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3.2 The evolution of the EU

The EU has grown out of a series of intergovernmental political initiatives which have been expressed in a number of treaties. These treaties form the building blocks that give authority and power to the institutions and law-making bodies of the EU. The process is evolutionary, as treaties are reviewed and amended to reflect both the changing membership and the vision of the EU.

The EU is founded on several treaties:

  1. The treaty that established t
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Modelling pollution
The cleaning up of the North American Great Lakes is one of the great success stories in pollution control. Modelling gives scientists the tools to forecast outcomes in a complex environment: how will the pollution accumulate? what will happen if pollution is stopped all together? The two video tracks on this album demonstrate how modelling techniques provide a scientific basis for the political decisions that have revived the area. The material forms part of course MSXR209 Mathematical Modellin
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Introduction

In this course we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group.

Please note that this course is presented through a series of PDF documents.

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

Introduction

Do you want to improve your ability to subtract one number from another, especially if decimals are involved, without having to rely on a calculator? This course will help you get to grips with subtraction and give you some practice in doing it.

You can start with some practice in subtracting small numbers in your head if you want to. Then we will show you how to subtract bigger numbers on paper. Finally we look at how to subtract decimal numbers.

You don’t need to complete th
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Introduction to group theory
This free course is an introduction to group theory, one of the three main branches of pure mathematics. Section 1 looks at the set of symmetries of a two-dimensional figure which are then viewed as functions. Section 2 introduces an algebraic notation for recording symmetries and calculating composites and inverses of symmetries. Section 3 introduces definitions and looks at how to check axioms for a group. Section 4 looks at how to prove that some properties of groups already looked at are gen
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Try some yourself

Activity 22

Carry out the following calculations, without using a calculator.

  • (a) A million pound lottery prize minus a three hundred pound administrative charge.


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3.1 Arithmetic with whole numbers

When you are adding or subtracting whole numbers, an important thing to keep in mind is the place value of the figures. It is often a good idea to set out the numbers in columns before doing the arithmetic.

Example 11

  • (a) There are 4
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7 Subtracting decimals by lining them up

Subtracting whole numbers such as 52 from 375 is fairly straightforward. Subtracting decimal numbers such as 6.892 from 223.6 uses the same process but with one extra step – you have to line the decimal points up first.

Rather than arranging your two numbers so that they line up on the right-hand side, you need to line up the decimal points, regardless of how many numbers there are after the decimal point. In the example below, the top number has one number after the decimal point. It
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2.11 Further exercises

Exercise 28

Let z1 = 2 + 3i and z2 = 1 − 4i. Find z1 + z2, z1z2, z1z
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2.10 The complex exponential function

Consider the real exponential function f (x) = ex (that is, f (x) = exp x). We now extend the definition of this function to define a function f(z) = ez whose domain and codomain are Author(s): The Open University

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2.2 The complex plane

Just as there is a one-one correspondence between the real numbers and the points on the real line, so there is a one-one correspondence between the complex numbers and the points in the plane. This correspondence is given by

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Acknowledgements

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

Course image: Stuart Rankin in Flickr made available under Creative Commons Attribution-NonCommercial 2.0 Licence.

All materials included in this course are
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Try some yourself

Question 1

Find the area of each of these shapes.

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Environment: journeys through a changing world
Genetically, mountain gorillas are amongst our closest living relatives, and also one of the world's most endangered species. Half the world's remaining population survive in the forests of Uganda. This album explores the challenges facing conservationists at Bwindi Impenetrable National Park. Known for its exceptional biodiversity, the Park became a major tourist destination when it opened for gorilla tourism in 1993. The problem is, because the Park lies in the heart of one of the most densely
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Water in Ethiopia
In Ethiopia, 65% of people don’t have access to clean drinking water. In this album we take a glimpse at the struggles Ethiopians go through each day, just to survive. We look closely at the different methods used to improve the quality of life in the rural highlands as well as the conflict between neighbouring farming villages attempting to share the same water supply. This material forms part of the course U116 Environment: journeys through a changing world.Author(s): The OpenLearn team

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Environmental responsibility
Complex questions of environmental responsibility are increasingly raised in times of change and uncertainty. The tracks on this album illustrate the need for nurturing a fresh sense of care for our environment as well as more appropriate forms of accountability. We demonstrate the necessity of addressing issues of entitlements, rights, obligations and duties if we are to critically and carefully shape our values in doing environmental responsibility and being environmentally responsible. This m
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Enacting European Citizenship (ENACT)
The term 'European citizenship' triggers an immediate association with the European Union, its member states, and people who are citizens of those states. This free course, Enacting European Citizenship (ENACT), develops another way of thinking about European citizenship whereby European citizenship need not be granted by the state, limited to the territory within the EU borders or acted out by people who are already citizens. Author(s): Creator not set

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The frozen planet
This free course is a general introduction to the frozen planet, including the temperature in the polar regions; the energy from the Sun and the seasons; reading and understanding graphs and maps; and how the Arctic and Antarctic regions are defined. First published on Thu, 23 May 2019 as Author(s): Creator not set

Neighbourhood nature
There is a fascinating world of nature all around us which we can see if we know how to look for it. Wherever you live, be it in a city or the countryside, you will find areas that support a range of wildlife. This free course, Neighbourhood nature, will provide you with basic scientific and observational skills so that you can go into your local neighbourhood to discover the animals and plants in open spaces. You will learn how to observe, identify and record the wildlife around you, building u
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