By the end of this unit you should be able to:

Section 1 â€“ Real numbers

• explain the relationship between rational numbers and recurring decimals;

• explain the term irrational number and describe how such a number can be represented on a number line;

• find a rational and an irrational number between any two distinct real numbers;

Section 2 â€“ Inequalities

• solve inequalities by re
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You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes.

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Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big?

First express 1â€‰000â€‰000â€‰000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109
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Understanding how your calculator displays and handles very large and very small numbers is important if you are to interpret the results of calculations correctly. This section focuses on a way of representing numbers known as scientific notation.

Before you start put your calculator into the float mode, so it will display up to about 10 digits and return to the home screen ready to do some calculations.

What answer would you expect if you square 20 million? How man
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You are now going to use the four operation keys (on the bottom right-hand side of the TI-84 keyboard): , Author(s): The Open University

The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective.

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## Activity 15

1. Work through Sections 1.6 and 1.7 of the Calculator Book, using the method suggested above of glancing ahead-pressing on-glancing back, if you find it useful.

2. A num
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## Example 3

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Many people's ideas about what mathematics actually is are based upon their early experiences at school. The first two activities aim to help you recall formative experiences from childhood.

## Activity 1 Carl Jung's school days

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This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.

In order to complete this unit you will need
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4 Proofs in group theory

In Section 4 we prove that some of the properties of the groups appearing earlier in the unit are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse.

Click 'View document' below to open Section 4 (9 pages, 237KB).

Learning outcomes

By the end of this unit you should be able to:

• explain what is meant by a symmetry of a plane figure;

• specify symmetries of a bounded plane figure as rotations or reflections;

• describe some properties of the set of symmetries of a plane figure;

• explain the difference between direct and indirect symmetries;

• use a two-line symbol to represent a symmetry;

• describe geometrically th
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4 Two identities

Section 4 introduces some important mathematical theorems.

Click 'View document' below to open Section 4 (7 pages, 237KB).

View document<
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Learning outcomes

By the end of this unit you should be able to:

• Section 1: Sets

• use set notation;

• determine whether two given sets are equal and whether one given set is a subset of another;

• find the union, intersection and difference of two given sets.

• Section 2: Functions

• determine the image of a given function;

• determine whether a given function is one-one
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4.3 Section summary

The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view of analysis the importance of this is that we can talk of the closeness of two complex numbers. We can then define the limit of a sequence of complex numbers in a way which is almost identical to the definition of the limit of a real sequence. Another analogue of real analysis arises
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2.3 Section summary

In this section we have seen that the complex number system is the set RÂ Ã—Â R together with the operationsÂ +Â andÂ Ã—Â defined by

From this, one can justify the performance of ordinary algebraic operations on expressions of the form aÂ +Â ib
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2.1 Introduction

In this section we shall define the complex number system as the set RÂ Ã—Â R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis.

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4 New graphs from old

In Section 3 we consider how to sketch the graphs of more complicated functions, sometimes involving trigonometric functions. We look at graphs which are sums, quotients and composites of different functions, and at those which are defined by a different rule for different values of x.

Click 'View document' below to open Section 3 (7 pages, 133KB).

3.2 Using scientific notation

Scientific notation can be very useful when estimating the answers to calculations involving very large and/or small decimal numbers.

## Example 9

A lottery winner won Â£7851 000. He put the money straight into a deposit account which earns 7.5% interest per annum (i.e. each year). If he wanted to
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3.1 Expressing numbers in scientific notation

Earlier you looked at place values for numbers, and why they were called powers of ten.

 Place value 10 000 1000 100 10 1 Author(s): The Open UniversityLicense informationRelated contentExcept for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share Copyright 2009 University of Nottingham