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!)

Author(s): The Open University

As you have probably observed, there are many mathematical functions on your calculator, but most users only need to use a few of them regularly. This is an opportunity to be introduced to some of the more useful ones informally. Many functions are directly visible on the keys of the keyboard, but your calculator may have additional functions, e.g. on a MATH menu.

This section is designed to give you a whistle-stop introduction to some of the important functions on your calculato
Author(s): The Open University

Here are two short investigations involving large numbers for you to try. Please do not turn to the comments on these exercises until you have made some notes and had a go yourself.

## Exercise 13: Where did I come from?

Family tree
Author(s): The Open University

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
Author(s): The Open University

## 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
Author(s): The Open University

The aims of this section are to:

• help you clarify your own ideas of what mathematics is;

• give you experience of reading different types of written mathematics;

• give you an initial feel of how a mathematician views the world.

You have chosen to study a unit entitled ‘Mathematics everywhere’, but what exactly is mathematics? It sounds a simple enough question but, in fact, mathematics is not easy to
Author(s): The Open University

## Audio and video materials

These extracts are from M208 © 2006 The Open University.

## Unit image

Evan Leeson: http://www.flickr.com/photos/ecstaticist/1139453775/ [Details correct as of 30th June 2008]

All other materials contained within this unit originated at The Open University.

Except for third party materials and otherwise stated (see
Author(s): The Open University

Section 3 is an audio section. We begin by defining the terms group, Abelian group and order of a group. We then demonstrate how to check the group axioms, and we extend the examples of groups that we use to include groups of numbers – the modular arithmetics, the integers and the real numbers.

Click 'View document' below to open Section 3 (11 pages, 703KB).

In this unit we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group.

Please note that this unit is presented through a series of PDF documents.

This unit is an adapted extract from the Open Unviersity course Pure mathematics (M208)

Author(s): The Open University

Section 5 contains solutions to the exercises that appear throughout sections 1-4.

Click 'View document' below to open the solutions (13 pages, 500KB).

In Section 3 we examine the language used to express mathematical statements and proofs, and discuss various techniques for proving that a mathematical statement is true. These techniques include direct proof, proof by mathematical induction, proof by contradiction and proof by contraposition. We also illustrate the use of counter-examples to show that a statement is false.

Click 'View document' below to open Section 3 (17 pages, 374KB).

Author(s): The Open University

In Section 1 we discuss the idea of a set and describe some ways to define sets. We illustrate our discussion with sets of numbers and with geometrical sets of points in the plane. We also explain how to check whether two given sets are equal and whether one set is a subset of another. Finally, we introduce the set operations of union, intersection and difference.

Click 'View document' below to open Section 1 (16 pages, 389KB).

Author(s): The Open University

When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion.

In mathematics we try to avoid these difficulties by ex
Author(s): The Open University

Section 6 contains solutions to the exercises that appear throughout sections 1-5.

Click 'View document' below to open the solutions (13 pages, 232KB).

In Section 3 we consider how to sketch the graphs of more complicated functions, sometimes involving trigonometric functions. We look at graphs which are sums, quotients and composites of different functions, and at those which are defined by a different rule for different values of x.

Click 'View document' below to open Section 3 (7 pages, 133KB).

1 Find each of the following by hand, giving your answers both as a power of ten and as a decimal number. You will use these answers as a check on your calculator work in the next question.

• (a) 10
Author(s): The Open University

The notation in Example 6 is called power notation, or index notation. In a number such as 25, the 5 is called the power, or index, of the number.

The squares are particular examples of powers: 92, for example, can be thought of as ‘9 to the power 2’.

For most numbers, calcul
Author(s): The Open University

Experiments or surveys usually generate a lot of information from which it is possible to draw conclusions. Such information is called data. Data are often presented in newspapers or books.

One convenient way to present data is in a table. For instance, the nutrition panel on the back of a food packet:

### Nutrition Information

Author(s): The Open University

After studying this unit you should be able to:

• appreciate the concept of force, and understand and model forces such as weight, tension and friction;

• model objects as particles or as rigid bodies, and the forces that act on an object in equilibrium;

• use model strings, rods, pulleys and pivots in modelling systems involving forces;

• understand and use torques;

• model and solve a variety of problems involving systems in eq
Author(s): The Open University

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ Theorem: the magnitude of a v
Author(s): The Open University