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1 Symmetry in two dimensions

In Section 1 we discuss intuitive ideas of symmetry for a two-dimensional figure, and define the set of symmetries of such a figure. We then view these symmetries as functions that combine under composition, and show that the resulting structure has properties known as closure, identity, inverses and associativity. We use these properties to define a group in Author(s): The Open University

5 Solutions to the exercises

Section 5 contains solutions to the exercises that appear throughout sections 1-4.

Click 'View document' below to open the solutions (13 pages, 500KB).

1 Sets

In Section 1 we discuss the idea of a set and describe some ways to define sets. We illustrate our discussion with sets of numbers and with geometrical sets of points in the plane. We also explain how to check whether two given sets are equal and whether one set is a subset of another. Finally, we introduce the set operations of union, intersection and difference.

Click 'View document' below to open Section 1 (16 pages, 389KB).

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4.4 Self-assessment questions and problems

SAQ 25

Find the distance between the numbers 2 − i and 1 + 3i.

Answer

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2.2 Defining the complex number system

In complex analysis we are concerned with functions whose domains and codomains are subsets of the set of complex numbers. As you probably know, this structure is obtained from the set R × R by defining suitable operations of addition and multiplication. This reveals immediately one important difference between real analysis and complex analysis: in real analysis we are concerned with sets of real numbers, in complex analysis we are concerned with sets of ordered pairs
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6 Curves from parameters

In Section 5 we show how functions may be used to sketch curves in the plane, even when these curves are not necessarily the graphs of functions.

Click 'View document' below to open Section 5 (8 pages, 151KB).

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 £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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3.1.1 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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2.5 The power zero

Now, what happens if you divide a number by itself? You should get the answer 1. For example

so 100 must equal 1. This must be true for any number (except for 0, because there is a problem in defining 00). So, for example, 70 = 1, 123
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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

All materials included in this unit are derived from content originated at the Open University.


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References

Hart, K., Kerslake, D., Brown, M., Ruddock, G., Kuchemann, D. and McCartney, M. (eds) (1981) Children's Understanding of Mathematics 11-16, London, John Murray.
McCague, W. (2003) 'A mathematical look at a medieval cathedral', Math Horizons, April, pp.11-15 and p.31. See also http://www.maa.org.
Phill
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1.2.3 Process

Mathematical processes are different from content in that they overarch the subject and are not thought of as hierarchical. A list of processes could contain:

  • problem-solving (including investigating);

  • mathematical modelling;

  • reasoning;

  • communicating;

  • making connections (including applying mathematics); and

  • using tools.

Each of the six processes liste
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1.4.5 M is for Method

Method is about the way in which a piece of information is produced. This is quite a complex area as different types of information are produced in different ways. These are a few suggestions to look out for:

Opinions – A lot of information is based on the opinion of individuals. They may or not be experts in their field (see P for Provenance) but the key message is to be clear that it is just an opinion and must be valued as such.

Research – You don’t have t
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4.4.1 Partnerships for sustainable consumption

Moderate NGOs, progressive businesses and government all have a stake in seeing roundtable partnerships come up with practical steps that can bring sustainability closer. One area that has attracted the attention of all these players is consumption. Directing or limiting consumption is politically difficult for even the NGOs to promote. Similarly, ‘voluntary simplicity’ of the sort lived at Findhorn eco-village (Author(s): The Open University

3.3 Green governance needs citizens

The term ‘good governance’ implies that ‘ordinary people’ will be involved in deciding what to do, trying to make it happen, and deciding whether it has happened (debate, implementation, monitoring). But what, in practical terms, might citizen involvement in the governance of an issue such as climate change mean? Citizen involvement in decisions and actions can mean anything from filling in a questionnaire to joining a demonstration to sitting on a committee. One helpful approach is A
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5.2 Lighter living costs and constraints

The costs of ‘light living’ actions need, of course, also to be considered. Some actions involve no cost or save money, for example, less flying, shopping or meat eating, or can even make money, such as letting out a spare room to increase household occupancy. Others are low cost with a rapid payback time; for example, replacing an incandescent light bulb with a low-energy compact fluorescent lamp (CFL) should pay back the new lamp's cost in lower electricity bills in about
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2.3 Contested science: a case study

For complex issues such as global climate change, there are many opportunities for scientists to take issue with the findings of their colleagues. They can disagree about the procedures for gathering data, the completeness or coverage of the data, how the data are analysed and interpreted, and then finally the conclusions. The assumptions that shape a particular piece of research and inform the kind of questions that will be asked can be no less contentious than the quality of the data gather
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Learning outcomes

Having studied this unit you should:

  • understand the physical basis of the natural greenhouse effect, including the meaning of the term radiative forcing;

  • know something of the way various human activities are increasing emmissions of the natural greenhouse gases, and are also contributing to sulphate aerosols in the troposphere;

  • be aware of the difficulties involved in the detection of any unusual global warming ‘signal’ above the ‘background
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1.4 Nature matters in terms of a critical systems literacy

The systems philosopher and social planner Werner Ulrich has long argued for a more ethically informed idea of systems. Before looking at Ulrich's ideas, however, it is worth returning to examine the relevance of the earlier Moore and Martell readings to this subject.

One of the hallmarks of systems thinking is a recognition of the limits of holism, relating to the problem of aesthetic framing expressed by Ronald Moore (2006, p. 263):

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1.3.8 Summary of section

  • During the 1970s and 1980s, countries such as Hong Kong, Singapore and Taiwan benefited from their low-cost advantages in the new global division of labour. Now, however, the gap between rich and poor nations is wider and competition in the world economy greater, prompting campaigning groups to argue that contemporary low-wage economies do not have the options for economic development that their predecessors had.

  • In the face of market fragment
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