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

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

Beautiful pictorial summary by @ThuyNganVu:

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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The live question setup with stats on-screen is powered by Itempool.

Curious about other animations?

Music by Vincent Rubinetti.
Download the music on Bandcamp:

Stream the music on Spotify:

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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Various social media stuffs:

[bluebrown]

Summary of the confusion on problem 2. TL;DR, given the spirit of the question it should have also specified that f(x) > 0 for all x, and I'm clearly prone to deep confusion while trying to juggle the many balls of live broadcasting.


The question asks about a function f satisfying f(a + b) = f(a)f(b). The relevant part of the question was whether f(-1) must be 1 / f(1).


- It was originally graded in a way to suggest that this _does_ indeed need to be true.
- While "explaining" it, I realized that the explanation required that f(0) = 1, and questioned whether this is necessarily true. I even said, "oh, you could just scale it". This thought of scaling is not correct, since (c*f(a))(c*f(b)) = c^2 f(a + b), so the lhs and rhs doesn't scale the same way.
- At 27:48, Sam points out that the constant function f(x) = 0 doesn't work, which I misread as asking about whether f(x) = 0 for some x (sorry Sam!).
- Crispin points out that f(0) must be 1 because f(x) = f(x + 0) = f(x)*f(0). This is true, as long as f(x) does not equal 0 everywhere, but at that point, I was still weirdly blind to that edge case and enthusiastically accepted this as a reason that the question was originally graded correctly.
- At 49:07, more of the discussion that has been happening on twitter is brought to screen, where Eric correctly points out that Crispin's proof doesn't work for the constant function f(x) = 0, which Sam had said all along much earlier and I misread.


Thanks for the good discussion, and the tolerance of some live befuddlement.

[VadimKudim]

As a non-native English speaker, I'm really grateful that someone finally explained what WTF means.

[microhoarray]

When he wrote Wtf = Whats the function
I waited for him to laugh (cause i did) bu he didn’t. This guy is good on camera

[halyon]

37:22 Grant: π² is same as g

Me: satisfied engineer noises

[OAmus]

PLEASE KEEP THESE LECTURES COMING! They're wonderful and are demistifying a lot for me - way past highschool :)

[Harsh-Singh-.]

I think " Maths for the Curious " is a better title for this series than 'Highschool maths'.

As I don't know if this lecture would specifically help in highschool exams etc.
But I think it is for anyone who is curious and wants to learn math with a little more intuition and creativity.
For example: I work in theoretical physics, and have already learnt lots of Maths. Still going back and understanding math fundamentals in this beautiful way accentuates my understanding.
Which is why I believe that " Maths for the Curious " is a better title, age no bar.

[davidgustavsson]

The memory rule I learned for the digits of e is
{Everyone knows how it starts}{Ibsen's birth year}{Again}{The angles of a right isosceles triangle}
2.7 1828 1828 459045

[peeyushkelkar]

Grant: *drinks water
Me: Write that down!

[raedinsmore]

His excitement is contagious. And seeing him get flustered because he was live and didn't want to say the wrong thing makes me feel better about how I freeze up or get flustered. We're all human!

[itays]

I remember a couple years ago, during a particularly boring physics lesson, I messed around with my calculator and typed the sum over the factorials to see what i would get. To my surprise, i found out that the answer was 2.718... so i thought to myself "wait, it can't be", i took the ln of it and sure enough it was 1. I was blown away, and i thought i found a secret way to calculate e. Later I took calc 1 and discovered that i discovered nothing new, but still, that feeling when I accidentally stumbled upon this formula for e was really something else, and throughout the last 2 years of my bachelor's math degree, I only ever felt that way once again, the feeling of pride that i discovered something new and beautiful.

[JustTIEriffic]

Video Timeline
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
Water drinks at 0:17:10 & 0:27:45 & 0:40:05
Edits: Moved water drinks to the bottom, spelling errors, these timestamps should be for after the video is trimmed at "Welcome!"

[WriteWordsMakeMagic]

I love how you just deadpan the WTF acronym

[chandrikadevib]

What I admire about you, is how you try to teach from a perspective of someone who doesn't know it. Most often, teachers forget how they felt while learning it and it becomes harder for them to explain than it was to learn.

[stewartmoore]

Genuine question: Why aren't we taught this concept intuitively at institutions we literally pay thousands for each year? Why is it that we have to come to a free source to learn these things deeply?

[bayleev]

In the proof of the necessity that f(0) = 1, the hole in the logic was after the step f(0)f(x) = f(x). This results in f(0) = 1 if and only if f(x) is not zero (so that we don't have to divide by 0). This means that, while true that f(x) = 0 is an exception, it's also the only exception.

Also at 24:03, where you say that you could scale the function to get a different result for f(0), that wouldn't work because it would no longer satisfy f(x+y)=f(x)f(y). Multiplying the two outputs would result in your scaling factor squared on the right hand side, while the left hand side would only have a single scaling factor.

With that said, I really enjoyed this lecture! Can't wait for the next one :)

[nobodysfool]

I’m a middle-aged financial engineer and learn from your lectures— and I was definitely paying attention to math classes in high school. Watching your videos and the beautiful new perspective you cast on sometimes elementary topics is like re-watching a classic movie or re-reading a classic novel and getting whole new appreciation for the material as if you were reading it for the very first time.

[capilover]

I don't know why, but this lecture seemed too short to me, even though it was only 10 minutes shorter than the previous ones. Apparently, I've become addicted to your awesome videos, Grant! Well done!

[gladiusilluminatus]

I just want to say that this series is absolutely awesome. I never thought of myself as a math person much less a math nerd and yet I just sat here for hours and have learned and understood so many things I never grasped before so Thank you! I really wish for this to become a regular thing even after the world returns to "normal". Would I not already be a Patreon this series would have definitely earned it. As a student of I often struggle to understand math and as of now firmly belong in the category of didn't understand it but just went along with it as well as I could but you are really changing my perspective on maths and are showing me that instead of frustrating it can be fun and interesting.


EDIT:
Okay it seems I accidentally canceled my Patreon subscription last time I cleared up my payments! This grave mistake has been remedied. So in a way this video DID earn my subscription (again).

[onlycheeseextracheese]

I'm a mechanical engineering graduate who's in the midst of studying for my math subject GRE in the hopes of pursuing academia and go back to grad school for pure mathematics. Your videos have not only helped me throughout the last year with grasping abstract ideas conceptually, but you've also helped me gain a whole new appreciation for mathematics as a whole. Thank you sir. You are a treasure to this world.

[uhgs]

OMG you just answered me a question wich confused me for 2 years now and I already gave up to understand why e^i*pi should make sense. It's because it's actually not e^i*pi but exp(i*pi) where exp(x) not necesseraly is equal to the thing we have in mind when we see e^x. THANK YOU!