This unit explores a real-world system – the Great Lakes – where mathematical modelling has been used to understand what is happening and to predict what will happen if changes are made. The system concerned is extremely complex but, by keeping things as simple as possible, sufficient information will be extracted to allow a mathematical model of the system to be obtained.

This unit is an adapted extract from the course Author(s): The Open University

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

## Exercise 29

In this exercise, take

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In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if then it can be shown that E is bounded above by 3, but it is not so easy to guess the least upper bound of E.

In such cases, it i
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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).

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## Exercise 7

Arrange the following numbers in increasing order:

• (a) 7/36, 3/20, 1/6, 7/45, 11/60;

• (b) Author(s): The Open University

We can do arithmetic with recurring decimals by first converting the decimals to fractions. However, it is not obvious how to do arithmetic with non-recurring decimals. For example, assuming that we can represent and Author(s): The Open University

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
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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.

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

Many people's ideas about what mathematics actually is are based upon their early experiences at school. The first two activities aim to help you recall formative experiences from childhood.

## Activity 1 Carl Jung's school days

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This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the Calculator Book, Tapping into Mathematics With the TI-83 Graphics Calculator. The unit ends by asking you to reflect on the process of studying mathematics.

In order to complete this unit you will need
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4 Proofs in group theory

In Section 4 we prove that some of the properties of the groups appearing earlier in the unit are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse.

Click 'View document' below to open Section 4 (9 pages, 237KB).

Learning outcomes

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

• explain what is meant by a symmetry of a plane figure;

• specify symmetries of a bounded plane figure as rotations or reflections;

• describe some properties of the set of symmetries of a plane figure;

• explain the difference between direct and indirect symmetries;

• use a two-line symbol to represent a symmetry;

• describe geometrically th
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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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Learning outcomes

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

• Section 1: Sets

• use set notation;

• determine whether two given sets are equal and whether one given set is a subset of another;

• find the union, intersection and difference of two given sets.

• Section 2: Functions

• determine the image of a given function;

• determine whether a given function is one-one
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4.3 Section summary

The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view of analysis the importance of this is that we can talk of the closeness of two complex numbers. We can then define the limit of a sequence of complex numbers in a way which is almost identical to the definition of the limit of a real sequence. Another analogue of real analysis arises
Author(s): The Open University

Learning outcomes

After studying this unit 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.

Author(s): The Open University

Acknowledgements

## Audio Materials

These extracts are from M208 © 2006 The Open University.

All material contained within this unit originated at The Open University.

Author(s): The Open University

2 Real functions

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 'View document' below to open Section 1 (12 pages, 1.8MB).