At the end of Section 1, we discussed the decimals and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least Upper Bound Property of Author(s): The Open University

We have seen that the set [0, 2) has no maximum element. However, [0, 2) has many upper bounds, for example, 2, 3, 3.5 and 157.1. Among all these upper bounds, the number 2 is the least upper bound because any number less than 2 is not an upper bound of [0, 2).

Author(s): The Open University

To practise using the techniques described in the audio, we suggest that you now try the following exercises.

## Exercise 18

Use the Binomial Theorem to prove that
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In this section we show you how to prove inequalities of various types. We use the rules for rearranging inequalities given in Section 2, and also other rules which enable us to deduce â€˜new inequalities from oldâ€™. We met the first such rule in Author(s): The Open University

Now we consider inequalities involving the modulus of a real number. Recall that if a , then its modulus, or abso
Author(s): The Open University

1.1 Rational numbers

The set of natural numbers is the set of integers is and the set of rational numbers is Author(s): The Open University

7.2 Square roots

Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively).

Now find
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6 Solutions to the exercises

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

Click 'View document' below to open the solutions (15 pages, 468KB).

4 Two identities

Section 4 introduces some important mathematical theorems.

Click 'View document' below to open Section 4 (7 pages, 237KB).

View document<
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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 RÂ Ã—Â R. We also know that elements of RÂ Ã—Â R can be represented as points in a plane. It seems reasonable to ask what insight can be obtained by representing complex numbers as p
Author(s): The Open University

1 Overview

A fundamental concept in mathematics is that of a function.

Consider, for example, the function f defined by

This is an example of a real function, because it associates with a given real number x the real number 2x2 âˆ’ 1: it maps real numbers to real n
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4 Open Mark quiz

Now try the quizÂ  and see if there are any areas you need to work on.

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1.1.1 Try some yourself

1 On the plan of the bathroom in Example 1, what is the width of the window and
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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.

Author(s): The Open University

Learning outcomes

After studying this unit you should:

• know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;

• understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;

• know ways to operate on and combine vectors.

Author(s): The Open University

Introduction

This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.

This is an adapted extract from the Open University course Mathematical methods and models (MST209)
Author(s): The Open University

1 Differential equations

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, 1.4 MB).

Learning outcomes

After studying this unit you should:

• be able to solve homogeneous second-order equations;

• know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;

• know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.

Author(s): The Open University

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.

Author(s): The Open University

1 Modelling with Fourier series

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, 0.6 MB).