In this section we shall define the complex number system as the set RÂ Ã—Â 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.

Author(s): The Open University

Section 6 contains solutions to the exercises that appear throughout sections 1-5.

Click the link below to open the solutions (13 pages, 232KB).

Section 6

Author(s): The Open University

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).

Section 5

Author(s): The Open University

Discount can be calculated in the same way as an increase by a percentage. For example, Â£8 with 15% discount means you actually pay

Â Â Â£8 less (15% of Â£8)

Â Â 15% of 8 = Ã— 8 =
Author(s): The Open University

Fractions and decimals can also be converted to percentages, by multiplying by 100%.

So, for example, 0.17, 0.3 and can be expressed as percentages as follows:

Â Â 0.17 Ã— 100% = 17%;

<
Author(s): The Open University

## Activity 3

A local supermarket sells a popular breakfast cereal in a â€˜Large Packâ€™ and â€˜New Extra Large Packâ€™. They are both being sold at â€˜knock downâ€™ prices. The large pack contains 450 g of cereal priced at Â
Author(s): The Open University

Ratios crop up often in official statistics. The government wants the teacherâ€“pupil ratio in schools to be increased to one teacher to thirty pupils or less. The birth rate has fallen: the ratio of children to women of child bearing age has gone down. It used to be 2.4 to 1, and now it is 1.9 to 1. Predictions for the ratio of working adults to retired adults is disturbing. Predictions are, that by 2030 the ratio will be two working adults to every retired person, instead of three to one no
Author(s): The Open University

The content acknowledged below is Proprietary (see and conditions made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence) and used under licence.

Course image: rod
Author(s): The Open University

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 listed here repre
Author(s): The Open University

World in transition: Managing Resources
Do you take your access to water for granted? The Peruvian and Tanzanian communities featured in this album certainly donâ€™t. This album examines how development agencies can empower communities to help themselves by introducing simple technologies, and facilitate the sharing of ideas through education. In the Andean mountains, scarce supplies of water and agricultural challenges give rise to conflict; but the changes engineered by development agencies can start to show a way out of poverty. Me
Author(s): The OpenLearn team

Understanding the environment: Learning and communication
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: Learning and communication, is to encourage the shift away from reductionist and human centred thinking towards a holistic and ecological worldview.
Author(s): Creator not set

Introducing the environment: Ecology and ecosystems
What is ecology and why is it important to our understanding of the world around us? This free course, Introducing the environment: Ecology and ecosystems, looks at how we can study ecosystems to explore the effect that humans are having on the environment. First published on Fri, 18 Mar 2016 as Author(s): Creator not set

Climate change
Climate change is a key issue on today's social and political agenda. This free course explores the basic science that underpins climate change and global warming. First published on Tue, 30 Oct 2018 as Climate change. To find out more visit The Open University's Author(s): Creator not set

Effects of pollutants on the aquatic environment
Effects of pollutants on the aquatic environment is a free course. It begins with an introduction to water and goes on to briefly outline the major sources of water pollution (these being sewage works, manufacturing and industrial plants, the farming and animal husbandry sectors, landfill sites and urban surface water run-off). It considers the major water pollutants and describes the effects they have on water. Author(s): Creator not set

Edwards, P. N. and Schneider, S. H. (2001) Self-governance and peer review in science-for-policy: the case of the IPCC Second Assessment Report, in Miller, C. and Edwards, P. N. (eds) Changing the Atmosphere: Expert Knowledge and Environmental Governance. Cambridge, MA: MIT Press. Available from: http://stephenschneider.stanford.edu/Mediarology/Mediarology.html (accessed 10 May 2007).
IPCC (2000) Land Use, L
Author(s): The Open University

2.6.2 The role of modelling studies

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 'internal' interactions that generate short-term natural variability in the real world. They provide modellers with a means of carrying out 'virtual' experiments on the climate system. In the present context, an important aim of these experiments is to identify the 'signal' of a human influence on climate, so studies typically involve 'feeding' into
Author(s): The Open University

2.1 Preamble

Here are some quotes from the 'Summary for Policymakers' (SPM) included in the report from the scientific working group in the IPCC TAR (IPCC, 2001a):

• The Earth's climate system has demonstrably changed on both global and regional scales since the pre-industrial era, with some of these changes attributable to human activities.

• Globally, it is very likely that the 1990s was the warmest decade and 1998 the warmest year in the instrumental
Author(s): The Open University

1.2 A framing paradox: experiencing nature with cognitive tools

Whilst language tools are helpful in conveying meaning in conversation amongst humans, establishing what matters in â€˜conversationâ€™ between human and non-human nature, or amongst non-human living entities, requires different cognitive tools. Cognition refers to the way in which external information from the environment is processed. As sentient beings, humans and some other animals are able to experience wellbeing and suffering. In the next reading, Ronald Moore examines how we engage with
Author(s): The Open University

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
Author(s): The Open University

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## Study another free course

There are more thanÂ 800 coursesÂ on OpenLearnÂ for you t
Author(s): The Open University