4.1 Introduction

We have seen in SAQ 18 of Section 3.4 how some sets of points of the complex plane can be described algebraically in terms of operations on C. We now use the modulus function to take this a step further by defining discs in the complex plane. As we shall see, discs are extensively used in arguments involving l
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3.4 Self-assessment questions and problems

SAQ 13

Find |z| and Arg z in each of the following cases.

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    3.3 Section summary

    In this section we have seen a correspondence between complex numbers and points in the plane using Cartesian coordinates; the real part of the complex number is represented on the real axis (“horizontal”) and the imaginary part on the imaginary axis (“vertical”). We can also use polar coordinates (r,θ) in which case r, the modulus of a non-zero complex number z is positive and θ is an argument of z, defined only to within an additive intege
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    3.2 Relationship between complex numbers and points in the plane

    We have seen in Section 2.2 that the complex number system is obtained by defining arithmetic operations on the set R × R. We also know that elements of R × R can be represented as points in a plane. It seems reasonable to ask what insight can be obtained by representing complex numbers as
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    2.3 Section summary

    In this section we have seen that the complex number system is the set R × R together with the operations + and × defined by

    From this, one can justify the performance of ordinary algebraic operations on expressions of the form
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    1 Introduction

    You have almost certainly met complex numbers before, but you may well not have had much experience in manipulating them. In this course we provide you with an opportunity to gain confidence in working with complex numbers by working through a number of suitable problems.

    Perhaps the most striking difference between real numbers and complex numbers is the fact that complex numbers have a two-dimensional character, arising from our definition of a complex number as an ordered pair of
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    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.

    Don't miss out:

    If reading
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    Keep on learning

    Study another free course

    There are more than 800 courses on OpenLearn for you to
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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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    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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    3.4 Decreasing by a percentage

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

      £8 less (15% of £8)

      15% of 8 = × 8 =
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