This unit explores a real-world system â€“ the Great Lakes â€“ 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

Having discussed nth roots, we are now in a position to define the expression ax, where a is positive and x is a rational power (or exponent).

## Definition

If aÂ >Â 0, m 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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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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Now we consider inequalities involving the modulus of a real number. Recall that if a , then its modulus, or abso
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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

Learning outcomes

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

Section 1 â€“ Real numbers

• explain the relationship between rational numbers and recurring decimals;

• explain the term irrational number and describe how such a number can be represented on a number line;

• find a rational and an irrational number between any two distinct real numbers;

Section 2 â€“ Inequalities

• solve inequalities by re
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8.2.2 The screen

You can see the calculations that you have entered as well as the answers. This means you can easily check whether you have made any mistakes.

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6.2 Getting the feel of big and small numbers

Very small and very large numbers can be difficult to comprehend. Nothing in our everyday experience helps us to get a good feel for them. For example numbers such as 1099 are so big that if Figure 1 was drawn to scale, you would be dealing with enormous distances. How big is big?

First express 1â€‰000â€‰000â€‰000 in scientific notation as 109. Next, to find out how many times bigger 1099 is, use your calculator to divide 1099 by 109
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6.1 Scientific notation

Understanding how your calculator displays and handles very large and very small numbers is important if you are to interpret the results of calculations correctly. This section focuses on a way of representing numbers known as scientific notation.

Before you start put your calculator into the float mode, so it will display up to about 10 digits and return to the home screen ready to do some calculations.

What answer would you expect if you square 20 million? How man
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2.1 The four rules of arithmetic

You are now going to use the four operation keys (on the bottom right-hand side of the TI-84 keyboard): , Author(s): The Open University

3 Aims

The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective.

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Pressing onwards

## Activity 15

1. Work through Sections 1.6 and 1.7 of the Calculator Book, using the method suggested above of glancing ahead-pressing on-glancing back, if you find it useful.

2. A num
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Does it make sense?

## Example 3

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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

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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).

View document<
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