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).

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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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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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Acknowledgements

Audio Materials

These extracts are from M208 © 2006 The Open University.

 

All material contained within this unit originated at The Open University.


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2 Real functions

In Section 1 we formally define real functions and describe how they may arise when we try to solve equations. We remind you of some basic real functions and their graphs, and describe how some of the properties of these functions are featured in their graphs.

Click 'View document' below to open Section 1 (12 pages, 1.8MB).

Introduction

Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply ‘joining up the dots’ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t
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Acknowledgements

All written material contained within this unit originated at the Open University.

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1. Join the 200,000 students currently studyi
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3.2.1 Try some yourself

1 Use the method outlined in Example 9 to estimate each of the following, and then use yo
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3.1.1 Try some yourself

1 Express each of the following numbers in scientific notation.

  • (a) Light travels 9460 700 000 000 km in a year.

  • (b) The average distance from the centre of the Earth to the centre o
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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.

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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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1.6.3 Mailing lists and newsgroups

Mailing or discussion lists are email-based discussion groups. When you send an email to a mailing list address, it is sent automatically to all the other members of the list.

The majority of academic-related mailing lists in the UK are maintained by Jiscmail. You will find details of joining these mailing lists on the Jiscmail website. Mailing lists are useful for getting in touch with like-minded colleagues. They are also handy for keeping up to date with current thinking and research
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6.4 International retributive justice

A further difference between communitarians and cosmopolitans arises over the question of retributive justice. Communitarians think that it is the responsibility of each state to uphold justice. Collectively, states can pursue international justice through the auspices of the UN, and are answerable to each other, to public opinion and to NGOs. However, there is no basis for claims to universal jurisdiction, and to deal with matters not found in specific states (such as piracy), or that cross
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5.5 Feminist critiques of international rights

The second source of criticisms that we would like to explore comes from feminist critiques. Some feminists argue that the universal notion of rights makes invisible the special problems faced by women as a group, and that, thereby, specific articles of the various human rights declarations and conventions reinforce traditional gender roles in the family and the workplace. This criticism comes in at least two forms.

The first is that rights for women (as for other disadvantaged groups)
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2.1 Background to the idea of international rights

The UN Charter and the Declaration form part of a post-Second World War international settlement which established, on the one side, the formal legitimating ideology of the international system, national self-determination and sovereign equality and, on the other, the ideology of universal human rights. The appeal of this set of claims was the hope that different peoples could live together in peace and security. It was an attempt to accommodate difference (through the idea of national self-d
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5 Conclusion

The issue of climate change draws attention to the power of human activity to transform the planet in its entirety, and it is brought into sharp focus by the predicament of low-lying islands like Tuvalu. As we have seen in this unit, the issue of rising sea level and other potential impacts of changing global climate also point to the transformations in the physical world that occur even without human influence. Oceanic islands provide a particularly cogent reminder that the living things wit
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