Dive into a comprehensive video lecture exploring the fascinating connections between compound interest, the mathematical constant e, and circular motion. Uncover the concept of imaginary interest rates and their relationship to springs and Hooke's law. Follow along as complex mathematical ideas are broken down and visualized through engaging animations and interactive polls. Explore topics such as continuous compounding, the definition of e, and the behavior of springs in physics. Gain insights into quaternions, rotation in multiple dimensions, and the beauty of mathematical connections. Benefit from a detailed timeline, allowing easy navigation through various segments of this in-depth mathematical exploration.
Overview
Syllabus
Welcome.
Q1: Prompt (Would you take an imaginary interest rate).
"e to the pi i for dummies" video shoutout.
Q1: Results.
Q2: Prompt (two banks, two rates).
Ask: Beauty of connections in math.
Q2: Results.
Desmos for Q2.
Q3: Prompt (savings growth rate, 6% every 6mo).
Q3: Results.
Desmos graph explored.
Breaking down an interest rate.
An interesting interest equation.
Q4: Prompt (100*(1+0.12/n)^2 as n → ∞).
Ask: Quaternions.
Q4: Results.
Explaining Q4.
Defining e.
The definition of e from previous lectures.
The imaginary interest rate.
Graphing this relationship.
The imaginary interest rate animation.
Compounding continuously with i.
The spring & Hooke's law.
Q5: Prompt (Δx & Δv for a spring).
Ask: Rotation in for multiple dimensions.
Q5: Results.
Rewriting the spring's position.
Bringing it all together.
Ask: Hints on last lecture's homework.
Closing Remarks.
Taught by
3Blue1Brown
