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Mathematical Induction

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This book is about how mathematical induction is a powerful tool for proving results in mathematics. This book will discuss the basic steps of mathematical induction, how to use it to solve problems and provide examples of how it can be used. The limitations of mathematical induction will also be explored, as well as some alternative methods for proving results. This book accompanies the video series: Mathematical Induction (The Complete Step-by-Step Guide) published on the Dave4Math YouTube channel.  In detail this book covers: 

  1. What is mathematical induction and why is it important 
  2. The basic steps of mathematical induction 
  3. How to use mathematical induction to solve problems 
  4. Examples of how to use mathematical induction 
  5. The well-ordering axiom, and strong induction
  6. The solution to the mathematical induction equivalence problem

So if you’re looking to become a pro at using mathematical induction, read on!

ISBN: 978-1-951290-06-1 SKU: d4m001

Description

Where did mathematical induction come from?

The principle of induction was first proposed by the Greek philosopher Aristotle, who said that “we may infer from particular premises to a universal conclusion”. However, it was not until the 17th century that mathematicians began using induction as a formal method of proof. The German mathematician Gottfried Leibniz was the first to use induction to prove that every natural number is the sum of two squares. In the 19th century, the British mathematician Augustus De Morgan formalized the principle of induction and used it to prove a variety of results in Number Theory.

How do I go about teaching it?

In this book, I concentrate on examples that demonstrate how to use mathematical induction to prove a statement is true for all natural numbers. In addition, the well-ordering axiom and the principle of mathematical induction are proven to be logically equivalent. Moreover, we show that both forms of mathematical induction and the well-ordering axiom are logically equivalent. We also discuss arithmetic and geometric progressions as examples of how to use induction. In the end, I provide the reader with lots of examples for them to practice on their own.

How is this book is organized?

 Let’s face it. Mathematical Induction is not the easiest topic to learn as a student.

That’s why in this book, you’re going to work through several examples of writing a proof by Mathematical Induction as a beginner. Then, you’ll concentrate on cases that demonstrate how to use induction to prove a statement is true for all natural numbers. 

  1. The first chapter gives a broad overview of the topic. It’s best to read this first and then again last, after reading all the rest of the chapters. 
  2. The second chapter goes slowly and gives the first examples. If you already have seen some examples of mathematical induction, then you may want to skip this chapter. 
  3. The well-ordering axiom and the principle of mathematical induction are closely related statements. If you already know what the well-ordering of the natural numbers is, then you can skip this chapter. 
  4. Here, we get more examples of mathematical induction. I recommend that you practice each example before (or after) reading each example in the chapter. 
  5. This chapter goes into more detail about the statement of Mathematical Induction. For those students who are not yet pros at writing proofs, this chapter is helpful. 
  6. In this chapter, we see some applications of writing proof by mathematical induction regarding arithmetic and geometric progressions. 
  7. Just when you might be getting the hang of induction, let’s change it up a bit. The strong form of induction is covered in this chapter (and why we want it). 
  8. The powerful conclusion of the series is the statement that all three are equivalent to each other: well-ordering, mathematical induction, and strong induction. Find out why in this chapter.
  9. To help you get more practice, try each of the exercises on your own. Then find out how you did.

Enjoy the book, Dave.

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