Look at the line *l*, which cuts two parallel lines *m* and *n*.

If you trace the lines at one of the intersections in the diagram below and place them over the lines at the other interse

Two straight lines that do not intersect, no matter how far they are extended, are said to be **parallel**. Arrows are used to indicate parallel lines.

## Question 1

Find all the remaining angles in each of the diagrams below.

1.3.4 Vertically opposite angles

When two straight lines cross, they form four angles. In the diagram below, these angles are labelled *α*, *β*, *θ* and *φ* and referred to as alpha, beta, theta and phi. The angles opposite each other are equal. They are called **vertically opposite** angles. Here *α* and *β* are a pair of vertically opposite angles, as are *θ* and *φ*. Although such angles are called ‘vertically opposite’, they do not need to be vertically above and bel

## Question 1

A company carried out a survey, recording how staff in a particular office spent their working time. The table shows the average number of minutes spent in each hour on various activities.

## Question 1

Calculate all the angles at the centres of these objects.

4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem

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 expressi

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

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

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

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 **

**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.**

**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**

**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.**

**Section 6 contains solutions to the exercises that appear throughout sections 1-5.**

**Click the link below to open the solutions (13 pages, 232KB).**

**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).**

**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**

**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).**

**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).**

**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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