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2.10.1 Mean and standard deviation for repeated measurements

In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation exemplified by Table 2, the sort to use is the mean (or more strictly the ‘arithmetic mean’) For a set of measurements, this is de
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Real functions and graphs
Sometimes the best way to understand a set of data is to sketch a simple graph. This exercise can reveal hidden trends and meanings not clear from just looking at the numbers. In this unit you will review the various approaches to sketching graphs and learn some more advanced techniques. First published on Tue, 28 Jun 2011 as
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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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2.3 Inequalities involving modulus signs

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

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9 When to use the calculator

Despite the list of advantages given, here is a word of warning: a calculator is not a substitute for a brain! Even when you are using your calculator, you will still need to sort out what calculation to do to get the answer to a particular problem. However skilled you are at using your calculator, if you do the wrong sum, you will get the wrong answer. The phrase ‘garbage in, garbage out’ applies just as much to calculators as to computers. Your calculator is just that – a calculator!<
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Flexibility

The calculator is very useful for ordinary arithmetic and yet it can also perform many functions commonly associated with a computer and deal with quite advanced mathematics. It is useful for both beginners and experts alike, because it has a variety of modes of operation.


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8.2.5 Memory

The calculator retains numbers, formulas and programs which you have stored in it, even when it is turned off. You can recall them when you need them and so save time by not having to enter the same information again.


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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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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 1 000 000 000 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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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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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).

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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2.1 Introduction

In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis.


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5 Hyperbolic functions

In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different.

Click 'View document' below to open Section 4 (5 pages, 104KB).

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