2017-09-11 22:53:35

Math Made Simple: Number Theory and The Secrets of Numbers

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Learn The Secrets Of The Integers And Prime Numbers And Gain A New Perspective On Mathematics.

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973 students enrolled

What Will I Learn?

- Learn about Prime Numbers
- Understand Factorization at an Advanced Level
- Introduce the Algebra of Congruences
- Look into Diophantine Equations
- Understand Primitive Roots
- Understand Quadratic Reciprocity Theorem

Requirements

- Algebra 1
- NO CALCULUS NEEDED

Description

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For thousands of years, mathematicians have been curious about numbers. Where did they come from? What properties do they have? The truth is that you will never learn the secrets of numbers until you take Number Theory, and all you need is a curious mind to understand (no prerequisites to this course!). One of the biggest problems in history has been: how to you factor a number into prime factors? Well, 26=13*2, but try factoring 1432479... Not so easy now? What if I asked what the next prime number was after 1432479 is? These are tough questions and their answers involve a different type of math - one that you don't need calculus for - but you need a lot of curiosity.

In this course, you will look into the secrets of the integers and the many properties that they hold. You do not need Calculus or any advanced mathematics to understand this course, however this is an advanced mathematics course. The material will not involve "solve for..." problems. This course is designed to prove things, and so most of the lectures will cover proofs, and not problem solving. This is a great introduction to what pure mathematics actually deals with, and what many modern professors research.

**Learn and Master Arithmetic, Prime Numbers and Factorization**

- Solving Diophantine Equations
- Quadratic Reciprocity Theorems and Legendre Symbol
- Continued Fractions
- Primitive Roots
- Extended Euclidean Algorithm
- Advanced Modern Factorization
- Algebra of Prime Numbers
- Pythagorean Triples

**See the Algebra of Modern Mathematics**

This is not just a basic math course. This course offers over 7 hours of content that will blow your mind. You will learn more material than most Universities offer in their own Number Theory courses. We go into depth on everything with clear examples that helps you understand.

How in depth do we go? Take for example the proof of Wilson's Theorem. We will prove it Once, Twice, Three times each in a different way. And that's just one theorem in this course.

So what are you waiting for?

Who is the target audience?

- College Students interested in Mathematics
- Any student interested in modern Mathematics
- Students in a Number Theory College Course
- Students who have taken a Number Theory College Course

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Curriculum For This Course

51 Lectures

09:17:35
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Introduction Video
1 Lecture
01:15

Preview
01:15

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Introduction to Number Theory
4 Lectures
33:42

In this introduction, I introduce myself and give a little information about the course. Again, my email is kdamours@asu.edu and you can email me for any math questions at any time.

Preview
06:47

This lecture covers all of the needed material that will be used in Number Theory. By watching this video, you will have all of the tools to understand what is happening in this course.

Basic Notation

13:17

We will go through the introductory material one more time to make sure that you understood everything.

Introductory Notations, Definitions and Axioms

6 questions

Induction is a tool that will be used in a couple of lectures throughout number theory, and so it is important to understand this method of proving in order to understand Lecture 4 and maybe other math courses as well.

Induction

13:38

Review the material from the induction section of the course.

Induction Quiz

4 questions

Don't doubt yourself! Most elementary number theory problems are solved by students who have never done research before. Find a problem on this pdf or any website that has open number theory problems, and start trying them!

Open Problems In Number Theory

103 pages

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Number Theory
46 Lectures
06:49:38

Preview
17:56

Let's do a quick review of Fibonacci numbers

Fibonacci

2 questions

This lecture introduces a new perspective on the integers and how numbers relate to each other by division.

Divisibility

09:59

Chapter 1 Homework

1 page

In this lecture, we prove the division algorithm which is essentially long division, but with a more precise approach.

Division Algorithm

15:34

This tests divisibility and the division algorithm

Divisibility and Division Algorithm

2 questions

We will discuss in full detail everything about prime numbers and their properties.

Prime Numbers

06:19

The greatest common divisor has many important properties that will be used later on. In this lecture, we introduce these properties.

Greatest Common Divisors

16:16

Chapter 3 Homework

1 page

The Euclidean Algorithm is one of the fastest methods of calculating the gcd of any two numbers. This algorithm will be explained in it's entirety.

The Euclidean Algorithm

09:42

The Fundamental Theorem of Arithmetic says that any number can be factored into prime numbers. Though the message is simple, the proof is involved and we will go through it completely.

The Fundamental Theorem of Arithmetic

08:41

Factorization has been a very tricky process for hundreds of years, and mathematicians still don't have a grasp on it. We will describe some basic methods of factoring quickly and we will describe some of the fastest methods known to date.

Factorization Methods

17:20

Test your factorization skills.

Can You Factor Numbers?

3 questions

In this lecture, we introduce a new type of algebra.

Congruences Part 1

19:33

This is a continuation of the last lecture.

Congruences Part 2

07:26

Now, we will attempt to solve some algebraic systems in this new algebra.

Linear Congruences Part 1

08:14

This is a continuation of the last lecture.

Linear Congruences Part 2

14:13

This is a continuation of the last lecture.

Linear Congruences Part 3

08:40

In this lecture, we go back to the extended euclidean algorithm, and make an additional step that will allow us to know more information about the gcd. We will use this to find multiplicative inverses.

Preview
13:30

We will go through some examples of algebraic systems and solve them using the tools that we had learned.

Solving Congruences

05:01

In this lecture, we introduce the importance of the Chinese Remainder Theorem.

Chinese Remainder Theorem Part 1

14:10

In this lecture, we introduce the Chinese Remainder Theorem.

Chinese Remainder Theorem Part 2

18:01

This is a continuation of the last lecture.

Chinese Remainder Theorem Part 3

08:51

In this lecture, we use the knowledge of solving linear congruences and the Chinese Remainder Theorem to solve polynomial congruences.

Solving Polynomial Congruences

08:23

Chapter 4 Homework

1 page

See what you got with linear congruences.

Test Your Congruence Skills

1 question

You have 75 minutes to complete this test.

Test 1

1 page

In this lecture, we introduce Wilson's Theorem, and prove it in two different ways.

Wilson's Theorem

17:51

In this lecture, we state and prove Fermat's Little Theorem and use it to prove Wilson's Theorem one more time.

Fermat's Little Theorem

12:56

Psuedoprimes are numbers that act like prime numbers, but they are not. We try to characterize these numbers.

Psuedoprimes

06:43

In this lecture, we give a generalization of Fermat's Little Theorem.

Euler's Generalization of Fermat's Little Theorem

13:57

In this lecture, we learn how to compute exponentiation very quickly.

Successive Squaring

16:56

Review the last few lectures with one simple problem.

Euler's Theorem and Successive Squaring Review

1 question

Chapter 6 Homework

1 page

We will introduce Order a little soon because it relates so much to Euler's Theorem.

Order

07:39

In this lecture, we investigate everything about the totient function.

The Euler-Phi Function Part 1

16:13

This is a continuation of the last lecture.

Euler-Phi Function Part 2

05:58

In this lecture, we take a look at a few more multiplicative functions like sigma and tow.

Other Multiplicative Functions

12:54

Here, we look at types of numbers that hold many unique properties and reveal many modern open problems in number theory.

Perfect Numbers and Mersenne Primes

06:31

Chapter 7 Homework

1 page

In this lecture, I introduce security and what it has to do with number theory (hint: the answer is everything).

Cryptography

07:34

Here, we find numbers that have an order equal to the totient function, and why they are so important.

Primitive Roots

13:20

Chapter 9 Homework

1 page

In this lecture, we characterize perfect squares and how some numbers like 2 can sometimes be perfect squares.

Quadratic (Non)residues

13:26

In this lecture, we introduce all of the properties of quadratic (non)residues.

The Law of Quadratic Reciprocity

09:21

In this lecture, we bring back a very old concept called continued fractions which haven't been touched since the 19th century.

Continued Fractions

09:10

Chapter 11 Homework

1 page

In this lecture, we discuss very modern number theory pertaining to right triangles and we discuss Fermat's Last Theorem.

Pythagorean Triples

09:14

Chapter 13 Homework

1 page

This covers everything from the last several lectures

Final Quiz

3 questions

You have 75 minutes to complete this test.

Test 2

1 page

You have finally completed number theory in its entirety! Congrats.

Conclusion

02:06

About the Instructor