What is Euler's formula actually saying? | Lockdown math ep. 4

2020-04-28

[public] 540K views, 30.2K likes, 159 dislikes audio only

What does it mean to compute e^{pi i}?

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Beautiful pictorial summary by @ThuyNganVu:

https://twitter.com/ThuyNganVu/status/1258220129327800320

https://twitter.com/ThuyNganVu/status/1258220541686628353

Not on the "homework" to show that exp(x + y) = exp(x) * exp(y). This gets a little more intricate if you start asking seriously about whether the series really converge, what they converge to, and how exactly you define a product with infinitely many terms. For anyone curious about the technical details, what you would want to show is that the Cauchy Product of the series for exp(x) and exp(y) converges to the product of the values exp(x) and exp(y) for any particular x and y. That requires the Merten's Theorem.

Thanks to these viewers for their contributions to translations

Hebrew: Omer Tuchfeld

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Video Timeline (Thanks to user "Just TIEriffic")

0:00:00 Welcome

0:00:20 Ending Animation Preview

0:01:15 Reminders from previous lecture

0:03:30 Q1: Prompt (Relationship with e^iθ=…)

0:05:40 Q1: Results

0:07:15 WTF, Whats The Function

0:10:00 Exploring exp(x)

0:11:45 Exploring exp(x) in Python

0:14:45 Important exp(x) property

0:15:55 Q2: Prompt (Given f(a+b) = f(a)f(b)…)

0:17:30 Ask: Which is more interesting, special cases or the general case

0:20:00 Q2: Results

0:23:50 Will a zero break Q2?

0:25:40 The e^x convention

0:27:10 Q3: Prompt (i^2 = -1, i^n = -1)

0:27:45 Ask: Zero does not break Q2

0:30:20 Q3: Results

0:31:05 Comparison to Rotation

0:33:00 Visualizing this relationship

0:36:50 The special case of π

0:39:20 Periodic nature of this relationship

0:39:40 Q4: Prompt (e^3i)

0:41:35 Q4: Results

0:43:55 Explaining the celebrity equation

0:45:55 Homework / Things to think about

0:49:15 Ask: Zero does break Q2.

0:50:30 Closing Remarks

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