Mathematics Read online

  (c) 1701 ÷ 7

  (d) 1752 ÷ 8

  8 Find the quotients and the remainders for the following divisions.

  (a) 8 ÷ 3

  (b) 23 ÷ 5

  (c) 101 ÷ 8

  1.3 Order of operations

  Sometimes you will see a calculation in the form 8 − (2 × 3), where brackets, that is the symbols (and), are put around part of the calculation. Brackets are there to tell you to carry out the calculation inside them first. So 8 − (2 × 3) = 8 − 6 = 2.

  If you had to find the answer to 8 + 5 − 3, you might wonder whether to do the addition first or the subtraction first.

  If you do the addition first, 8 + 5 − 3 = 13 − 3 = 10.

  If you do the subtraction first, 8 + 5 − 3 = 8 + 2 = 10.

  In this case, it makes no difference which operation you do first.

  If, however, you had to find the answer to 3 + 4 × 2, your answer will depend on whether you do the addition first or the multiplication first.

  If you do the addition first, 3 + 4 × 2 = 7 × 2 = 14.

  If you do the multiplication first, 3 + 4 × 2 = 3 + 8 = 11.

  To avoid confusion, mathematicians have decided that, if there are no brackets, multiplication and division are carried out before addition and subtraction. Such a rule is called a convention: clearly the opposite decision could be made, but it has been agreed that the calculation 3 + 4 × 2 should be carried out as 3 + 4 × 2 = 3 + 8 = 11.

  There are a number of these conventions, often remembered by the word ‘BiDMAS’, which stands for

  Brackets,

  Indicies,

  Divide,

  Multiply,

  Add,

  Subtract.

  This is the order in which arithmetic operations must be carried out. Most calculators have the ‘BiDMAS’ convention programmed into them, and so automatically carry out operations in the correct order.

  Spotlight

  When a calculation involves just multiplication/division or addition/subtraction, then you can carry it out in any order. It often helps to reorder a calculation to make it easier to perform. You must keep the operation with its original number.

  For example, 2 × 24 × 5 ÷ 3 = 2 × 5 × 24 ÷ 3 = 10 × 24 ÷ 3 = 10 × 8 = 80

  Example 1.3.1

  Carry out the following calculations.

  (a) 5 × (6 − 2)

  (b) (10 + 2) ÷ 4

  (c) 20 − 3 × 5

  (d) 24 ÷ 6 + 2

  (e) 4 × (8 − 3) + 2

  (a) 5 × (6 − 2) = 5 × 4 = 20 brackets first

  (b) (10 + 2) ÷ 4 = 12 ÷ 4 = 3 brackets first

  (c) 20 − 3 × 5 = 20 − 15 = 5 multiply before subtracting

  (d) 24 ÷ 6 + 2 = 4 + 2 = 6 division before addition

  (e) 4 × (8 − 3) + 2 = 4 × 5 + 2 = 20 + 2 = 22 brackets first

  multiply before adding

  Try the calculations in Example 1.3.1 on your calculator to see if it uses the ‘BiDMAS’ convention and carries them out correctly.

  Exercise 1.3

  In each part of question 1, use the ‘BiDMAS’ convention to carry out the calculation.

  1 (a) (3 + 4) × 5

  (b) 3 + 4 × 5

  (c) (7 − 3) × 2

  (d) 7 − 3 × 2

  (e) (12 ÷ 6) ÷ 2

  (f) 12 ÷ (6 ÷ 2)

  (g) 20 ÷ 4 + 1

  (h) 3 × 5 + 4 × 2

  (i) 4 × (7 − 2) + 6

  (j) 6 × 4 − 2 × 8

  (k) 24 ÷ (4 + 2) − 1

  (l) (7 + 3) × (9 − 2)

  (m) 10 + (8 + 4) ÷ 3

  (n) 9 + 5 × (8 − 2)

  (o) 15 ÷ 3 − 16 ÷ 8

  (p) 5 + 6 × 3 − 1

  (q) (5 + 6) × 3 − 1

  (r) 5 + 6 × (3 − 1)

  1.4 Problems which use arithmetic

  In Exercise 1.4, you have to decide which arithmetic operation to use. If you are not sure, it can sometimes be helpful to think of a similar question with smaller numbers.

  Example 1.4.1

  An egg box holds six eggs. How many eggs boxes are needed to pack 500 eggs, and how many eggs are there in the last box?

  You need to find how many sixes there are in 500. If you use a calculator you get 500 ÷ 6 = 83.333 333 3.

  This means that 84 boxes are needed. 83 full boxes hold 83 × 6 eggs, which is 498 eggs. So there are 2 eggs in the last box.

  Spotlight

  Whether you need to round up or down depends on the context of the question. In the last example, you rounded up as you needed 84 boxes altogether. The answer to the question: ‘How many pens costing £6 each can be bought for £500?’ is 83, as 83 × £6 = £498. So there is not enough money to buy 84 pens.

  Exercise 1.4

  1 In a school, 457 of the pupils are boys and 536 are girls. Work out the number of pupils in the school.

  2 How many hours are there in 7 days?

  3 The sum of two numbers is 3472. One of the numbers is 1968. Find the other number.

  4 The product of two numbers is 161. One of the numbers is 7. Find the other number.

  5 A car travels 34 miles on one gallon of petrol. How far will it travel on 18 gallons?

  6 How many pieces of metal 6 cm long can be cut from a bar 117 cm long? What length of metal is left over?

  7 A box holds 24 jars of coffee. How many boxes will be needed to hold 768 jars?

  8 At the start of a journey, a car’s milometer read 34 652. After the journey, it read 34 841. How long was the journey?

  9 There are 365 days in a year. How many days are there in 28 years?

  10 A coach can carry 48 passengers. How many coaches are needed to carry 1100 passengers? How many seats will be unoccupied?

  1.5 Special numbers

  The numbers 0, 1, 2, 3, 4,…are called whole numbers.

  The numbers 1, 2, 3, 4,…are called natural numbers. They are also called counting numbers or positive whole numbers.

  The numbers…, −3, −2, −1, 0, 1, 2, 3,…are called integers. These are the positive and negative whole numbers, together with 0.

  The natural numbers can be split into even and odd numbers.

  An even number is one into which 2 divides exactly. The first six even numbers are 2, 4, 6, 8, 10 and 12. You can tell whether a number is even by looking at its last digit. If the last digit is 2, 4, 6, 8 or 0, then the number is even. The last digit in the number 784 is 4, so 784 is even.

  An odd number is one into which 2 does not divide exactly; that is, it is a natural number which is not even. The first six odd numbers are 1, 3, 5, 7, 9 and 11. You can tell whether a number is odd by looking at its last digit. If the last digit is 1, 3, 5, 7 or 9, then the number is odd. In other words, if the last digit is odd, then the number is odd. The last digit in the number 837 is odd, so 837 is odd.

  A square number is a natural number multiplied by itself. It is sometimes called a perfect square or simply a square, and can be shown as a square of dots. The first four square numbers are 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and 4 × 4 = 16 (Figure 1.1).

  Figure 1.1

  To get a square number you multiply a whole number by itself. So the fifth square number is 5 × 5 = 25.

  A triangle number can be illustrated as a triangle of dots (Figure 1.2). The first four triangle numbers are 1, 3, 6 and 10.

  Figure 1.2

  Each triangle number is obtained from the previous one by adding one more row of dots down the diagonal. The fifth triangle number is found by adding 5 to the fourth triangle number, so the fifth triangle number is 15.

  A rectangle number can be shown as a rectangle of dots. A single dot or a line of dots is not regarded as a rectangle. Figure 1.3 shows the rectangle number 12 illustrated in two ways.

  Figure 1.3

  A cube number can be shown as a cube of dots (Figure 1.4). The first three cube numbers are 1, 8 and 27 which can be written 1 = 1 × 1 × 1, 8 = 2 × 2 × 2 and 27 = 3 × 3 × 3. To get a cube number you multiply a whole number by itself, and then by itsel
f again. So the fourth cube number is 4 × 4 × 4 = 64.

  Figure 1.4

  Exercise 1.5

  1 From the list 34, 67, 112, 568, 741 and 4366, write down

  (a) all the even numbers,

  (b) all the odd numbers.

  2 Write down the first ten square numbers.

  3 Write down the first ten triangle numbers.

  4 Write down the first ten rectangle numbers.

  5 Write down the first ten cube numbers.

  6 From the list of numbers 20, 25, 28, 33, 36 and 37, write down

  (a) all the square numbers,

  (b) all the triangle numbers,

  (c) all the rectangle numbers.

  7 The difference between consecutive square numbers is always the same type of number. What type of number is it?

  8 The sum of the first two triangle numbers is 4 (=1 + 3).

  (a) Find the sums of three more pairs of consecutive triangle numbers.

  (b) What do you notice about your answers?

  9 Find the sums of the first two, three, four and five cube numbers. What do you notice about your answers?

  10 (a) Write down a rectangle number, other than 12, for which there are two different patterns of dots. Draw the patterns.

  (b) Write down a rectangle number for which there are three different patterns of dots.

  (c) Write down a rectangle number for which there are four different patterns of dots.

  1.6 Multiples, factors and primes

  Figure 1.5 shows the first five numbers in the 8 times table. The numbers on the right, 8, 16, 24, 32 and 40, are called multiples of 8. Notice that 8 itself, which is 1 × 8 or 8 × 1, is a multiple of 8.

  Figure 1.5

  The factors of a number are the numbers which divide exactly into it. So, the factors of 18 are 1, 2, 3, 6, 9 and 18. Notice that 1 is a factor of every number and that every number is a factor of itself.

  Example 1.6.1

  Write down all the factors of (a) 12, (b) 11.

  (a) The factors of 12 are 1, 2, 3, 4, 6 and 12.

  (b) The factors of 11 are 1 and 11.

  Spotlight

  It is easy to find all the factors of a number by looking for pairs of numbers that multiply to give that number. For example, 1 × 18 = 18, 2 × 9 = 18, 3 × 6 = 18, so the factors of 18 are 1, 2, 3, 6, 9 and 18.

  The number 11 has only two factors and is an example of a prime number. A prime number, sometimes simply called a prime, is a number greater than 1 which has exactly two factors, itself and 1.

  Thus, the first eight prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19.

  Spotlight

  2 is the only even prime number.

  Is 1 a prime number? The number 1 has only one factor (itself), and so it is not a prime number since the definition states that a prime number must have two factors.

  Exercise 1.6

  1 Write down the first five multiples of 7.

  2 (a) Write down the first ten multiples of 5.

  (b) How can you tell that 675 is also a multiple of 5?

  3 Write down all the factors of 20.

  4 (a) Find the sum of the factors of 28, apart from 28 itself.

  (b) Comment on your answer.

  5 Write down all the factors of 23.

  6 (a) The number 4 has three factors (1, 2 and 4). Find five more numbers, each of which has an odd number of factors.

  (b) What do you notice about your answers to part (a)?

  7 Write down the four prime numbers between 20 and 40.

  8 2 is the only even prime number. Explain why there cannot be any others.

  9 Write down the largest number which is a factor of 20 and is also a factor of 30.

  10 Write down the smallest number which is a multiple of 6 and is also a multiple of 8.

  11 Write down the factors of 30 which are also prime numbers. (These are called the prime factors of 30.)

  12 Write down the prime factors of 35.

  13 The prime factors of 6 are 2 and 3. Find another number which has 2 and 3 as its only prime factors.

  Key ideas

  • The decimal system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

  • Sum means the result of adding two or more numbers together.

  • Difference means the result of subtracting a smaller number from a larger number.

  • Product means the result of multiplying two or more numbers together.

  • Quotient means the whole number resulting from the division of two numbers, i.e. when 14 is divided by 3, the quotient is 4 and the remainder is 2.

  • The order operations should be performed in can be remembered using the mnemonic BiDMAS, which stands for:

  • Natural numbers or counting numbers or positive whole numbers are the numbers 1, 2, 3, 4,…

  • Integers are the numbers…, −3, −2, −1, 0, 1, 2, 3,…

  • Even numbers are exactly divisible by 2, and are the numbers 2, 4, 6, 8,…

  • Odd numbers are not exactly divisible by 2, and are the numbers 1, 3, 5, 7,…

  • Square numbers are the result of multiplying a natural number by itself. They include the numbers 1, 4, 9, 16,…

  • Triangle numbers can be shown as a triangle of dots. They include the numbers 1, 3, 6, 10,…

  • Rectangle numbers are any number that can be shown as a rectangle of dots.

  • Cube numbers can be shown as a cube of dots. The first three cube numbers are 1, 8 and 27.

  • The multiples of a number are the answers to that number’s times table. So the multiples of 4 are 4, 8, 12, 16, 20,…

  • A factor of a number is any number which divides exactly into that number. Every number has 1 and itself as factors. The factors of 10 are 1, 2, 5 and 10.

  • Prime numbers are numbers with exactly two factors. The first five prime numbers are 2, 3, 5, 7 and 11.

  2

  Angles

  In this chapter you will learn:

  • about different types of angles

  • how to measure and draw angles

  • angle facts and how to use them

  • about parallel lines

  • about bearings.

  2.1 Introduction

  The study of angles is one branch of geometry, which is concerned also with points, lines, surfaces and solids. Geometry probably originated in Ancient Egypt, where it was applied, for example, to land surveying and navigation. Indeed, the name ‘geometry’ is derived from the Greek for ‘earth measure’.

  It was Greek mathematicians, notably Euclid, who developed a theoretical foundation for geometry. He began his classic treatise Elements with definitions of basic geometrical concepts including angles, the subject of this chapter.

  2.2 Angles

  An angle is formed when two straight lines meet, as in Figure 2.1. The point where the lines meet is called the vertex of the angle. The size of the angle is the amount of turn from one line to the other, and is not affected by the lengths of the lines. The angles in Figure 2.2 are all equal.

  Figure 2.1

  Figure 2.2

  If you begin by facing in a certain direction and then turn round until you are facing in the original direction again, as in Figure 2.3, you will have made a complete turn (or revolution). One complete turn is divided into 360 degrees, which is written as 360°.

  Figure 2.3

  In a half turn, Figure 2.4, therefore, there are 180°. In a quarter turn there are 90°, which is called a right angle. It is often marked on diagrams with a small square, as shown in Figure 2.5. If the angle between two lines is a right angle, the lines are said to be perpendicular.

  Figure 2.4

  Figure 2.5

  Apart from angles of 90° and 180°, all other sizes of angles fall into one of three categories, shown in Figure 2.6.

  Figure 2.6

  An angle which is less than 90° is called an acute angle.

  An angle which is between 90° and 180° is called an obtuse angle.

  An angle which is greate
r than 180° is called a reflex angle.

  Spotlight

  A useful memory aid is ‘a-cute little angle’ to remind you that angles less than 90° are acute.

  There are three ways of naming angles.

  One way is to use a letter inside the angle, as with the angle marked x in Figure 2.7. Usually, a small letter, rather than a capital, is used.

  Figure 2.7

  In Figure 2.8, the angle could be called angle B, but you should do this only if there is no risk of confusion. The symbols ∠ and ∧ are both used to represent ‘angle’, so angle B could be shortened to ∠B or B̂. It could be argued that there are two angles at B, one of them acute and one of them reflex, but you should assume that the acute angle is intended unless you are told otherwise.

  Figure 2.8

  Alternatively, the angle in Figure 2.8 could be called angle ABC (or angle CBA). You could shorten this to ∠ABC or . The middle letter is always the vertex of the angle.

  Exercise 2.1

  1 How many degrees does the hour hand of a clock turn in 20 minutes?

  2 Work out the size of the angle the second hand of a clock turns through in 1 second.

  3 Work out the size of the angle the hour hand of a clock turns through in 35 minutes.

  4 Work out the size of the angle the second hand of a clock turns through in 55 seconds.

  5 Work out the size of the angle between South and West when it is measured (a) clockwise, (b) anticlockwise.