Dave4Math / Courses / The Natural Numbers

The Natural Numbers

Master the natural numbers. Learn the well-ordering axiom, mathematical induction, Fibonacci numbers, and Presburger and Peano arithmetic. By the time you've finished this course, you'll be excellent at applying mathematical induction, and you'll have a better understanding of the formal languages behind the numbers.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Imagine a professor walks into a classroom and starts listing rules. The professor doesn’t give you any context and so you start writing it down, assuming you’ll have to memorize it later. Well, this type of math class is still around, and can be quite boring. Now things can be different! Introducing Dave’s online math courses.  In this course, The Natural Numbers, Dave doesn’t spoon feed, he always asks you relevant questions to help you understand the important questions.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in the natural numbers.
  • Students who are enrolled in Number Theory and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about mathematical induction, Fibonacci numbers, Presburger’s arithmetic, and Peano’s arithmetic.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

What You’ll Learn in The Natural Numbers

  • The Principle of Mathematical Induction
  • Well-Ordering Axiom
  • Examples Using Mathematical Induction
  • Induction Principles
  • Arithmetic and Geometric Progressions
  • The Strong Form of Induction
  • The Equivalence
  • Introduction to the Fibonacci Numbers
  • Fibonacci Identities
  • More Examples on Fibonacci Numbers
  • The Euler-Binet Formula
  • The Fibonacci Prime Conjecture
  • The Growth of the Fibonacci Sequence
  • Axioms for the Natural Numbers
  • The Successor Function
  • The Universal Property of Natural Numbers
  • The Axioms for Presburger Arithmetic
  • Addition
  • The Good News for Presburger Arithmetic
  • The Bad News for Presburger Arithmetic
  • The Axioms for Peano’s Arithmetic
  • Multiplication
  • The Good News for Peano’s Arithmetic
  • The Bad News for Peano’s Arithmetic

Course Description

This course focuses on the Natural Numbers, especially the Principle of Mathematical Induction. So, we concentrate on the well-ordering axiom, applying mathematical induction, and the strong form of mathematical induction. After we work out many examples, we then work through that induction, the well-ordering axiom, and strong induction are logically equivalent. Next, the infamous Fibonacci numbers are covered in detail, including proving several Fibonacci identities using mathematical induction.

After that, we begin a more formal approach to the natural numbers. In other words, we take the natural numbers system and form a list of axioms and show that the natural numbers are universal with these properties.

In addition, in this course, we extend the list of axioms to include a total function called addition. Then we prove several properties of addition, showing that this total function is the usual addition that we are familiar with. As a result, this system is called Presburger Arithmetic, and we discuss its logical foundations.

Finally, in this course, we extend our natural numbers system to involve both addition and multiplication. In doing so, this system is called Peano Arithmetic, which is the basis for the standard natural number system. After that, we discuss its logical foundations.

I recommend the prerequisite course Logic and Proof. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free number theory articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.