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3 Aims

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.


Author(s): The Open University

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Pressing onwards

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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1.1 Mathematics and you

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

Read
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Introduction

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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2.1 Introduction

In this section we shall define the complex number system as the set R × 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).

3.1 Expressing numbers in scientific notation

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

Place value10 0001000100101Author(s): The Open University

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2.5.1 Try some yourself

1 What are the following?

  • (a) 10

  • (b) 01

  • (c) 20

  • (d) 02


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2.4.1 Try some yourself

1 Write the following as a number to a single power:

  • (a) 26 ÷ 22

  • (b) 1010 ÷ 107

  • (c) 78 ÷ 74


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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

3.4.1 Try some yourself

1 Write down the coordinates of the point P on each of the graphs below and interpret these coordinates in terms of the labels on the axes.

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1.1.1 Try some yourself

1 On the plan of the bathroom in Example 1, what is the width of the window and
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1 Modelling static problems

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. Once you have completed the workbook and exercises return to this page and watch the video below, ‘The arch never sleeps’, which discusses a practical application of some of the ideas in workbook.

Click 'View document' to open the workbook (PDF, 0.8
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1.3 Designing alternative programmes and curricula

Assuming that both the content of mathematics and the processes need to be included in programmes and curricula, the problem becomes one of how a suitable curriculum can be structured. One possibility is to construct a very specific curriculum with clearly defined objectives for both content and processes separately, and possibly with suggested learning activities. However, content and process are two complementary ways of viewing the subject.

An alternative is to see the curriculum in
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Learning outcomes

After studying this unit you should:

  • be able to perform basic algebraic manipulation with complex numbers;

  • understand the geometric interpretation of complex numbers;

  • know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.


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