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Unformatted text preview: Unit 1 Algebra Welcome to MST124 Welcome to MST124 In this module you’ll learn the essential ideas and techniques that underpin university-level study in mathematics and mathematical subjects such as physics, engineering and economics. You’ll also develop your skills in communicating mathematics. Here are some of the topics that you’ll meet. Vectors are quantities that have both a size and a direction. For example, a ship on the ocean moves not only with a particular speed, but also in a particular direction. Speed in a particular direction is a vector quantity known as velocity. Calculus is a fundamental topic in mathematics that’s concerned with quantities that change continuously. If you know that an object is moving at a constant speed, then it’s straightforward to work out how much distance it covers in any given period of time. It’s not so easy to do this if the object’s speed is changing – for example, if it’s accelerating, as a falling object does. Calculus can be used to deal with situations like this. The motion of a ship consists of speed and direction Matrices are rectangular arrays of numbers – for example, any rectangular table of numbers forms a matrix. Matrices have many applications, which involve performing operations on them that are similar to the operations that you perform on individual numbers. For example, you can add, subtract and multiply matrices. Sequences are lists of numbers. Sequences whose numbers have a connecting mathematical relationship arise in many different contexts. For example, if you invest £100 at a 5% rate of interest paid annually, then the value in pounds of your investment at the beginning of each year forms the sequence 100, 105, 110.25, 115.76, 121.55, . . .. The complex numbers include all the real numbers that you know about already, and also many ‘imaginary’ numbers, such as the square root of −1. Amazingly, they provide a simple way to deal with some types of complicated mathematics that arise in practical problems. You’ll see that not only do the topics above have important practical applications, but they’re also intriguing areas of study in their own right. One of the main aims of the first few units of MST124 is to make sure that you’re confident with the basic skills in algebra, graphs, trigonometry, indices and logarithms that you’ll need. The mathematics in the later units of the module depends heavily on these basic skills, and you’ll find it much easier and much quicker to study and understand if you can work with all the basic skills fluently and correctly. To help you attain confidence with these skills, the first few units of the module include many revision topics, as well as some new ones. Which parts, and how much, of the revision material you’ll need to study will depend on your mathematical background – different students start MST124 with widely differing previous mathematical experiences. When you’re deciding which revision topics you need to study, remember that The speed of a falling ball increases as it falls 3 Unit 1 Algebra even though you’ll have met most of them before, you won’t necessarily have acquired the ‘at your fingertips’ fluency in working with them that you’ll need. Where that’s the case, you’ll benefit significantly from working carefully through the revision material. Information for joint MST124 and MST125 students If you are studying Essential mathematics 2 (MST125) with the same start date as MST124, then you should not study the MST124 units on the dates shown on the main MST124 study planner. Instead, you should follow the MST124 and MST125 joint study planner, which is available from the MST124 and MST125 websites. This is important because you will not be prepared to study many of the topics in MST125 if you have not already studied the related topics in MST124. The MST124 and MST125 joint study planner ensures that you study the units of the two modules interleaved in the correct order. The MST124 assignment cut-off dates shown in the MST124 and MST125 joint study planner are the same as those shown on the MST124 study planner. Introduction The main topic of this first unit is basic algebra, the most important of the essential mathematical skills that you’ll need. You’ll find it difficult to work through many of the units in the module, particularly the calculus units, if you’re not able to manipulate algebraic expressions and equations fluently and accurately. So it’s worth spending some time now practising your algebra skills. This unit gives you the opportunity to do that. The unit covers a lot of topics quite rapidly, in the expectation that you’ll be fairly familiar with much of the material. You should use it as a resource to help you make sure that your algebra skills are as good as they can be. You may not need to study all the topics – you should concentrate on those in which you need practice. For many students these will be the topics in Sections 3 to 6. A good strategy might be to read through the whole unit, doing the activities on the topics in which you know you need practice. For the topics in which you think you don’t need practice, try one or two of the later parts of each activity to make sure – there may be gaps and rustiness in your algebra skills of which you’re unaware. Remember to check all your answers against the correct answers provided (these are at the end of the unit in the print book, and can be obtained by pressing the ‘show solution’ buttons in some screen versions). As with all the units in this module, further practice questions are available in both the online practice quiz and the exercise booklet for the unit. 4 1 Numbers Working through the revision material in this unit should also help you to clarify your thinking about algebra. For example, you might know what to do with a particular type of algebraic expression or equation, but you might not know, or might have forgotten, why this is a valid thing to do. If you can clearly understand the ‘why’, then you’ll be in a much better position to decide whether you can apply the same sort of technique to a slightly different situation, which is the sort of thing that you’ll need to do as you study more mathematics. Some of the topics in the unit may seem very easy – basic algebra is revised starting from the simplest ideas. Others may seem quite challenging – some of the algebraic expressions and equations that you’re asked to manipulate may be more complicated than those that you’ve dealt with before, particularly the ones involving algebraic fractions and indices. The final section of the unit, Section 6, describes some basic principles of communicating mathematics in writing. This will be important throughout your study of this module and in any further mathematical modules that you study. If you find that much of the content of this unit (and/or Unit 2) is unfamiliar to you, then contact your tutor and/or Learner Support Team as soon as possible, to discuss what to do. The word ‘algebra’ is derived from the title of the treatise al-Kit¯ ab al-mukhtas.ar f¯i h.is¯ ab al-jabr wa’l-muq¯ abala (Compendium on calculation by completion and reduction), written by the Islamic mathematician Muh.ammad ibn M¯ us¯a al-Khw¯ arizm¯i in around 825. This treatise deals with solving linear and quadratic equations, but it doesn’t use algebra in the modern sense, as no letters or other symbols are used to represent numbers. Modern, symbolic algebra emerged in the 1500s and 1600s. 1 Numbers In this section you’ll revise different types of numbers, and some basic skills associated with working with numbers. 5 Unit 1 Algebra 1.1 Types of numbers We’ll make a start by briefly reviewing some different types of numbers. Remember that all the definitions given here, and all the other definitions and important facts and techniques given in the module, are also set out in the Handbook, so you can refer to them easily. The natural numbers, also known as the positive integers, are the counting numbers, 1, 2, 3, . . . . (The symbol ‘. . . ’ here is called an ellipsis and is used when something has been left out. You can read it as ‘dot, dot, dot’. In some texts the natural numbers are defined to be 0, 1, 2, 3, . . . .) The natural numbers, together with their negatives and zero, form the integers: . . . , −3, −2, −1, 0, 1, 2, 3, . . . . The Latin word integer consists of the prefix in, meaning ‘not’, attached to the root of tangere, meaning ‘to touch’. So it literally means ‘untouched’, in the sense of ‘whole’. The rational numbers are the numbers that can be written in the form integer ; integer that is, as an integer divided by an integer. For example, all the following numbers are rational numbers: 3 4, 2 13 , 4, −4, − 78 , 0.16, 7.374. You can check this by writing them in the form above, as follows: 3 7 4 −4 −8 16 7374 , , , , , , . 4 3 1 1 7 100 1000 The real numbers include all the rational numbers, and many other numbers as well. A useful way to think of the real numbers is to envisage them as lying along a straight line that extends infinitely far in each direction, called the number line or the real line. Every point on the number line corresponds to a real number, and every real number corresponds to a point on the line. Some points on the line correspond to rational numbers, while others correspond to numbers that are not rational, which are known as irrational numbers. Figure 1 shows some numbers on the number line. 6 1 − −4 −3 −2 √ 2 − 25 −1 √ 2 1 2 0 1 9 2 e π 2 3 Numbers 4 5 Figure 1 Some numbers on the number line √ √ Four of the numbers√marked in Figure 1 are irrational, namely − 2, 2, e and π. The number 2 is the positive square root of 2, that is, the positive number that when multiplied by itself √ gives the answer 2. Its value is approximately 1.41. The number − 2 is the negative of this number. The numbers e and π are two important constants that occur frequently in mathematics. You probably know that π is the number obtained by dividing the circumference of any circle by its diameter (see Figure 2). Its value is approximately 3.14. (The symbol π is a lower-case Greek letter, pronounced ‘pie’.) The constant e has value approximately 2.72, and you’ll learn more about it in this module, starting in Unit 3. √ √ To check that the numbers − 2, 2, e and π are irrational, you have to prove that they can’t be written as an integer √ divided by an integer. If you’d like to √ see how this can be done for 2, then look at the document A proof that 2 is irrational on the module website. Proving that e and π can’t be written as an integer divided by an integer is more difficult, and outside the scope of this module. circumference diameter Figure 2 The circumference and diameter of a circle Every rational number can be written as a decimal number. To do this, integer you divide the top number of the fraction of the form by the integer bottom number. For example, 1 8 = 0.125, 2 3 = 0.666 666 666 . . . , 83 74 = 1.1 216 216 216 216 . . . . As you can see, the decimal form of 18 is terminating: it has only a finite number of digits after the decimal point. The decimal forms of both 23 and 83 74 are recurring: each of them has a block of one or more digits after the decimal point that repeats indefinitely. There are two alternative notations for indicating a recurring decimal: you can either put a dot above the first and last digit of the repeating block, or you can put a line above the whole repeating block. For example, 2 ˙ and 3 = 0.666 666 666 . . . = 0.6 = 0.6, 83 ˙ ˙ 74 = 1.1 216 216 216 216 . . . = 1.1216 = 1.1216 . In fact, the decimal form of every rational number is either terminating or recurring. Also, every terminating or recurring decimal can be written as an integer divided by an integer and is therefore a rational number. If you’d like to know why these facts hold, then look at the document Decimal forms of rational numbers on the module website. 7 Unit 1 Algebra The decimal numbers that are neither terminating nor recurring – that is, those that are infinitely long but have no block of digits that repeats indefinitely – are the irrational numbers. This gives you another way to distinguish between the rational and irrational numbers, summarised below. Decimal forms of rational and irrational numbers The rational numbers are the decimal numbers that terminate or recur. The irrational numbers are the decimal numbers with an infinite number of digits after the decimal point but with no block of digits that repeats indefinitely. So, for example, the irrational number π has a decimal expansion that is infinitely long and has no block of digits that repeats indefinitely. Here are its first 40 digits: π = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 . . . . You might like to watch the one-minute video clip entitled The decimal expansion of π, available on the module website. In 2006, a Japanese retired engineer and mental health counsellor, Akira Haraguchi, recited the first 100 000 digits of π from memory. It took him 16 hours. Natural numbers Integers Rational numbers Real numbers Figure 3 Types of numbers 8 Figure 3 is a summary of the types of numbers mentioned in this subsection. It illustrates that all the natural numbers are also integers, all the integers are also rational numbers, and all the rational numbers are also real numbers. In Unit 12 you’ll learn about yet another type of number. The complex numbers include all the numbers in Figure 3, and also many ‘imaginary’ numbers, such as the square root of −1. The idea of imaginary numbers might seem strange, but these numbers are the foundation of a great deal of interesting and useful mathematics. They provide a natural, elegant way to work with seemingly complicated mathematics, and have many practical applications. 1 Numbers 1.2 Working with numbers In this subsection, you’ll revise some basic skills associated with working with numbers. It’s easy to make mistakes with these particular skills, and people often do! So you should find it helpful to review and practise them. Before doing so, notice the label ‘(1)’ on the right of the next paragraph. It’s used later in the text to refer back to the contents of the line in which it appears. Labels like this are used throughout the module. The BIDMAS rules When you evaluate (find the value of) an expression such as 200 − 3 × (1 + 5 × 23 ) + 7, (1) it’s important to remember the following convention for the order of the operations, so that you get the right answer. Order of operations: BIDMAS Carry out mathematical operations in the following order. B I D M A S brackets indices (powers and roots) 0 divisions same precedence multiplications 0 additions same precedence subtractions Where operations have the same precedence, work from left to right. As you can see, the I in the BIDMAS rules refers to ‘indices (powers and roots)’. Remember that raising a number to a power means multiplying it by itself a specified number of times. For example, 23 (2 to the power 3) means three 2s multiplied together: 23 = 2 × 2 × 2. In particular, squaring and cubing a number mean raising it to the powers 2 and 3, respectively. When you write an expression such as 23 , you’re using index notation. Taking a root of a number means taking its square root, for example, or another type of root. Roots are revised in Subsection 4.1. If you type an expression like expression (1) into a calculator of the type recommended in the MST124 guide, then it will be evaluated according to the BIDMAS rules. However, it’s essential that you understand and remember the rules yourself. For example, you’ll need to use them when you work with algebra. Example 1 reminds you how to use the BIDMAS rules. It also illustrates another feature that you’ll see throughout the module. Some of the worked examples include lines of blue text, marked with the following icons . 9 Unit 1 Algebra This text tells you what someone doing the mathematics might be thinking, but wouldn’t write down. It should help you understand how you might do a similar calculation yourself. Example 1 Using the BIDMAS rules Evaluate the expression 200 − 3 × (1 + 5 × 23 ) + 7 without using your calculator. Solution The brackets have the highest precedence, so start by evaluating what’s inside them. Within the brackets, first deal with the power, then do the multiplication, then the addition. 200 − 3 × (1 + 5 × 23 ) + 7 = 200 − 3 × (1 + 5 × 8) + 7 = 200 − 3 × (1 + 40) + 7 = 200 − 3 × 41 + 7 Now do the multiplication, then the addition and subtraction from left to right. = 200 − 123 + 7 = 77 + 7 = 84 You can practise using the BIDMAS rules in the next activity. Remember that where division is indicated using fraction notation, the horizontal line not only indicates division but also acts as brackets for the expressions above and below the line. For example, 1+2 1 + 32 means (1 + 2) , (1 + 32 ) that is, (1 + 2) ÷ (1 + 32 ). In a line of text, this expression would normally be written as (1 + 2)/(1 + 32 ), with a slash replacing the horizontal line. The brackets are needed here because 1 + 2/1 + 32 would be interpreted as 1 + (2/1) + 32 . Part (b) of the activity involves algebraic expressions. Remember that multiplication signs are usually omitted when doing algebra – quantities that are multiplied are usually just written next to each other instead (though, for example, 3 × 4 can’t be written as 34). 10 1 Activity 1 Numbers Using the BIDMAS rules (a) Evaluate the following expressions without using your calculator. (i) 23 − 2 × 3 + (4 − 2) (ii) 2 − 1 2 ×4 (iii) 4 × 32 1+2 (vi) 1 − 2/32 1 + 32 (b) Evaluate the following expressions when a = 3 and b = 5, without using your calculator. * + b 2 (i) 3(b − a) (ii) a + b(2a + b) (iii) a + 9 (iv) 30/(ab) a (iv) 2 + 22 (v) Rounding When you use your calculator to carry out a calculation, you often need to round the result. There are various ways to round a number. Sometimes it’s appropriate to round to a particular number of decimal places (often abbreviated to ‘d.p.’). The decimal places of a number are the positions of the digits to the right of the decimal point. You can also round to the nearest whole number, or to the nearest 10, or to the nearest 100, for example. More often, it’s appropriate to round to a particular number of significant figures (often abbreviated to ‘s.f.’ or ‘sig. figs.’). The first significant figure of a number is the first non-zero digit (from the left), the next significant figure is the next digit along (whether zero or not), and so on. Once you’ve decided where to round a number, you need to look at the digit immediately after where you want to round. You round up if this digit is 5 or more, and round down otherwise. When you round a number, you should state how it’s been rounded, in brackets after the rounded number, as illustrated in the next example. Notice the ‘play button’ icon next to the following example. It indicates that the example has an associated tutorial clip – a short video in which a tutor works through the example and explains it. You can watch the clip, which is available on the module website, instead of reading through the worked example. Many other examples in the module have tutorial clips, indicated by the same icon. 11 Unit 1 Algebra Example 2 Rounding numbers Round the following numbers as indicated. (a) 0.0238 to three decimal places (b) 50 629 to three significant figures (c) 0.002 958 2 to two significant figures Solution (a) Look at the digit after the first three decimal places: 0. 023 8. It’s 8, which is 5 or more, so round up. 0.0238 = 0.024 (to 3 d.p.) (b) Look at the digit after the first three significant figures: 506 29. It’s 2, which is less than 5, so round down. 50 629 = 50 600 (to 3 s.f.) (c) Look at the digit after the first two significant figures: 0.00 29 582. It’s 5, which is 5 or more, so round up. 0.002 958 2 = 0.0030 (to 2 s.f.) Notice that in Example 2(c), a 0 is included after the 3 to make it clear that the number is rounded to two significant figures. You shoul...
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