This course is concerned with a special class of topological spaces called surfaces. Common examples of surfaces are the sphere and the cylinder; less common, though probably still familiar, are the torus and the MÃ¶bius band. Other surfaces, such as the projective plane and the Klein bottle, may be unfamiliar, but they crop up in many places in mathematics. Our aim is to classify surfaces â€“ that is, to produce criteria that allow us to determine whether two given surfaces are
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In the OpenLearn unit on Developing modelling skills (MSXR209_3), the idea of revising a model was introduced. In this unit you will be taken through the whole modelling process in detail, from creating a first simple model, through evaluating it, to the subsequent revision of the model by changing one of the assumptions. The new aspect here is the emphasis on a revised model, which comes in Section 2. The problem that will be examined is one based on heat transfer.

This unit, the four
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The content acknowledged below is Proprietary (see terms and conditions) and is available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

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

Author(s): The Open University

All written material contained within this unit originated at the Open University.

## Audio Materials

The audio extract is taken from M208 Â© Copyright 2006 The Open University.

## Unit image

http://www.flickr.com/photos/re_birf/69485963/ [Details correct as of 9th June 2008]

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

Just as we usually take for granted the basic arithmetical operations with real numbers, so we usually assume that, given any positive real number a, there is a unique positive real number bÂ =Â  such that b2Â =Â a. We now discuss the justification
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## Exercise 21

Use the Triangle Inequality to prove that
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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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The audio provided below illustrates various methods for proving inequalities. In addition to the techniques already described for proving inequalities, we use mathematical induction and the Binomial Theorem, restated below.

## Theorem 3.1 Binomial Theorem

1. If x
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3.2 Inequalities involving integers

In analysis we often need to prove inequalities involving an integer n. It is a common convention in mathematics that the symbol n is used to denote an integer (frequently a natural number).

It is often possible to deal with inequalities involving n using the rules of rearrangement given in Section 2. Here
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3.1 Triangle Inequality

Our next inequality is also used to deduce â€˜new inequalities from oldâ€™. It involves the absolute values of three real numbers a, b and aÂ +Â b, and is called the Triangle Inequality. As you will see, the Triangle Inequality has many applications in the analysis units.

## Triangle Inequality

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3 Proving inequalities

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

2.1 Rearranging inequalities

Much of analysis is concerned with inequalities of various kinds; the aim of this section and the next is to provide practice in their manipulation.

The fundamental rule, on which much manipulation of inequalities is based, is that the statement aÂ <Â b means exactly the same as the statement b âˆ’aÂ >Â 0.

### Author(s): The Open UniversityLicense informationRelated contentExcept for third party materials and/or otherwise stated (see terms and conditions) the content in OpenLearn is released for use under the terms of the Creative Commons Attribution-NonCommercial-Share

1.6 Further exercises

## Exercise 7

Arrange the following numbers in increasing order:

• (a)Â 7/36, 3/20, 1/6, 7/45, 11/60;

• (b)Â Author(s): The Open University

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

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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1.3 Irrational numbers

There is no rational number which satisfies the equation x2Â =Â 2. A number which is not rational is called irrational. There are many other mathematical quantities which cannot be described exactly by rational numbers; for example, where m and n are natural numbers and
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10 Conclusion

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
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8.1 Benefits of using a calculator

A calculator can help you learn mathematics â€“ it is not a substitute for learning. In fact it can help you see the underlying mathematics in many ways, as in the previous section. Here are some other examples of how it can help you to learn mathematics:

• Instead of getting engrossed in performing long, sometimes tedious calculations, you can focus your attention on the problem you are trying to solve.

• You can work with more realistic
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7.6 Consolidation

You have probably learnt quite a lot about your calculator by now. So this may be a good time to pause and consolidate that knowledge. Speaking mathematics aloud and explaining concepts to somebody else are good ways to do this.

## Exercise 15: Speakeasy

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