Introduction

This unit provides an overview of the processes involved in developing models. It starts by explaining how to specify the purpose of the model and moves on to look at aspects involved in creating models, such as simplifying problems, choosing variables and parameters, formulating relationships and finding solutions. You will also look at interpreting results and evaluating models.

This unit, the third in a series of five, builds on the ideas introduced and developed in Modelling poll
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Unit Image

Wade_In_Tulsa, photos

All other materials included in thi
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1 Analysing skid marks

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, 0.2 MB).

Acknowledgements

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


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Introduction

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

5.1 Arithmetic with real numbers

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

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4.4 Further exercises

Exercise 29

In this exercise, take

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4.3 Least Upper Bound Property

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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4.2 Least upper and greatest lower bounds

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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1.6 Further exercises

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

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1.5 Arithmetic with real numbers

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

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1.4 Real numbers and their properties

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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1.2 Decimal representation of rational numbers

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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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

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8.2.4 Accuracy

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!)


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1.1 Mathematics and you

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

Read
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Introduction

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).

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