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Time: 20 hours
Level: Advanced
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 Introduction Resource- You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology,...
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- 1 Introduction Resource
- You have almost certainly met complex numbers before, but you may well not have had much experience in manipulating them. In this unit we provide you with an opportunity to gain confidence in working with...
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| | 2 The complex number system
- 2.1 Introduction Resource
- In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define...
- 2.2 Defining the complex number system Resource
- In complex analysis we are concerned with functions whose domains and codomains are subsets of the set of complex numbers. As you probably know, this structure is obtained from the set R × R by defining...
- 2.3 Section summary Resource
- In this section we have seen that the complex number system is the set R × R together with the operations + and × defined by
- 2.4 Self-assessment questions and problems Resource
- Self-assessment questions are intended to test your immediate comprehension of a reading section. If you have difficulty with them you should read again the appropriate parts of the material. Before checking...
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- 3.1 Introduction Resource
- In this section we shall develop the correspondence between C and R × R by obtaining a geometric representation of elements of C and operations on C. We shall define the polar form of a complex number...
- 3.2 Relationship between complex numbers and points in the plane Resource
- 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...
- 3.3 Section summary Resource
- 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”)...
- 3.4 Self-assessment questions and problems Resource
- Find |z| and Arg z in each of the following cases.
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| | 4 Sets of points in the complex plane
- 4.1 Introduction Resource
- 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...
- 4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem Resource
- You will no doubt recall that in real analysis extensive use is made of the modulus function
. It gives us a way of measuring the “closeness” of two numbers, which we exploit in writing...
- 4.3 Section summary Resource
- The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view...
- 4.4 Self-assessment questions and problems Resource
- Find the distance between the numbers 2 − i and 1 + 3i.
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| | References and Acknowledgements | |
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