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2.5 Modulus of a complex number

We also need the idea of the modulus of a complex number. Recall that the modulus of a real number x is defined by

For example, |7| = 7 and |−6| = 6.

In other words, |x| is the distance from the point x on the real line to the origin. We
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2.4 Complex conjugate

Many manipulations involving complex numbers, such as division, can be simplified by using the idea of a complex conjugate, which we now introduce.

Definition

The complex conjugate Author(s): The Open University

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2.3 Complex arithmetic

Arithmetical operations on complex numbers are carried out as for real numbers, except that we replace i2 by −1 wherever it occurs.

Example 1

Let z1 = 1 + 2i and z2 = 3
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2.1 What is a complex number?

We will now discuss complex numbers and their properties. We will show how they can be represented as points in the plane and state the Fundamental Theorem of Algebra: that any polynomial equation with complex coefficients has a solution which is a complex number. We will also define the function exp of a complex variable.

Earlier we mentioned several sets of numbers, including Author(s): The Open University

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1.3 Further exercises

Exercise 4

Solve the following linear equations.

  • (a)   5x + 8 = −2

  • (b)  
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1.2 Real numbers

The rational and irrational numbers together make up the real numbers. The set of real numbers is denoted by . Like rationals, irrational numbers can be represented by decimals, but unlike the decimals for rational numbers, those for irrationals are neither finite nor recurring. All such infinite non-recurr
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5.3 Neighbourhoods

We know that a polygon X is a surface and so each point x in X has a disc-like or half-disc-like neighbourhood. We shall show that a map f that identifies edges of a polygon to create an object Y automatically creates corresponding disc-like or half-disc-like neighbourhoods of each point y = f(x) of Y.

If x is in the interior of X, there is no difficulty: the point x has a disc-like neighbourhood U
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5.2.1 Proof

We check that Tf satisfies conditions (T1)–(T3) for a topology.

Since (T1)–(T3) are satisfied, Tf is a topology on I(X).

Thus (I(X),Tf) is a topological space. We give the topology Tf a sp
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5.2 The identification topology

Our aim is to show that the object that we produce when we identify some or all the edges of a polygon is a surface. Therefore, by the definition of a surface given in Section 2.5, we must show how it can be given the structure of a topological space, and that this space is Hausdorff. Furthermore, we must show that every point has
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5.1 Identifying edges of a polygon

In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re
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3.2.1 Remarks

  1. By ‘contains’, we mean that we can find part of the surface that is homeomorphic to a Möbius band. The edge of the Möbius band does not need to correspond to an edge at the surface, so that a surface without boundary can be non-orientable (as we shall shortly see).

  2. When seeking Möbius bands in a surface, it can be helpful to look at all possible closed curves on the surface and thicken these into bands.

  3. Remember, fro
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2.4.1 Remarks

  1. This theorem applies to all surfaces and not just to surfaces in space.

  2. This theorem tells us that the boundary number is a topological invariant for surfaces, i.e. a property that is invariant under homeomorphisms.

  3. It follows from the theorem that two surfaces with different boundary numbers cannot be homeomorphic. It does not follow that two surfaces with the same boundary number are homeomorphic –
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2.3.8 Sphere

Surfaces can be constructed in a similar way from plane figures other than polygons. For example, starting with a disc, we can fold the left-hand half over onto the right-hand half, and identify the edges labelled a, as shown in Figure 36; this is rather like zipping up a purse, or ‘crimping’ a Cornish pasti
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2.3.7 Two-fold torus

As the polygons become more complicated, so the identifications become more difficult to visualise. For example, what happens if we try to identify the edges of an octagon in pairs, as indicated by the edge labels and arrowheads in Figure 34? Author(s): The Open University

2.3.4 Klein bottle

There are two other surfaces that can be obtained by identifying both pairs of opposite edges of a rectangle. In one of these, shown in Figure 31, we first identify the edges AB and A'B', labelled a, in the direction shown by the arrowheads. This gives us a cylinder, as before. We then try to ident
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2.1 Surfaces in space

In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon by identifying edges. A more formal approach to surfaces is presented at the end of the section.

Figure 3 shows
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Learning outcomes

By the end of this unit you should be able to:

  • explain the terms surface, surface in space, disc-like neighbourhood and half-disc-like neighbourhood;

  • explain the terms n-fold torus, torus with n holes, Möbius band and Klein bottle;

  • explain what is meant by the boundary of a surface, and determine the boundary number of a given surface with boundary;

  • construct certa
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Introduction

This unit is concerned with a special class of topological spaces called surfaces. Common examples of surfaces are the sphere and the cylinder; less common, though probably still familiar, are the torus and the Möbius band. Other surfaces, such as the projective plane and the Klein bottle, may be unfamiliar, but they crop up in many places in mathematics. Our aim is to classify surfaces – that is, to produce criteria that allow us to determine whether two given surfaces are h
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Acknowledgements

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

All materials included in this unit are derived from content originated at the Open University.


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Modelling pollution in the Great Lakes: a review

The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. When prompted to watch the video for this unit, return to this page and watch the clips below. After you've watched the clips, return to the workbook.

Click 'View document' to open the workbook (PDF, 0.3 MB).

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