2.1 Introduction In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis.
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 the link below to open Section 5 (8 pages, 151KB).
4 New graphs from old In Section 3 we consider how to sketch the graphs of more complicated functions, sometimes involving trigonometric functions. We look at graphs which are sums, quotients and composites of different functions, and at those which are defined by a different rule for different values of x. Click the link below to open Section 3 (7 pages, 133KB).
3 Graph sketching In Section 2 we describe how the graphs of polynomial and rational functions may be sketched by analysing their behaviour – for example, by using techniques of calculus. We assume that you are familiar with basic calculus and that its use is valid. In particular, we assume that the graphs of the functions under consideration consist of smooth curves. Click the link below to open Section 2 (16 pages, 200KB).
3.3.1 Increasing by a percentage Our everyday experience of percentages includes percentage increases (like VAT at For example, £8 plus
%, or a service charge of 15%) and percentage decreases (such as a discount of 15%).
1.3.1 Try some yourself A local supermarket sells a popular breakfast cereal in a ‘Large Pack’ and ‘New Extra Large Pack’. They are both being sold at ‘knock down’ prices. The large pack contains 450 g of cereal priced at  Studying mammals: Food for thought Understanding the environment: Problems with the way we think 'Land grab': an environmental issue? Migration Exploring philosophy: faking nature Animals at the extremes: The desert environment Water in the UK An introduction to sustainable energy 1.4.1 Dealing meaningfully with holism Ulrich's primary observation is quite straightforward. Any system as a human construct is unable to capture the total complexity of interrelationships and interdependencies that make up the real world. This idea resonates with the paradox of framing referred to by Moore. It also resonates with Ilan Kapoor's reference to the work of Slavoj Žižek, quoted earlier: ‘Reality is what we (mistakenly) take to be wholeness or harmony, while the Real denotes the impossibility of wholeness’ (Kapoo 1.1 Framing nature using language tools By framing, I mean the structures and pre-assumptions that we consciously or unconsciously apply to a situation in order to make sense of it. So are there any differences between the way in which we frame nature in caring for environment and the way in which we frame it to provide accountability? What significance might this have, and what tools might be used to bridge the responsibilities of caring and accountability? Caring for environment makes manifest the informal aspects of 1.4.3 It's all down to connections For Iris Marion Young, the responsibility of those in North America and Europe towards distant others does indeed rest with their connections to injustices elsewhere, but it would be a mistake to stretch this line of reasoning too far. Although these connections, whether as a consumer, boardroom executive or shop manager, can establish a line of responsibility, as was claimed in Section 3.1, for Young this is only the starting point and not the end point of our involvement. We do not have to 2.2.3 Ecological economics Ecological economics, which formally came to prominence in the mid-1980s, represents a departure from reliance on the use of mainstream economic modelling. Instead, it branches out to actively engage with and incorporate the ethical, social and behavioural dimensions of environmental issues. In short, ecological economics attempts to provide an interdisciplinary approach to environmental issues, whereas environmental economics maintains the primacy of economic modelling. Mark Sag 2.4 Summarising conversation as what matters Brian Wynne suggests that fundamental dichotomies associated with environmental matters underpin modern society – society versus nature, the social versus the natural, social knowledge versus natural knowledge, expert knowledge versus lay knowledge (1996, p. 45). The metaphor of conversation helps to move us beyond these dichotomous constructs and allows us to focus more on the integral relationships enmeshed in nature matters, relationships that I would argue are central to environmental r 1.2 Connecting human and non-human nature Environmental responsibility – caring and generating accountability – requires interaction between human and non-human nature. For example, from a caring perspective what matters in climate change might constitute, say, the continued existence and protection of an arctic wilderness (Figure 3). But this necessarily involves a conn
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Who were our ancestors? How are apes and humans related? And where does the extinct Homo erectus fit into the puzzle? In this free course, Studying mammals: Food for thought, we will examine culture, tool use and social structure in both apes and humans to gain an understanding of where we come from and why we behave as we do. This is the tenth course in the Studying mammals series.
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There is increasing recognition that the reductionist mindset that is currently dominating society, rooted in unlimited economic growth unperceptive to its social and environmental impact, cannot resolve the converging environmental, social and economic crises we now face. The primary aim of this free course, Understanding the environment: Problems with the way we think, is to encourage the shift away from reductionist and human centred thinking towards a holistic and ecological worldview.
This free course, 'Land grab': an environmental issue?, explores how environmental problems are entangled with economic and political issues and offers tools for making sense of the complexity that results. The case of land grab illustrates how everyday issues such as food prices are caught up in connections that link different places, different people and their livelihoods across the globe; connections that are brought to life in the course through rich audio-visual material and interactive act
Migration is a free course looking at the migrations of animals, with special reference to birds, and also introducing the themes of movement, selection and homeostasis.
First published on Thu, 07 Sep 2017 as Migration. To find out more visit The Open University's Author(s):
Commercial exploitation of nature, such as mining, fracking, or generating hydro-electric power, often damages the way the natural environment looks. What if the environment could be restored to exactly how it looked before? Would that mean that no damage had been done, that the natural environment was as valuable as it had been before the commercial exploitation? This free course, Exploring philosophy: faking nature, examines ‘the restoration thesis’, and provides an insight into philosophi
Animal life has adapted to survive in the most unlikely and inhospitable habitats. This free course, Animals at the extremes: The desert environment, looks at the surprisingly diverse desert climates throughout the world and mammals, birds, lizards and amphibians that survive there. It splits these animals into three groups according to their strategy for survival: evaders, evaporators and endurers, then discusses how these strategies work on a biochemical and physiological level.
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Water is arguably the most important physical resource as it is the one that is essential to human survival. Understanding the global water cycle and how we use water is essential to planning a sustainable source of water for the future. Globally, there are many areas that do not have enough water to support the current population adequately. Decisions will have to be made on the best way to use water in a world where there is climate change. This free course looks at Water in the UK where water
The search for sustainable energy will dominate the twenty-first century. This free course, An introduction to sustainable energy, provides an introductory overview of the present energy systems and takes a brief look at where the world may find energy in the future - cleaner use of fossil fuels or renewable energy sources?
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