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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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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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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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Introduction

This unit is an adapted extract from the course Mathematical methods and models (MST209)

This unit lays the foundations of Newtonian mechanics and in particular the procedure for solving dynamics problems. The prerequisite skills needed for this unit are the ability to solve first- and second-order differential equations, a knowledge of vectors, and an understanding of the concept of a
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Introduction

This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.

This is an adapted extract from the Open University course Mathematical methods and models (MST209)
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Acknowledgements

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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7.2 Answers

Question 1

Although most people immediately think of economic globalisation, Section 1 shows how political, social/cultural and ecological globalisation are also significant in the context of global environmental change.

Question 2

The advocates of ‘business learns’ are optimistic about the global free-market civilisation they believe they are building, but they also believe business needs to heed environmental and social concerns for its own sake. ‘Radic
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5 Conclusion – new ways of looking at the world

There is a variety of new approaches or terms that are interlinked, and have been prominent throughout this book. All of them have played a part in this book's journey through the scientific, political, philosophical and social implications of climate change.

Governance of climate change is about: decision making under uncertainty; understanding and representing vulnerability even when vulnerabilities are difficult to assess or unknowable; and making every aspect of human
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4.2.3 Business needs sustainability

The second argument is more profound: long-term profitability, and the existence of business itself, is threatened if companies can't transform themselves. This assumes that although the costs of environmental and social impacts can be ignored for a period, in the context of globalisation of environmental, social and political processes, they will come back to haunt businesses, and ultimately threaten their survival. There are several communications and management tools that have been develop
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3.2.2 Good green governance in five easy steps

It would be a serious error to imagine that ‘government’ has evaporated: it still shapes many aspects of our lives from beginning to end (welfare, taxation, transport and, of course, the recording of births and deaths). Governments are the central negotiators of environmental-change policies at international level, and of their implementation at national and local level. Nevertheless, for many areas of life, governance is undeniably a better description both of new processes that are alre
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2.2 Vibrant civil societies and a networked globe

One thing is common to all three attempts to find a route to a sustainable economy and society: in different ways they all assume that people will get actively involved in making human societies more sustainable. But this transformation will not take place through the corporate world's promises, by local protectionism, a return to ‘strong states’ or the publication of numerous indicators. Any of the three positions outlined above requires interactions and feedbacks created by a vibrant
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2.2 Records of the Earth's temperature

To put the temperature records reported by the IPCC in context, we start with a longer-term geological perspective on the Earth's GMST.


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1.7 Summary

  1. Figure 12 summarises the ways in which the Earth's surface and atmosphere gain and lose energy. The main points are as follows:

     

    • A proportion (the planetary albedo) of the incoming shortwave radiation from the Sun is reflected (or scattered) directly back to space, mainly by clouds and the Earth's surface (especially snow and ice cover), but also by aerosols (e.g. dust, salt particles, etc.). Most of the re
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1.1 Introduction

At the beginning of the 21st century, terms such as the ‘greenhouse effect’, ‘greenhouse gases’ and ‘greenhouse warming’ are printed or spoken thousands of times a week in the context of climate change caused by human activities. This section is designed to consolidate your understanding of the basic science behind these terms, and then to review what is known about the human impact on the composition of the atmosphere since the dawn of the industrial age, commonly put (in this co
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Learning outcomes

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

  • understand why systems thinking might be useful and know something about how it can be applied in the context of environmental responsibility;

  • describe the significance of environmental pragmatism and cognitive justice as tools for supporting environmental policy and action.


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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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4.5 Sustainability of renewable energy sources

Renewable energy sources are generally sustainable in the sense that they cannot ‘run out’ – although, as noted above, both biomass and geothermal energy need wise management if they are to be used sustainably. For all of the other renewables, almost any realistic rate of exploitation by humans would be unlikely to approach their rate of replenishment by nature, though of course the use of all renewables is subject to various practical constraints.

Renewable energies are also rela
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2.4.1 The emergence of ecology

To help you gain a better perspective on the general progress of scientific development, Table 4 provides a schematic potted history of some of the major ideas coming from science that have sought to guide our actions. It includes both social and natural sciences, though the former really only became distinct ‘science
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2.1 Accounting for the consequences of environmental harm

The ethical tradition of consequentialism informs not only what matters from the perspective of caring for the environment, but also what matters from the perspective of accountability towards it. In eighteenth-century Europe, the actual environmental consequences of rapid economic development, triggered by the industrial revolutions taking place at that time, prompted an increasing concern for accountability. The most evident expression of this came with ideas of sustainable development
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