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This unit will help you understand more about real numbers and their properties. It...
This unit will help you understand more about real numbers and their properties. It will explain the relationship between real numbers and recurring decimals, explain irrational numbers and discuss inequalities. The unit will help you to use the Triangle Inequality, the Binomial Theorem and the Least Upper Bound Property.
By the end of this unit you should be able to:
- explain the relationship between rational numbers and recurring decimals;
- explain the term irrational number and describe how such a number can be represented on a number line;
- find a rational and an irrational number between any two distinct real numbers;
- solve inequalities by rearranging them into simpler equivalent forms;
- solve inequalities involving modulus signs;
- state and use the Triangle Inequality;
- use the Binomial Theorem and mathematical induction to prove inequalities which involve an integer n;
- explain the terms bounded above, bounded below and bounded;
- use the strategies for determining least upper bounds and greatest lower bounds;
- state the Least Upper Bound Property and the Greatest Lower Bound Property;
- explain how the Least Upper Bound Property is used to define arithmetical operations with real numbers;
- explain the meaning of rational powers.
- Learning outcomes
- 1 Real numbers
- 2 Inequalities
- 3 Proving inequalities
- 4 Least upper bounds
- 5 Manipulating real numbers
3.3 Worked examples
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
If x and n then
(The notation is also denoted by nCk.
By convention, 0! = 1, so Here, we also adopt the convention that 00 = 1.)
If a, b x and n , then