statement ‘x Ɛ ℤ^{+}’ means ‘x belongs to the set of positive integers’; that is, ‘x is a positive integer’. So ‘x Ɛ ℤ^{+}’ and ‘xis a positive integer
are two ways of saying the same thing.
You can also draw diagrams of these sets of numbers on a number line, as in Fig. 1.2. 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 number, the further it is to the right on the line.
Image not available in HTML versionFig. 1.2
The point which represents 0 is called the origin on the number line. It is usually denoted by the letter O.
But you will notice a snag. There are more real numbers than rational numbers. But in the figure ℝ andℚ, and ℝ^{+} and ℚ^{+}, look the same. This is because there are rational numbers as close as you like to any real number. For example, π Ɛ ℝ, but π is not a member of ℚ. However, 3.141 592 65 (which is the 8 decimal place approximation to π) is a member of ℚ, and you can’t distinguish π from 3.141 592 65 in the figure.
All the numbers that can be shown on a calculator are rational numbers. Calculators can’t handle irrational numbers. So when you key in √2 on your calculator, and it displays 1.414 213 562, this is only an approximation to √2. If you square the rational number 1.414 213 562, you get
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1.2 Notation for inequalities
You often want to compare one number with another and say which is the bigger. This comparison is expressed by using the inequality symbols >,<,≤ and ≥.
The symbol a > b means that a is greater than b. You can visualise this geometrically as in Fig. 1.3, which shows three number lines, with a to the right of b.
Image not available in HTML versionFig. 1.3
Notice that it does not matter whether a and b are positive or negative. The position of a and b in relation to zero on the number line is irrelevant. In all three lines, a > b. As an example, in the bottom line, 4 > 7.
Similarly, the symbol a < b means that a is less than b. You can visualise this geometrically on a number line, with a to the left of b.
Display matter not available in HTML versionThe symbol a ≥ b means ‘either a > b or a = b ’; that is, a is greater than or equal to, but not less than, b. Similarly, the symbol a ≥ b means ‘either a < b or a = b’ ; that is, a is less than or equal to, but not greater than, b.
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Some books use the symbols ≥ and ≤ in place of ≥ and ≤.
Since, for any two numbers a and b, a must be either greater than, equal to, or less than b, it follows that another way of writing
Display matter not available in HTML versionThe symbols < and > are called strict inequalities, and the symbols ≤ and ≥ are called weak inequalities.
Example 1.2.1
Write down the set of numbers x such that x єℕ and x <6.
Display matter not available in HTML versionExample 1.2.2
The points A and B on the number line represent the numbers −2 and 3. Use inequalities to describe the numbers represented by the line segment [AB] shown in Fig. 1.4.
Image not available in HTML versionThe notation [AB] is explained in Section .
All the points of the line segment, except A itself, are to the right of A. So the numbers they represent satisfy the inequality x≥2. They are also, except B itself, to the left of B, so the numbers satisfy x≤ 3.
You could also write x≥ 2 as  2≤ x. So  2 ≤ x and x ≤ 3.
These two inequalities can be combined 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 be both 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 numbers between 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 numbers 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 subtract) the same number on both sides of the equation
• you can multiply (or divide) both sides of the equation by the same number.
In this example, subtracting 7 from both sides and then dividing by 3 leads to the solution x = 4.
An inequality like x + 7 ≥  5 doesn’t have a single solution for x, but it can be replaced by a simpler statement about the value of x. To find this, you need rules for working with inequalities. These are similar to those for equations, but with one very important difference.
Adding or subtracting the same number on both sides
You can add or subtract the same number on both sides of an inequality. Justifying such a step involves showing that, for any number c, ‘if a ≥ b then a + c ≥ b + c’.
Image not available in HTML versionThis is saying that if a is to the right of b on the number line, then a + c is to the right of b + c. Fig. 1.5 shows that this is true whether c is positive or negative.
Since subtracting c is the same as adding  c, you can also subtract the same number from both sides.
Example 1.3.1
If x  3 ≤  4, what can you say about the value of x?
You can add 3 on both sides of the inequality, which gives
Display matter not available in HTML versionthat is
Display matter not available in HTML versionIn this example, the inequality x ≤  1 is called the solution of the inequality x  3 <  4. It is the simplest statement you can make about x which is equivalent to the given inequality.
Multiplying both sides by a positive number
You can multiply (or divide) both sides of an inequality by a positive number. That is, if a < b and c < 0, then ca < cb. Here is a justification.
As a < b, a is to the right of b on the number line.
As c < 0, ca and cb are enlargements of the positions of a and b relative to the number 0.
Fig. 1.6 shows that, whether a and b are positive or negative, ca is to the right of cb, so ca < cb.
Image not available in HTML versionExample 1.3.2
Solve the inequality 1 / 3x > 2.
Multiply both sides of the inequality by 3. This gives the equivalent inequality
Display matter not available in HTML versionthat is
Display matter not available in HTML versionMultiplying both sides by a negative number
If a > b, and you subtract a + b from both sides, then you get  b >  a, which is the same as  a <  b. This shows that if you multiply both sides of an inequality by  1, then you change the direction of the inequality.
Suppose that you wish to multiply the inequality a > b by  2. This is the same as multiplying  a <  b by 2, so  2a <  2b.
Image not available in HTML versionYou can also think of multiplying by  2 as reflecting the points corresponding to a and b in the origin, and then multiplying by 2 as an enlargement (see Fig. 1.7).
You can summarise this by saying that if you multiply (or divide) both sides of an inequality by a negative number, you must change the direction of the inequality. Thus if a > b and c < 0, then ca < cb.
Solve the inequality  3x < 21.
Example 1.3.3
In this example you need to divide both sides by  3. Remembering to change the direction of the inequality,  3x < 21 becomes x >  7.
Summary of operations on inequalities
• You can add or subtract a number on both sides of an inequality.
• You can multiply or divide an inequality by a positive number.
• You can multiply or divide an inequality by a negative number, but 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 > b and c > 0, both a  b and c are positive numbers, so c ({a  b}) is also positive. So ca  cb is positive, ca  cb > 0 and ca > cb.
Example 1.3.4
Solve the inequality
Display matter not available in HTML versionBegin by subtracting 3 from both sides, to get
Display matter not available in HTML versionNow multiply both sides by −2, remembering to change the direction of the inequality. This gives
Display matter not available in HTML versionExample 1.3.5
Solve the inequality 4  2x > 3x.
Begin by adding 2x on both sides, to get
Display matter not available in HTML versionDividing both sides by 5 then gives
Display matter not available in HTML versionThe answer is usually written with x on the left side, as
Display matter not available in HTML versionExercise 1A
Which of the number systems ℕ, ℤ, ℚ, ℝ, ℤ ^{+}, ℚ ^{+}, andℝ ^{+} contain the following numbers?
Display matter not available in HTML versionState in list form the sets of numbers x which satisfy the following.
Display matter not available in HTML versionSolve the following inequalities, where x єℝ.
Display matter not available in HTML version1.4 Modulus notation
Suppose that you want to find the difference between the heights of two children. With numerical information, the answer is quite straightforward: if their heights are 90 cm and 100 cm, you would answer 10 cm; and if their heights were 100 cm and 90 cm, you would still answer 10 cm.
But how would you answer the question if their heights were H cm and h cm? The answer is, it depends which is bigger: 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 0 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 number. This is a quantity which tells you the ‘size’ of a number regardless of its sign. For example, the modulus of 15 is 15, but the modulus of −15 is also 15.
The notation for modulus is to write the number between a pair of vertical lines. So you would write 15 = 15 and  15 = 15.
The modulus is sometimes called the ‘absolute value’ of the number. Some calculators have a key marked ‘abs’ which produces the modulus. If yours has, try using it with a variety of inputs.
The modulus can be defined formally like this:
The modulus of x, written x and pronounced ‘mod x’, is defined by
Display matter not available in HTML versionUsing 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 about numbers which are large numerically, but which are negative, such as  1000 or  1,000,000. These are ‘negative numbers with large modulus’.
For example, for large positive values of x, the value of 1/x is close to 0. The same is true for 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>1000, 1/x< 0.001.
1.5 Modulus on the number line
Image not available in HTML version
In Fig. 1.8 A and B are points on a number line with coordinates a and b. How can you express the distance AB in terms of a and b?
• If B is to the right of A, then b > a, so b  a > 0 and the distance is b  a.
• If B is to the left of A, then b < a, so b  a < 0 and the distance is a  b =  (b  a).
• If B and A coincide, then b = a, so b  a = 0 and the distance is 0.
You will recognise this as the definition of b  a.
The distance between points on the number line representing numbers a and b is b  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 about X?
There are two possibilities: 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 = 3 or x =  3, then X is 3 units from O, so x = 3.
A convenient way of summarising this is to write
Display matter not available in HTML versionSimilarly you can write
Display matter not available in HTML versionsince these are two different ways of saying that X is within 3 units of O on the number line.
In this example there is nothing special about the number 3. You could use the same argument with 3 replaced by any positive number a to show that
Display matter not available in HTML versionYou can get a useful generalisation by replacing x in this statement by x  k, where k is a constant:
If a is a positive number, x  k<=meansthat  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,
Display matter not available in HTML versionThe equivalence then becomes:
If a is a positive number,
x k<= ameansthat k  a <= x <= k + a.This result is used when you give a number correct to a certain number of decimal places. For example, to say that x = 3.87 ‘correct to 2 decimal places’ is in effect saying that x  3.87<= 0.005.
Image not available in HTML versionFig. 1.9 shows that x  3.87<= 0.005 means
Display matter not available in HTML versionor 3.865 <= x <= 3.875.
Exercise 1B
1 Write down the values of
Display matter not available in HTML version2 If x> 100, what can you say about 1/x^{2}? Write your answer using inequality notation.
4 The mass of a prize pumpkin is 4.76 kg correct to 2 decimal places. Denoting the mass by m kg, write this statement in mathematical form
(a) without modulus notation,
(b) using modulus notation.
4 The length and width of a rectangle are measured as 4.8 cm and 3.6 cm, both correct to 1 decimal place. Write a statement about the perimeter, P cm, in mathematical form
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 < x< b or a < x < b
Display matter not available in HTML version7 Rewrite the given inequalities using modulus notation. 1< x <2
Display matter not available in HTML version8 What can you say about x if x Ƹ ℤ?
1.6 Surds and their properties
When you have met expressions such as √ 2, √8 and √12 before, it is likely that you have used a calculator to express them in decimal form. You might have written Display matter not available in HTML version
Why is the statement ‘√ 2 = 1.414’ incorrect?
Expressions like √2 or 3∛9 are called surds. However, expressions like √4 or 3∛27, which are whole numbers, are not surds.
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