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Time: 20 hours
Level: Intermediate
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 Introduction Resource- This unit is devoted to the real numbers and their properties. In particular, we discuss inequalities, which play a crucial role in analysis.
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- 1.1 Rational numbers Resource
- The set of natural numbers is
the set of integers is
and the set of rational numbers is
- 1.2 Decimal representation of rational numbers Resource
- The decimal system enables us to represent all the natural numbers using only the ten integers
which are called digits. We now remind you of the basic facts about the representation of rational numbers...
- 1.3 Irrational numbers Resource
- There is no rational number which satisfies the equation x2 = 2. A number which is not rational is called irrational. There are many other mathematical quantities which cannot be described exactly by rational...
- 1.4 Real numbers and their properties Resource
- Together, the rational numbers (recurring decimals) and irrational numbers (non-recurring decimals) form the set of real numbers, denoted by .
- 1.5 Arithmetic with real numbers Resource
- 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...
- 1.6 Further exercises Resource
- Arrange the following numbers in increasing order:
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- 2.1 Rearranging inequalities Resource
- 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.
- 2.2 Solving inequalities Resource
- Solving an inequality involving an unknown real number x means determining those values of x for which the inequality holds; that is, finding the solution set of the inequality, usually given as a union...
- 2.3 Inequalities involving modulus signs Resource
- Now we consider inequalities involving the modulus of a real number. Recall that if a , then its modulus, or absolute value, |a| is defined by
- 2.4 Further exercises Resource
- Solve the following inequalities.
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- 3 Proving inequalities Resource
- 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...
- 3.1 Triangle Inequality Resource
- 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...
- 3.2 Inequalities involving integers Resource
- 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).
- 3.3 Worked examples Resource
- 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,...
- 3.4 Further exercises Resource
- Use the Triangle Inequality to prove that
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- 4.1 Upper bounds and lower bounds Resource
- Any finite set of real numbers has a greatest element (and a least element), but this property does not necessarily hold for infinite sets. For example, neither of the sets = {1, 2, 3, … } and [0, 2)...
- 4.2 Least upper and greatest lower bounds Resource
- 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...
- 4.3 Least Upper Bound Property Resource
- In the examples just given, it was straightforward to guess the values of sup E and inf E. Sometimes, however, this is not the case. For example, if
then it can be shown that E is bounded above by...
- 4.4 Further exercises Resource
- In this exercise, take
| | | | | 5 Manipulating real numbers
- 5.1 Arithmetic with real numbers Resource
- At the end of Section 1, we discussed the decimals
and asked whether it is possible to add and multiply these numbers to obtain another real number. We now explain how this can be done using the Least...
- 5.2 Existence of roots Resource
- 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...
- 5.3 Powers Resource
- 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).
| | | | | References and Acknowledgements | | |
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