Exercise 1

A vector a has magnitude |a|Â =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.

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Author(s): The Open University

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.

Author(s): The Open University

This unit is an adapted extract from the Open University course Complex analysis (M337)

This unit is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that
Author(s): The Open University

Journals and articles written by academics or experts are an excellent source of information. Journals are usually published monthly or quarterly, and contain a selection of articles providing details of recent research. Often they will also contain reviews of relevant books. They are usually published more quickly than books, and so are often more up to date.

To access content of journals, most publishers require a subscription. There are, however, some journals which you can freely ac
Author(s): The Open University

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

• appreciate how chemical processes in the rest of the world affect the Arctic environment and the species inhabiting it;

• recognise the physical processes that determine atmosphere and oceanic flows in the Arctic;

• appreciate the scientific research process and the use of scientific evidence;

• use quantitative scientific evidence to examine the link between atmospheric carbon dioxide levels a
Author(s): The Open University

Few people agree that individuals should take the main responsibility for tackling environmental issues. For example, in a 2007 poll of over 2000 UK citizens, 70% agreed that the government should take a lead in combating climate change, even if it means using the law to change people's behaviour. However, over 60% disagreed that there was nothing they could do to avert climate change and over half agreed that they would do more if others did more too, although 40% thought that recycling was
Author(s): The Open University

Carbon dioxide (CO2) is used as the basis for the carbon footprint because it is by far the main contributor to the enhanced greenhouse effect from human activity (mainly burning fossil fuels, clearing forests and making cement). So, often only CO2 is counted in the carbon footprint. However, for a more complete measure of the carbon footprint the other human-generated greenhouse gases are converted into a CO2 equivalent (in kilograms or tonnes CO2e
Author(s): The Open University

The carbon footprint is the annual amount of greenhouse gas emissions, mainly carbon dioxide, that result from the activities of an individual or a group of people, especially their use of energy and transport and consumption of goods and services. It's measured as the mass, in kilograms or tonnes per year, either of carbon dioxide (CO2) emissions alone, or of the carbon dioxide equivalent (CO2e) effect of other greenhouse gas emissions.

Author(s): The Open University

State-of-the-art models are designed to simulate the workings of the climate system (in so far as this is currently understood), and include the â€˜internalâ€™ interactions that generate short-term natural variability in the real world. They provide modellers with a means of carrying out â€˜virtualâ€™ experiments on the climate system. In the present context, an important aim of these experiments is to identify the â€˜signalâ€™ of a human influence on climate, so studies typically involve â€˜
Author(s): The Open University

Figure 36 (again adapted from the TAR) takes your thoughts on Question 11 on a stage. It gives estimates of the cumulative effect since pre-industrial times of the various climate change agents, with the contributions expressed in terms of radiative forcing. Note that the figure also includes yet another device for communicating the IPCC's confidence in a particular finding â€“ an indication of the â€˜level of scientific understandingâ€™ that accompanies each estimate. This reflects the autho
Author(s): The Open University

The notion of a link between climatic conditions and the behaviour of plants and animals (e.g. the growth of trees or coral) and the composition of natural communities or ecosystems (the type of vegetation in a given area, say) is fundamental to the use of proxy data to reconstruct past climates. Some examples of biological responses to recent climate change were included in Box 9. Here we should be wary of jumping to conclusions. Such changes involve complex living systems that can respond i
Author(s): The Open University

The indicators collected in Table 4 have been observed to change over large regions of the Earth during the 20th century. According to the TAR, there is now a good level of confidence that what is being recorded is the result of long-term change rather than short-term natural fluctuations. As we noted earlier (Section 2.2.2), the most recent period of warming has been almost global in extent, but particularly marked at high latitudes. So are the changes in Table 4 consistent with rising tempe
Author(s): The Open University

Capra, F. (1996) The Web of Life. Used by permission of Doubleday, a division of Random House, Inc., and, in the UK, reprinted by permission of HarperCollins Publishers Ltd.
Capra, F. (2002) The Hidden Connections: Integrating the Biological, Cognitive, and Social Dimension of Life into a Science of Sustainability. Used by permission of Doubleday, a division of Random House, Inc., and, in the UK, reprin
Author(s): The Open University

There are two points which are central to this line of thinking. One, according to Wolf (2004), is that the whole process, as odd as it may sound, is about mutual exploitation. Outside firms do indeed exploit the poor by taking advantage of the profitable opportunities that a pool of cheap labour represents. But Indonesian or Chinese workers, for instance, could be said to exploit the incoming firms by extracting higher pay from them and taking advantage of opportunities that previousl
Author(s): The Open University

You have already glanced at Figure 1 and some of the worki
Author(s): The Open University

Many of the smaller branded goods on sale to consumers in Europe and North America â€“ the latest in clothing and footwear or the smart toys and electronic gadgets on offer â€“ are made in factory â€˜sweatshopsâ€™. Found in the backstreets of modern, Western cities, but more often than not a feature of the poorer parts of the world, factory sweatshops are an integral part of today's global economy. Increasingly, as you can see from Author(s): The Open University

Sweatshops and the exploitation of workers are often linked to the globalised production of â€˜big brandâ€™ labels. This unit examines how campaigners have successfully closed the distance between the brands and the sweatshops, while others argue that such production â€˜kick startsâ€™ economies into growth benefiting whole communities.

This unit is an adapted extract from the Open University course Author(s): The Open University

What are the possibilities for radical changes in our energy systems when viewed from a world perspective? There have been numerous studies of the various future options for the world's energy systems. One of the most recent and most comprehensive was produced in 1998 by the International Institute for Applied Systems Analysis (IIASA) and the World Energy Council (WEC), a version of which was published in 2000 as part of the United Nationsâ€™ World Energy Assessment (United Nations Dev
Author(s): The Open University

The UK's Royal Commission on Environmental Pollution produced (RCEP) its 22nd report Energy: the Changing Climate in June 2000. The Commission examined what changes would be needed in Britain's energy systems if, as suggested by the various reports of the Intergovernmental Panel on Climate Change (IPCC, 2001), it should prove necessary to reduce the country's emissions of greenhouse gases by about 60 per cent by 2050.

The Commission investigated the various possibilities very tho
Author(s): The Open University

The use of renewable energy usually involves environmental impacts of some kind, but these are normally lower than those of fossil or nuclear sources.

Approaches (a) and (b) are essentially â€˜supply-sideâ€™ measures â€“ applied at the supply end of the long chain that leads from primary energy production to useful energy consumption.

Author(s): The Open University