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).

Author(s): The Open University

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

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
Author(s): The Open University

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 â€˜garbage in, garbage outâ€™ applies just as much to calculators as to computers. Your calculator is just that â€“ a calculator!<
Author(s): The Open University

Most aspects of the calculator are straightforward to use. Calculations are entered on the screen in the same order as you would write them down. More complicated mathematical functions and features are also reasonably intuitive, and there are â€˜escapeâ€™ mechanisms, so that you can explore without worrying about how you will get back to where you were.

Author(s): The Open University

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.

Author(s): The Open University

Earlier you met the square function and on most calculators the square root is the second function on the same key. Look to see if this is the case for your calculator and check the calculator handbook on how to use this function. In many cases you will need to press the square root key before the number, instead of afterwards, as for the square key. This is the case on the TI-84. Check that you can find the square root of 25 and of 0.49 (you should get 5 and .7 respectively).

Now find
Author(s): The Open University

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.

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

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

Author(s): The Open University

Having set out on her mathematical journey, Dawn suddenly remembered that she had forgotten to pack any sandwiches

There are many re
Author(s): The Open University

In Section 2 we develop an algebraic notation for recording symmetries, and demonstrate how to use the notation to calculate composites of symmetries and the inverse of a symmetry.

Click 'View document' below to open Section 2 (9 pages, 504KB).

## Unit image

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

Author(s): The Open University

In Section 2 we give the general definition of a function, and illustrate how functions can be used to describe a variety of mathematical concepts, such as transformations of the plane. We discuss the idea of composing two functions, and the idea of forming the inverse of a function.

Click 'View document' below to open Section 2 (16 pages, 366KB).

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

1. Join the 200,000 students currently studying withThe Open University.

Author(s): The Open University

Many problems are best studied by working with real functions, and the properties of real functions are often revealed most clearly by their graphs. Learning to sketch such graphs is therefore a useful skill, even though computer packages can now perform the task. Computers can plot many more points than can be plotted by hand, but simply â€˜joining up the dotsâ€™ can sometimes give a misleading picture, so an understanding of how such graphs may be obtained remains important. The object of t
Author(s): The Open University

Now try the quizÂ  and see if there are any areas you need to work on.

Author(s): The Open University

Earlier you looked at place values for numbers, and why they were called powers of ten.

 Place value 10 000 1000 100 10 1 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 What are the following?(a) 10(b) 01(c) 20(d) 02 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 Up to now only those points with positive or zero coordinates have been considered. But the system can be made to cope with points involving negative coordinates, such as (âˆ’2, 3) or (âˆ’2, âˆ’3). Just as a number line can be extended to deal with negative numbers, the x-axis and y-axis can be extended to deal with negative coordinates.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 Copyright 2009 University of Nottingham