Fundamental Engineering Mathematics (eBook)

A Student-Friendly Workbook

N Challis, H Gretton (Autoren)

eBook Download: PDF | EPUB

2008 | 1. Auflage
296 Seiten
Elsevier Science (Verlag)
978-0-85709-939-6 (ISBN)

Infos

This student friendly workbook addresses mathematical topics using SONG - a combination of Symbolic, Oral, Numerical and Graphical approaches. The text helps to develop key skills, communication both written and oral, the use of information technology, problem solving and mathematical modelling. The overall structure aims to help students take responsibility for their own learning, by emphasizing the use of self-assessment, thereby enabling them to become critical, reflective and continuing learners - an essential skill in this fast-changing world.The material in this book has been successfully used by the authors over many years of teaching the subject at Sheffield Hallam University. Their SONG approach is somewhat broader than the traditionally symbolic based approach and readers will find it more in the same vein as the Calculus Reform movement in the USA. - Addresses mathematical topics using SONG - a combination of Symbolic, Oral, Numerical and Graphical approaches - Helps to develop key skills, communication both written and oral, the use of information technology, problem solving and mathematical modelling - Encourages students to take responsibility for their own learning by emphasizing the use of self-assessment

Neil Challis was born in Cambridge, UK. He studied mathematics at the University of Bristol and subsequently worked for some years as a mathematician in the British Gas Engineering Research Station at Killingworth. Since 1977, he has worked in the Mathematics Group, Sheffield Hallam University, UK and is currently head of that group. He obtained a PhD in mathematics from the University of Sheffield in 1988 and has taught mathematics to a wide variety of students, across the spectrum from first year engineers and other non-mathematicians who need access to mathematical ideas, techniques and thinking, to final year single honours mathematics students.

Textauszug

1

Numbers, Graphics and Algebra

“Can you do addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know,” said Alice. “I lost count.” Lewis Carroll: Through the Looking Glass

1.1 Numbers, Graphics and Algebra

The purpose of these first few chapters is to get you to look with fresh eyes at some basic mathematical ideas that you might think you already know all about. You will see how technology can help you to do that and can help to develop a deeper understanding of those ideas. You need this understanding and “feel” for what numbers and pictures can tell you about the world. You will almost certainly see “doing maths” differently from the way you saw it earlier in your life.

This chapter does not give a complete treatment for beginners in its topics. It provides a fresh look at, and reminder of, a mixture of ideas that you will find useful. You will think again about numbers and how they behave and how you can very simply represent them by letters – the idea of algebra. You will revisit simple geometry, as well as the ideas of trigonometry, describing the links between distances and angles. You will also develop ideas about the strong links between numbers, algebra and pictures – the pictures are what bring the numbers to life.

As you work through this chapter, and those which follow, remember that mathematics is a “doing”, not a “watching” activity, so make sure you keep on doing the exercises and activities as you go.

1.2 WHAT NUMBERS ARE

First you need to remind yourself of the basic terminology of numbers – bearing in mind that in mathematics some words have a precise meaning which is more specialised than in standard English.

Natural numbers are the positive whole numbers 1, 2, 3 … (0 may be included).

Integers are all whole (including negative) numbers … -2, -1, 0, 1, 2, 3 … .

Rational numbers are those which can be written as fractions in the form b where a and b are integers.

Some important and useful numbers cannot be written in rational form − e.g. the special numbers π ≈ 3.14159‥, e ≈ 2.71828‥ and ≈1.41421‥, which you will meet later, if you have not already done so. Such numbers are called irrational and occur with infinitely many non-repeating decimal places.

When the rational and irrational numbers are combined, they are called real numbers and these are what you will normally think of as decimal numbers.

It is useful to think of numbers graphically being represented by their position on a horizontal straight line, with distance away from a fixed point О governed by their value, and with negative numbers to the left and positive to the right (see Figure 1.1).

Figure 1.1 The number line

1.3 HOW NUMBERS (AND LETTERS) BEHAVE

You will be familiar with the arithmetic operations – add, subtract, multiply, divide, exponentiation (raise to a power), and the use of brackets. You should of course be used to using your calculator (graphic or otherwise) to do such things, but you will find that even with technology, if we do not all agree to use the same rules and language, then you can have serious problems. You must communicate properly, not only with other people, but also with machines.

First let us ask you a question in words – answer it without using your calculator: “What is one plus two times three?” DO IT NOW!

Some of you may get the answer 7, and others may get 9. Can you explain why two answers are possible? Check it on your calculator. You see that we need to agree the order of operations: if you do the addition first, you get 9; doing the multiplication first gives 7.

Thus, in this section, we give you the chance to remind yourself of the rules. We must all agree to use BODMAS, which some remember as Please Excuse My Dear Aunt Sally, to tell us the order in which to carry out operations:

BODMAS (Brackets/Order/Division/Multiplication/Addition/Substraction)

PEMDAS (Please Excuse My Dear Aunt Sally -Parentheses/Exponents/Multiplication/Division/Addition/Subtraction).

This means that brackets can override any other order of operation by forcing calculation of quantities in brackets first; then powers are done; then multiply and divide at the same level of priority working from left to right, and, finally, add and subtract working from left to right.

DO THIS NOW! 1.1

Investigate the order in which your calculator does numerical operations, by looking at the following expressions (note we use * here for “multiply” and / for “divide”, just as many technologies do). Recall two negative quantities multiplied together make a positive. Set your calculator accuracy (number of decimal places) to a sensible level before you start. First, predict without technology which of the expressions in column A is equal to the corresponding expression in column B. Look particularly at the effect of brackets. Then use your calculator to check whether you are right, explaining any discrepancies.

A

B

1 + 2

2 + 1

1 – 2

2 – 1

(1 + 2) + 3

1 + (2 + 3)

(1 – 2) – 3

1 – (2 – 3)

3 * 4

4 * 3

3 /4

4 /3

7 + 3*2

(7 + 3)*2

− 3 + 4

−(3 + 4)

6/3 + 4

6/(3 + 4)

(3/4)/5

3/(4/5)

Notice if you have a graphic calculator, it probably has two different keys. One will be the unary, or leading minus, often made with three pixels, as in (-1). The other is the take-away minus sign – which often has five pixels. See Figure 1.2 to see the unary minus in action. Sort this out on your personal technology NOW.

Figure 1.2 The unary minus in action

DO THIS NOW! 1.2

Predict by hand the value your calculator will give for the following expression, and then check by doing it on your calculator, explaining, step by step, what your calculator has done to evaluate this and what rule, as well as BODMAS, it is using:2*3*4/5/6/7 + 8 – 6 = ?

1.4 FRACTIONS, DECIMALS AND SCIENTIFIC NOTATION

Although you probably studied fractions years ago, we find that many people have forgotten the rules, or worse still, think they remember them but do not. It is true that you can do fractional number calculations on your calculator, and get your answer as either a fraction or a decimal, and that you can deal with algebraic fractions using a CAS (Computer Algebra System) such as Derive. However, it is part of getting a feel for what is happening, to be able to do some simple manipulations with algebraic expressions involving fractions, so here is an opportunity to do just that.

First, here is a quick reminder of the rules:

To add or subtract two fractions, put them over a common denominator (bottom) so that you are adding or subtracting like with like.

Example 1.4.1 2+23=36+46=76and23−12=46−36=16

To multiply two fractions, multiply the numerators (tops) and multiply the denominators (bottoms). To divide two fractions, just turn the second one upside down and multiply!

Example 1.4.2 2×23=1×22×3=26=13and12÷23=12×32=1×32×2=34

Note you can simplify a fraction by multiplying or dividing top and bottom by the same number:

Example 1.4.3 9=13, dividing top and bottom by 3. This is commonly called

cancelling. However remember that you cannot simply add or subtract numbers on top and bottom – a quick example should convince you of that:

Example 1.4.4 +19+1=510=12 is not the same as 9. That is to say you cannot cancel the + 1 terms if they are added (or subtracted).

DO THIS NOW! 1.3

Evaluate the following expressions by hand, giving your answer as a single fraction. In each case, check with your calculator, considering carefully how you must type things in to take account of My...

Produkt-Details

• Erscheint lt. Verlag: 1.1.2008
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-85709-939-6 / 0857099396
• ISBN-13: 978-0-85709-939-6 / 9780857099396