Mathematics: Proofs by induction

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Mathematics: Proofs by induction

Requirements

  • Being able to perform arithmetical operations (addition, subtraction, multiplication, division) on real numbers, in particular fractions.
  • Being familiar with basic rules for computations with real numbers such as the commutativity law, the associativity law (for addition and multiplication) and the distributivity law (F.O.I.L. method / parenthesis multiplication; the square of a sum and of a difference).
  • Being familiar with the rules for powers with natural exponents (the zero-power rule and the product rule).
  • Being familiar with the concept of divisibility for natural numbers.
  • Being able to handle variable expressions and understanding that all the regular rules for computations with real numbers also hold for algebraic expressions.
  • Being interested in mathematics or at least feeling the need of studying and understanding it.

Description

How would you prove that a theorem or a formula is true for *all* natural numbers? Try it for n=0, n=1, n=2, etc? It seems to be a lot of work, or even completely impossible, as there are infinitely many natural numbers!

Don’t worry, there is a solution to this problem. This solution is called “proof by induction” and this is the subject of this short (and free) course. The Induction Principle is often compared to the “domino effect”, which will be illustrated in the course. (This is also the reason for our course image.)

In this course you will learn how induction proofs work, when to apply them (and when not), and how to conduct them. You will get an illustration of this method on a variety of examples: some formulas, some inequalities, some statements about divisibility of natural numbers. You will also get some information about other courses where you can see some theory, and more advanced proofs based on the same principle.

Sadly, there is no possibility of asking question in free courses, but you can ask me questions about this subject via the QA function in my other course: “Precalculus 1: Basis notions”, where the topic of proofs by induction is covered, both theoretically (Peano’s axioms) and practically, with several examples.

Who this course is for:

  • High school students who want to learn conducting proofs by induction.
  • University or college students who have discovered that they need to master proofs by induction for some university level courses, and they want to re-learn this method of proving theorems.
  • Everybody who likes mathematics and want to learn more of it.


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