4.4 Self-assessment questions and problems

## SAQ 25

Find the distance between the numbers 2 âˆ’ *i* and 1Â +Â *3i*.

## Answer

In this section we have seen that the complex number system is the set **R**Â Ã—Â **R** together with the operationsÂ +Â andÂ Ã—Â defined by

From this, one can justify the performance of ordinary algebraic operations on expressions of the form *a*Â +Â *ib*

Scientific notation can be very useful when estimating the answers to calculations involving very large and/or small decimal numbers.

## Example 9

A lottery winner won Â£7851 000. He put the money straight into a deposit account which earns 7.5% interest per annum (i.e. each year). If he wanted to

**1** The new home owners from Example 4 above want to price grass seed, as well as the turf (transport only). The best buy seems to be loose seed, which says â€˜1 kilo covers 80 m^{2}â€™. They wonder what length the side of an 80 m^{2Author(s): The Open University}

**1** Without using your calculator, find the following:

(a) 10

^{2}(b) 100

^{2}(c) 0.1

^{2}(d) 0.01

^{2}(e)

**1** Evaluate the following:

(a) 6

^{2}(b) 0.5

^{2}(c) 1.5

^{2}

## Answer<

Up to now only those points with positive or zero coordinates have been considered. But the system can be made to cope with points involving negative coordinates, such as (^{âˆ’}2, 3) or (^{âˆ’}2, ^{âˆ’}3). Just as a number line can be extended to deal with negative numbers, the *x*-axis and *y*-axis can be extended to deal with negative coordinates.

**1** The frequency diagram below shows the numbers of people in different age groups in a sample of the UK population.

(a) What is the width of each age group?

(b) Which age group conta

This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.

This unit is an adapted extract from the course *Mathematical methods and models* (MST209

## 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.

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

*n*th roots of complex numbers and the solutions of simple polynomial equations.

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

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

Keywords are significant words which define the subject you are looking for. The importance of keywords is illustrated by the fact that there is a whole industry around providing advice to companies on how to select keywords for their websites that are likely to make it to the top of results lists generated by search engines. We often choose keywords as part of an iterative process; usually if we don't hit on the right search terms straight off, most of us tweak them as we go along based on t

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

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

6.3 The role of active citizens and communities

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

1.1.1 Carbon dioxide and other greenhouse gases

Carbon dioxide (CO_{2}) 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 CO_{2} 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 CO_{2} equivalent (in kilograms or tonnes CO_{2}e

1.1 What is the carbon footprint and why is it important?

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 (CO

_{2}) emissions alone, or of the

**(CO**

*carbon dioxide equivalent*_{2}e) effect of other greenhouse gas emissions.

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 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€˜