Introduction to analysis
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Learn the fundamentals of analysis and inequalities in this free course. Discover rational and irrational numbers, solving inequalities, and proving inequalities. Explore the concept of least upper bounds and how they define arithmetic operations on real numbers. Enhance your mathematical skills today.
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.
Course learning outcomes
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.
