How to visualize quaternions, a 4d number system, in our 3d world

Euler's formula, e^{pi i} = -1, is one of the most famous expressions in math, but why on earth is this true? A few perspectives from the field of group theory can make this formula a bit more intuitive.

How to visualize quaternions, a 4d number system, in our 3d world

Introduction to the derivatives of polynomial terms and trigonometric functions thought about geometrically and intuitively. The goal is for these formulas to feel like something the student could have discovered, rather than something...

How to think about this 4d number system in our 3d space.

Euler's formula, e^{pi i} = -1, is one of the most famous expressions in math, but why on earth is this true? A few perspectives from the field of group theory can make this formula a bit more intuitive.

What is the math behind quantum computers? And why are quantum computers so amazing? Find out on this episode of Infinite Series.

A proof of the Wallis product for pi, together with some neat tricks using complex numbers to analyze circle geometry.

There are a few special right triangles many of us learn about in school, like the 3-4-5 triangle or the 5-12-13 triangle. Is there a way to understand all triplets of numbers (a, b, c) that satisfy a^2 + b^2 = c^2? There is! And it...

A new and more circularly proof of a famous infinite product for pi.

Introduction to the derivatives of polynomial terms and trigonometric functions thought about geometrically and intuitively. The goal is for these formulas to feel like something the student could have discovered, rather than something...

There are a few special right triangles many of us learn about in school, like the 3-4-5 triangle or the 5-12-13 triangle. Is there a way to understand all triplets of numbers (a, b, c) that satisfy a^2 + b^2 = c^2? There is! And it uses...

Timestamps<br/>
0:00 - Puzzle statement and<br/> motivation
<br/>4:31 - Simpler example6:51 - The gener<br/>ating function11:52 - Evaluation t<br/>ricks
17:24 - Roots of unity
26:31 - Recap and final trick
30:13 - Takeaways

Introduction to the derivatives of polynomial terms and trigonometric functions thought about geometrically and intuitively. The goal is for these formulas to feel like something the student could have discovered, rather than something...

What is the Riemann zeta function? What is analytic continuation? This video lays out the complex analysis needed to answer these questions.

What is the Riemann zeta function? What is analytic continuation? This video lays out the complex analysis needed to answer these questions.

New ReviewIn this video we are going to understand how we can identify and remember all of the Pythagorean identities

New ReviewIn this video we are going to go over how to determine the other quadrants of the unit circle

New ReviewIn this video we are going to explore a great question that many students struggle with in solving trigonometric equations.

New ReviewWhen learning about sine and cosine it can be tricky to remember what they mean and how they are useful. In this video I hope to show you exactly what you need to know.

New ReviewWhen you need to graph the tangent function, not always does it need to be exact with multiple points. Sometimes we just want to know how to graph something quickly.

New ReviewWhen you need to remember how to graph the sine and cosine graphs quickly there is one thing you should remember. In this video that is what I want to explore with you.

New ReviewWhen you are learning about the Unit Circle the most important part is the first quadrant. In this video I will explore how the first quadrant is created and where the values come from.

New ReviewIn this video I am going to work through how to evaluate a basic trig function using the unit circle and one more difficult