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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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2.3.1 Try some yourself

1 Write the following as one number to a single power:

  • (a) 23 × 24

  • (b) 32 × 34

  • (c) 42 × 43 × 44<
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Learning outcomes

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

  • evaluate the squares, cubes and other powers of positive and negative numbers with or without your calculator;

  • estimate square roots and calculate them using your calculator;

  • describe the power notation for expressing numbers;

  • use your calculator to find powers of numbers;

  • multiply and divide powers of the same number;

  • understand and apply negative powers, t
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3.1 Positive coordinates

For many towns and cities, an individual book of street maps called an A to Z has been produced. You can look up the name of a street in the index, and it will give you the page number of the map that contains the street, plus the grid reference square for the street. There are different conventions for these grid references. You may have met several of these.


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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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Learning outcomes

After studying this unit you should:

  • be able to solve homogeneous second-order equations;

  • know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;

  • know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.


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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

All materials included in this unit are derived from content originated at the Open University.


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Acknowledgements

All materials included in this unit are derived from content originated at the Open University.


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1.4.1 Bearings

In the following subsections, we apply the vector ideas introduced so far to displacements and velocities. The examples will feature directions referred to points of the compass, known as bearings.

The direction of Leeds relative to Bristol can be described as ‘15° to the East of due North’, or N 15° E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions North (N), South (S), East (E)
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1.1: Converting to component form

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
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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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1.3.1 Introduction

You can find a lot of information about the maths and statistics on the internet.

To find this information you might choose to use:

  • search engines and subject gateways;

  • books and electronic books;

  • databases;

  • journals;

  • encyclopedias

  • internet resources


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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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4.3 Green from the grassroots up

People who demand a radical break with the business-dominated path of economic globalisation believe that the claims of the mainstream business community are at best hopelessly inadequate, and at worst deceitful. However, they know they have to come up with some answers of their own. This section outlines ideas that seek to underpin a transition to green economies owned and run at grassroots level. Sounds ambitious? Author(s): The Open University

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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Introduction

Human societies have to take urgent action to end their dependence on fossil fuels. They also have to prepare to adapt to the uncertainties inherent in global environmental changes, particularly climatic ones. We have to alter the whole path of our development and decision making in order to make our societies both environmentally adaptable and sustainable. This unit takes on the task of trying to chart some of the ways in which this might come about.

The context for these changes by g
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