Introduction to analysis
This free course is an introduction to analysis which looks at real numbers and their properties, with a particular emphasis on inequalities. Section 1 starts by revising rational numbers and their decimal representations. Then, real numbers are introduced as infinite decimals. Section 2 looks at rules for manipulating inequalities and finding the solution set of an inequality. Section 3 looks at various techniques for proving inequalities. Section 4 introduces the concept of a least upper bound. Section 5 looks at how least upper bounds can be used to define arithmetic operations on the set of real numbers.
After studying this course, you should be able to:
- Find a rational and an irrational number between any two distinct real numbers
- Solve inequalities by rearranging into simpler equivalent forms and solve inequalities involving modulus signs
- State and use the Triangle Inequality and use the Binomial Theorem and mathematical induction to prove inequalities which involve an integer n
- Use the strategies introduced for determining least upper bounds and greatest lower bounds
- Explain how the least upper bound property is used to define arithmetical operations with real numbers and explain the meaning of rational powers.