 Figure 7 shows the fossilised remains of a type of echinoderm called a crinoid (‘cry-noyed’). Although crinoids occur today, they were far more common in the Palaeozoic and Mesozoic Eras. Most crinoids feed by bending their umbrella-like arrangement of flexible appendages (called ‘arms’) downstream so as to catch a current, rather as in an umbrella being caught in the wind. Tube feet (multipurpose tentacles) on the arms gather food particles suspended in the water, which are th
Author(s): The Open University

We have already listed some of the main modifiable risk factors, such as smoking and excessive alcohol consumption.

## SAQ 1.2

Question: Try to recall two ot
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Cardiovascular diseases are the main cause of premature death (before the age of 75) in the UK, across Europe and the USA – indeed, across many parts of the world (Figures 3 and Author(s): The Open University

According to Equation 1, probability is defined as a fraction. However a fraction such as may also be expressed as a decimal number or as a percentage:

Author(s): The Open University

You may have met complex numbers before, but not had experience in manipulating them. This course gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The course includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.

This OpenLearn course provides a sample of Level 3 study in Author(s): The Open University

Numbers
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property. First published on Wed, 2
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This course is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets. The other topic, is that of dealing with data presented in tabular form. You are, no doubt, familiar with such tables: they are common in the media and in reports an
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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 Möbius 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 – that is, to produce criteria that allow us to determine whether two given surfaces are
Author(s): The Open University

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 © Copyright 2006 The Open University.

## Unit image

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

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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 a > 0, m [Image_Link]http://ocw.open.edu/ope
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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 b = such that b2 = a. We now discuss the justification
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## Exercise 21

Use the Triangle Inequality to prove that 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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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
Author(s): The Open University              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
Author(s): The Open University              3.1 Triangle Inequality

Our next inequality is also used to deduce ‘new inequalities from old’. It involves the absolute values of three real numbers a, b and a + b, and is called the Triangle Inequality. As you will see, the Triangle Inequality has many applications in the analysis units.

## Triangle Inequality

Author(s): The Open University              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 ‘new inequalities from old’. 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 a < b means exactly the same as the statement ba > 0.