Learning outcomes

After studying this course, you should be able to:

  • perform basic algebraic manipulation with complex numbers

  • understand the geometric interpretation of complex numbers

  • know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.


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Acknowledgements

These extracts are from M208 © 2006 The Open University.

All material contained within this course originated at The Open University.

Course image: Matt in Flickr made available under Creative Commons Attribution 2.0 Licence.

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Conclusion

This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.


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7 Solutions to the exercises

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


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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).

Section 5


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Acknowledgements

These extracts are from M208 © 2006 The Open University.

All other materials contained within this course originated at The Open University.

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Common
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5 Hyperbolic functions

In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

Click the link below to open Section 4 (5 pages, 104KB).

Section 4
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2 Real functions

In Section 1 we formally define real functions and describe how they may arise when we try to solve equations. We remind you of some basic real functions and their graphs, and describe how some of the properties of these functions are featured in their graphs.

Click the link below to open Section 1 (12 pages, 1.8MB).

Section 1
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Learning outcomes

After studying this course, you should be able to:

  • understand the definition of a real function

  • use the notation for intervals of the real line

  • recognise and use the graphs of the basic functions described in the audio section

  • understand the effect on a graph of translations, scalings, rotations and reflections

  • understand how the shape of a graph of a function features properties of the function such as increasing, decr
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Keep on learning

Study another free course

There are more than 800 courses on OpenLearn for you t
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Conclusion

This free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.


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3.4.1 Try some yourself

Activity 24

A new train operator boasts ‘Train times reduced by 12%’. Decrease 90 minutes by 12%. Give your answer as minutes and seconds.

Answer
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3.3.1 Increasing by a percentage

Our everyday experience of percentages includes percentage increases (like VAT at %, or a service charge of 15%) and percentage decreases (such as a discount of 15%).

For example, £8 plus
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3.2 Converting to a percentage

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:

  0.17 × 100% = 17%;

<
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3.1.1 Try some yourself

Activity 17

Express each of the following percentages as fractions:

  • (a) 40%

  • (b) 8%

  • (c) 70%

  • (d)
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3.1 What are percentages?

Percentages are used, particularly in newspaper articles, to indicate fractions (as in ‘64% of the population voted’) or to indicate changes (as in ‘an increase of 4%’).

Percentages often indicate proportions. For example, labels in clothes indicate the various proportions of different yarns in the fabric. ‘Per cent’ means ‘per hundred’ and is denoted by the symbol %. 100% is the same as the whole, or one hundred per hundred.

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2.3.1 Try some yourself

Activity 14

A piece of computer software is to be developed by a team of programmers. It is estimated that a team of four people would take a year. Which of the following times is the length of time taken by three program
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2.3 Inverse proportion

In Section 2.2 you saw that direct proportion described relationships between two quantities, where as one increased, so did the other. Sometimes as one quantity increases the other decreases instead of increasing. This is called indirect proportion. Team tasks are often an example of this. The ti
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