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2.8 Polar form

You have seen that the complex number x + iy corresponds to the point (x, y) in the complex plane. This correspondence enables us to give an alternative description of complex numbers, using so-called polar form. This form is particularly useful when we discuss properties related to multiplication and division of complex numbers.

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2.7 Arithmetical properties of complex numbers

The set of complex numbers satisfies all the properties previously given for arithmetic in . We state (but do not prove) these prope
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2.6 Division of complex numbers

The second of the conjugate–modulus properties enables us to find reciprocals of complex numbers and to divide one complex number by another, as shown in the next example. As for real numbers, we cannot find a reciprocal of zero, nor divide any complex number by zero.

Example 2

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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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2.2 The complex plane

Just as there is a one-one correspondence between the real numbers and the points on the real line, so there is a one-one correspondence between the complex numbers and the points in the plane. This correspondence is given by

Thus we can represent points in the plane by compl
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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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1.1 Rational numbers

In OpenLearn unit M208_5 Mathematical language you met the sets

  • = {1, 2, 3, …}, the natural numbers;


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Learning outcomes

After studying this unit you should be able to:

  • understand the arithmetical properties of the rational and real numbers;

  • understand the definition of a complex number;

  • perform arithmetical operations with complex numbers;

  • represent complex numbers as points in the complex plane;

  • determine the polar form of a complex number;

  • use de Moivre's Theorem to find the nth roots o
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Introduction

In this unit we look at some different systems of numbers, and the rules for combining numbers in these systems. For each system we consider the question of which elements have additive and/or multiplicative inverses in the system. We look at solving certain equations in the system, such as linear, quadratic and other polynomial equations.

In Section 1 we start by revising the notation used for the rational numbers and the real numbers, and we list their arithmetical prop
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5.3.3 Torus with 1 hole

In our last example, we consider a pentagon with two pairs of edges identified. As we saw in Section 2.3, identification of the edges produces a torus with a hole. In this case there are five vertex-neighbourhoods to fit together, as shown in Author(s): The Open University

5.3.2 Klein bottle

If x lies on an edge, then each of the two points in [x] has a half-disc-like neighbourhood (see Figure 107). When we identify edges, these neighbourhoods fit together to form disc-like neighbourhoods in the Klein bottle.

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5.3.1 Torus

If x lies on an edge, then each of the two points in [x] has a half-disc-like neighbourhood. When we identify edges, these neighbourhoods fit together to form disc-like neighbourhoods on the torus, as Figure 105 shows.

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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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