This course is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that this is very useful in understanding the arithmetic properties introduced in Section 1.

In Section 3 we discuss methods of finding nth roots of complex numbers and the solutions of simple
Author(s): The Open University

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.

Click 'View document' to open the workbook (PDF, 0.4 MB).

Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of real numbers, denoted by .

As with rational numbers, we can determine which of two real numbers is greater by comparing their decimals and noticing the first pair of corresponding digits
Author(s): The Open University

The decimal system enables us to represent all the natural numbers using only the ten integers which are called digits. We now remind you of the basic facts about the representation of rational numbers by decimals.

Author(s): The Open University

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

Grateful acknowledgement is made to the following sources for permission to reproduce material in this unit:

The content ackno
Author(s): The Open University

This unit has introduced you to some aspects of using a scientific or graphics calculator. However, in many ways, it has only scratched the surface. Hopefully your calculator will be your friend throughout your study of mathematics and beyond. Like any friend, you will get to know it better and appreciate its advantages as you become more familiar with it. Don't expect to know everything at the beginning. You may find the instruction booklet, or other help facility, a bit hard going to begin
Author(s): The Open University

Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase â€˜garbage in, garbage outâ€™ applies just as much to calculators as to computers. Your calculator is just that â€“ a calculator!<
Author(s): The Open University

The calculator does not make mistakes in the way that human brains tend to. Human fingers do, however, make mistakes sometimes; and the calculator may not be doing what you think you have told it to do. So correcting errors and estimating the approximate size of answers are important skills in double-checking your calculator calculations. (Just as they are for checking calculations done in your head or on paper!)

Author(s): The Open University

As you have probably observed, there are many mathematical functions on your calculator, but most users only need to use a few of them regularly. This is an opportunity to be introduced to some of the more useful ones informally. Many functions are directly visible on the keys of the keyboard, but your calculator may have additional functions, e.g. on a MATH menu.

This section is designed to give you a whistle-stop introduction to some of the important functions on your calculato
Author(s): The Open University

Here are two short investigations involving large numbers for you to try. Please do not turn to the comments on these exercises until you have made some notes and had a go yourself.

## Exercise 13: Where did I come from?

Family tree
Author(s): The Open University

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

Click 'View document' below to open the solutions (13 pages, 232KB).

The notation in Example 6 is called power notation, or index notation. In a number such as 25, the 5 is called the power, or index, of the number.

The squares are particular examples of powers: 92, for example, can be thought of as â€˜9 to the power 2â€™.

For most numbers, calcul
Author(s): The Open University

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

## Example 3

Experiments or surveys usually generate a lot of information from which it is possible to draw conclusions. Such information is called data. Data are often presented in newspapers or books.

One convenient way to present data is in a table. For instance, the nutrition panel on the back of a food packet:

### Nutrition Information

Author(s): The Open University

## Exercise 1

A vector a has magnitude |a|Â =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.

<
Author(s): The Open University

The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector

If P and Q represent places on the ground, then it is natural to use a bearing to describe the direct
Author(s): The Open University

In the following subsections, we apply the vector ideas introduced so far to displacements and velocities. The examples will feature directions referred to points of the compass, known as bearings.

The direction of Leeds relative to Bristol can be described as â€˜15Â° to the East of due Northâ€™, or NÂ 15Â°Â E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions NorthÂ (N), SouthÂ (S), EastÂ (E)
Author(s): The Open University

The following activity illustrates how the conversion processes outlined in the preceding sections may come in useful. If two vectors are given in geometric form, and their sum is sought in the same form, one approach is to convert each of the vectors into component form, add their corresponding components, and then convert the sum back to geometric form.

Author(s): The Open University

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector aÂ =Â a 1 iÂ +Â a 2 j, what are its magnitude |a| and direction Î¸?

The first part of this question is dealt with using Pythagorasâ€™ Theorem: the magnitude of a v
Author(s): The Open University

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

Author(s): The Open University