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2.10.1 Mean and standard deviation for repeated measurements

In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the mean (or more strictly the ‘arithmetic mean’) For a set of measurements, this is de
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2.9 Repeated measurements

Scientists are always concerned with the reliability and precision of their data, and this is the prime reason for them to repeat measurements many times. Consider the photograph shown in Figure 6, which is the result of X-ray diffraction (see Box 5). To determine the atomic structure of the substance that produced this pattern, it would be necessary to measure the diameters of the fuzzy rings. At one time, this would probably have been done with an instrument called a travelling microscope;
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2.1 Preamble

Statistical information is a familiar aspect of modern life, which features routinely in, for example, news reports, sports commentaries and advertising. Scientists who have collected large amounts of data by either counting or measuring quantities also rely on statistical techniques to help them make sense of these data. Suppose you had information collected from, say, three thousand patients, all with the same medical condition but undergoing a variety of treatments. First you would need te
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Introduction

In this course we look at some different systems of numbers, and the rules for combining numbers in these systems. For each system we consider the question of which elements have additive and/or multiplicative inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.

In Section 1 we start by revising the notation used for the rational numbers and the real numbers, and we list their arithmetical pr
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Introduction

For many centuries, ancient Egypt was seen as the source of wisdom and knowledge, about mathematics as well as other things. There was a long classical Greek tradition to this effect, and in later centuries the indecipherability of the hieroglyphs did nothing to dispel this belief. But since the early nineteenth century, when the deciphering of the Rosetta Stone by Young and Champollion enabled rapid progress to be made in translating extant Egyptian texts, the picture has changed to reveal a
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Introduction

This course is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that this is very useful in understanding the arithmetic properties introduced in Section 1.

In Section 3 we discuss methods of finding nth roots of complex numbers and the solutions of simple
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Modelling heat transfer

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook.

Click 'View document' to open the workbook (PDF, 0.4 MB).

1.4 Real numbers and their properties

Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of real numbers, denoted by .

As with rational numbers, we can determine which of two real numbers is greater by comparing their decimals and noticing the first pair of corresponding digits
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1.2 Decimal representation of rational numbers

The decimal system enables us to represent all the natural numbers using only the ten integers which are called digits. We now remind you of the basic facts about the representation of rational numbers by decimals.


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Acknowledgements

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit:

The content ackno
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10 Conclusion

This unit has introduced you to some aspects of using a scientific or graphics calculator. However, in many ways, it has only scratched the surface. Hopefully your calculator will be your friend throughout your study of mathematics and beyond. Like any friend, you will get to know it better and appreciate its advantages as you become more familiar with it. Don't expect to know everything at the beginning. You may find the instruction booklet, or other help facility, a bit hard going to begin
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9 When to use the calculator

Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!<
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8.2.4 Accuracy

The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!)


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7.1 Introduction

As you have probably observed, there are many mathematical functions on your calculator, but most users only need to use a few of them regularly. This is an opportunity to be introduced to some of the more useful ones informally. Many functions are directly visible on the keys of the keyboard, but your calculator may have additional functions, e.g. on a MATH menu.

This section is designed to give you a whistle-stop introduction to some of the important functions on your calculato
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6.3 More short investigations

Here are two short investigations involving large numbers for you to try. Please do not turn to the comments on these exercises until you have made some notes and had a go yourself.

Exercise 13: Where did I come from?

Family tree
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7 Solutions to the exercises

Section 6 contains solutions to the exercises that appear throughout sections 1-5.

Click 'View document' below to open the solutions (13 pages, 232KB).

2.2 Power notation

The notation in Example 6 is called power notation, or index notation. In a number such as 25, the 5 is called the power, or index, of the number.

The squares are particular examples of powers: 92, for example, can be thought of as ‘9 to the power 2’.

For most numbers, calcul
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1.3 Square roots

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

Example 3

2.1 Tables

Experiments or surveys usually generate a lot of information from which it is possible to draw conclusions. Such information is called data. Data are often presented in newspapers or books.

One convenient way to present data is in a table. For instance, the nutrition panel on the back of a food packet:

Nutrition Information

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1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
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