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An introduction to complex numbers
This unit looks at complex numbers. You will learn how they are defined, examine their geometric representation and then move on to looking at the methods for finding the nth roots of complex numbers and the solutions to simple polynominal equations. First published on Mon, 13 Jun 2011 as Author(s): Creator not set

1.4 Real numbers and their properties

Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of real numbers, denoted by .

As with rational numbers, we can determine which of two real numbers is greater by comparing their decimals and noticing the first pair of corresponding digits
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1.1 Rational numbers

The set of natural numbers is the set of integers is and the set of rational numbers is Author(s): The Open University

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8.2.2 The screen

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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7.2 Square roots

Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively).

Now find
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4 Open Mark quiz

Now try the quiz  and see if there are any areas you need to work on.


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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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Learning outcomes

After studying this unit you should:

  • know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;

  • understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;

  • know ways to operate on and combine vectors.


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1.4 An overview of the global energy budget

Figure 12 incorporates the additional factors considered in Section 1.3, including the non-radiative energy transfers across the surface-air boundary (green arrow). Essentially a more detailed version of Figure 7, this figure gives quantified estimates of the globally averaged energy budget for the whole Earth-atmosphere system, and its component parts. Question 3 should help you to find your way around Figure 12, and to draw together many of the key points developed so far in this chapter. M
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2.3 Citizens in conversation with nature and experts

Before leaving office in 2008, Sir David King (the ex-Chief Scientific Advisor to the UK Government) introduced an ethical code for scientists. This drew particularly on his experience in working across the scientific–political divide on issues of climate change. The code comprises three attributes of scientific endeavour: rigour, representation and responsibility (Author(s): The Open University

2.2 Environmental pragmatism: positioning expert support

I believe that the principal task for an environmental pragmatism is not to reengage the … debates in environmental ethics but rather to impress upon environmental philosophers the need to take up the largely empirical question of what morally motivates humans to change their attitudes, behaviours, and policy preferences toward those more supportive of long-term environmental sustainability.

(Light, 2002, p. 446)


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References

Anon (2002) ‘The Windicator’, Windpower Monthly, January, p. 50.
Blake, William (1994) The Marriage of Heaven and Hell, Dover Publications.
Boyle, G. (1966, 2003) Renewable Energy, Oxford, Oxford University Press in association with the Open University.
BP (2002) BP Statistical Review of World Energy [on
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2.2.2 Environmental economics and green consumerism

In economic terms, green consumerism is typically expressed using measures based on the willingness to pay (WTP) principle. As mentioned above, this takes two main forms: eco-taxation, in which environmental costs are estimated and added to the price of commodities (e.g. vehicles with high carbon emissions); and eco-labelling, in which products are labelled with relevant environmental information, such as is now required by the food industry and governments in many industrialised count
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2.2.1 Environmental economics

Environmental economics emerged as a sub-discipline in the 1960s, following a tradition that began in the early twentieth century with ‘agricultural’ economics and continued in the 1950s with ‘resource’ economics. In each case, natural resources are treated as environmental assets in the same way as other resource inputs, using the classical mainstream supply and demand economic models. David Pearce, who at one stage was at the forefront of environmental economics and was an ac
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2.2 Economic valuation: towards ecological economics

The blue whale could have supplied indefinitely a sustainable yield of 6000 individuals a year.

This is one of the earliest references to sustainability in the literature, taken from the 1971 edition of the science journal Nature (cited in Senge et al., 2006, p. 45). Here, the blue whale is given instrumental value – a means of measuring not the survival of the blue whale for its intrinsic v
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1.3 The influence of environmental ethics: value and care

Religious ethics can play a significant role in shaping appropriate narratives that provide for a lived ethic – that is, the obligations and entitlements associated with human relationships with Nature that embody what’s good and what’s right. But how might other ethical traditions help towards developing a lived ethic? To what extent has the emergence of environmental ethics since the 1970s influenced a lived ethic commensurate with developing care for the environment?

Andrew Lig
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1.2 The influence of narratives and spiritual traditions

In his 1974 publication Man’s Responsibility for Nature, John Passmore – an Australian philosopher who pioneered a concern for developing a change of attitude towards the environment – argues from an explicitly anthropocentric perspective. He suggests that the special ties between parents and children provide the basis for continual development of obligations amongst humans, which can then translate into a more responsible engagement with the environment.

People normally ca
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6 Further reading

The models being used in research take such simple energy flows and increase the ‘granularity’ of the components used, to build complex time sequences.

You may like to see Information Sheet 8 at the website of the Climatic Research Unit for a summary of how these have developed.

Click on 'View document' to see charts mentioned in the activity below

2 A 4.6 billion-year history

Climate change is a natural process of warming and cooling that has occurred all through the Earth's history. Throughout geological time there have been ‘hot-house’ periods and ice ages. In order to understand the current situation, it is necessary to have some sense of context and perspective, from historical and geological time-scales. The document below shows a chart showing a generalised temperature history of the Earth.

Click on 'View document' to see the chart

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