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.

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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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Power

The calculator is a powerful tool and has several modes of operation: arithmetic, statistical, graphical, programming, and so on. In this respect it is like a computer with a number of different packages. However, it is in general rather simpler to use than a computer.

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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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7.5 The pi key

The value of the mathematical constant , pronounced pi, is stored on scientific and graphics calculators. The TI-84 has as the second function on
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7.4 Reciprocals

There is a key on most scientific and graphics calculators which will give the reciprocal of a number. This is one over the number. So the reciprocal of 2 is or 0.5. The reciprocal of 4 is Author(s): The Open University

7.3 Square rooting a negative number

Another problem surfaces if you start with a negative number and try to find its square root. For example try to find the square root of âˆ’4 on your calculator. Depending upon how your calculator is set up, you may either get an error message or an unfamiliar number like 2i or 2j. This is because there is no real number which squared will give you the negative number âˆ’4. Every real number, whether positive or negative, has a positive square. There are some numbers, ca
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7.2 Square roots

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
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3.2 Keeping a record: a learning file

The term learning file is used to mean a record of your work in some sort of filing system. This may consist of a file, a box, note books, a filing cabinet, files on your computer or something else that suits you. Whatever the content, you will certainly need some way of organizing your written notes so that they stay together and in order.

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## Activity 13

Take a very quick look through Section 1.5 of the Calculator Book, entitled â€˜Everyday calculationsâ€™. Do not read it all yet!

Use the headings, subheadings, diagrams, and so on to give yourself an overview of
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The history of the calculator

Ever since recorded mathematics began, people have been making use of mathematical aids. Four thousand years ago, Babylonian scribes were consulting mathematical tables which included multiplication tables, tables of squares and square roots, and tables of reciprocals of numbers. These values were recorded as marks on clay tablets that were then baked hard in the sunâ€”and some have survived to the present day. (There are several originals to be seen in the British Museum.)

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2 Aims

The aims of this section are for you to:

• gain greater fluency, confidence and skill in using your calculator;

• begin to appreciate how the calculator can be used as a tool for learning mathematics;

• develop an effective means of working from the Calculator Book.

In order to complete this section you will need to have obtained a Texas Instruments TI-83 calculator and the book Tapping into M
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