# Imaginary interest rates | Lockdown math ep. 5

From 3Blue1Brown.

Compound interest, e, and how it relates to circles.
Brought to you by you: https://3b1b.co/ldm-thanks

Great Mathologer video:
https://youtu.be/-dhHrg-KbJ0

Beautiful pictorial summary by @ThuyNganVu:

My other videos on imaginary exponents:
https://youtu.be/v0YEaeIClKY
https://youtu.be/mvmuCPvRoWQ

Mistakes:
In the off-handed remarks on quaternions, I mentioned rotation in 4d would require 10 degrees of freedom. That’s wrong, what I should have said was it requires 6 degrees of freedom, and rotation in 5D is what requires 10 degrees of freedom.

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Video Timeline (Thanks to user "Just TIEriffic")
0:00:00 Welcome
0:00:55 Q1: Prompt (Would you take an imaginary interest rate)
0:02:05 "e to the pi i for dummies" video shoutout
0:02:45 Q1: Results
0:03:30 Q2: Prompt (two banks, two rates)
0:04:55 Ask: Beauty of connections in math
0:06:00 Q2: Results
0:07:05 Desmos for Q2
0:09:10 Q3: Prompt (savings growth rate, 6% every 6mo)
0:10:35 Q3: Results
0:12:35 Desmos graph explored
0:14:45 Breaking down an interest rate
0:18:00 An interesting interest equation
0:19:20 Q4: Prompt (100*(1+0.12/n)^2 as n → ∞)
0:22:35 Q4: Results
0:24:50 Explaining Q4
0:26:40 Defining e
0:28:40 The definition of e from previous lectures
0:30:45 The imaginary interest rate
0:32:35 Graphing this relationship
0:33:50 The imaginary interest rate animation
0:37:55 Compounding continuously with i
0:40:45 The spring & Hooke’s law
0:43:20 Q5: Prompt (Δx & Δv for a spring)
0:44:50 Ask: Rotation in for multiple dimensions
0:47:45 Q5: Results
0:49:50 Rewriting the spring’s position
0:55:00 Bringing it all together
0:59:00 Ask: Hints on last lecture’s homework
1:03:25 Closing Remarks
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