What is 2^π?

Figuring out what 2^π means

Resources on exponentiation:

Resources about the exponent algorithms:

Corrections:

[JayTemple]

On a broader version of this general idea, I remember my dad explaining that my calculator doesn't have trig tables in it. (Yes, I'm old enough to remember looking for trig values and logarithms in a table.) Once we covered power series in calculus class, it made sense, but there was a long gap before I made the connection between those and irrational exponents.

[Taric]

It's actually amazing the algorithmic optimizations from computer science and then the floating point and circuitry optimizations from computer engineering that make calculating a number like this so accessible and inexpensive for anyone to do.

[GMPranav]

π: *exists
e: Imma sneak into the video as well

[idkjustleavemebeplease]

Dude. You single handedly made fractional exponents make so much more sense in five seconds than high school and college. 1:00

[rail]

2:08 man jumped from concepts middle schoolers that have experience with exponents to college level calculus real quick

[natejack]

In my real analysis class last semester I remember talking about definitions of real numbers to the power of rational numbers. I remember asking "what about 3^pi or pi^pi?" Prof replied "what about it?". I replied, "I have no idea what that means". Lead to this conversation. First time I've ever felt smart for saying I don't know anything about a topic

[em.]

*groan* Can't BELIEVE nobody else has pointed out this obvious error to you yet, but the fourth "2" in your thumbnail is CLEARLY not accurate down to the full ratio! You need to enable transcendental mode on your editor so that it can render the length infinitely precisely. Beginner mistake!

[somerandomcsgoplayerlol]

thanks for posting videos like this. as a 14 year old sophomore, i shouldn't normally be interested in topics like these, but thanks to the ease of access it is made possible. very good explanation, i understood most of it, some of it i didnt, and that's on my part since I literally still am in high school math

[WEAVER]

Great presentation and conceptual delivery, I Can hardly wait to see what's brewing beneath the surface!

[Mutual_Information]

Excellent vid. Defining the exponential function as an infinite series seems to frequently be the way forward. It appears in a few theorems regarding the moments of the exponential family of distribution. 3Blue1Brown also has that great vid showing how it can make sense of matrices in the exponent.

Btw, your channel is growing fast! I try to keep an eye out on all the math edu channels (I have one as well), and this one has the knowledge and animations to go places. Well done - im sure you’ll crush it

[carlwindhorst]

Keep the great videos coming. Your channel has the potential to grow exponentially!

[edwinvlasics]

I'd argue that it's probably not better to call the integration version the "definition" of exponentiation (at least for real numbers). Just because it's computationally faster, doesn't make it a better definition. This is why nobody defines multiplication with Harvey and Van Der Hoeven's O(n log n) multiplication algorithm even though it's technically faster.

Edit: There's a lot of debate in the replies, and I think it all stems from different objectives, which is why we have different characterizations. My above comment naively repeated the definition I first learned for real numbers without further considerations as to it's advantages/disadvantages, which I now realize was presented first because it was the first definition that could be understood--as in a real analysis course, integration and power series are covered rather late. I'd argue that this definition is more natural for the real numbers, as real numbers themselves are sort of just limits of rational numbers (by one characterization of the real numbers, at least) and this is how many other operations are defined for the real numbers. The problem of well-definedness is a non-issue to me because the night after I wrote this comment I derived a relatively simple proof of well-definedness--one which would convince a fellow undergrad, but is really just a very standard proof, which might be delegated to the exercises section of a beginner analysis text.
The definition presented in the video definitely has no questions of well-definedness, however exponent laws may cause trouble (and before you note that the exponent laws were proven by SackVideo in the replies, it's rather hard to prove that for f:R->R such that f(x) = e^xe^y/e^(x+y), its derivative f'(x) = (e^xe^ye^(x+y) - e^xe^ye^(x+y)/(e^(x+y))^2 without using any exponent laws, which would result in circular reasoning). The definition in the video has the advantage to generalizing to many fields, including the noteworthy complex numbers. This is probably why beyond real analysis it's used more often.
For the question of which can prove more stuff... it's up to what you're proving. That's why we have different characterizations after all.
Hope i didn't miss anything or mess anything up

[toasteduranium]

I love your channel. I found it yesterday, and I hope you make more videos.

[martinepstein]

Great video. For me, the main selling point of defining exp(x) via series is how there's no issue with plugging in complex numbers. Also it makes things like matrix exponentials seem very natural.

Would you really say a calculator computes ln(x) by quadrature of 1/x? ln(x) has nice series representations too.

[ffff-odjb]

Amazing video, most explanations and steps were simple and easy to follow. Even though I don't understand each step, I still learned quite a bit.

[ImKinoNichtSabbeln]

(-2)^3 = exp ( 3 × ln(-2) ) = -8

Logarithms of negative numbers [e.g. ln(-2)] are "imaginary" (i.e. no "real" numbers), but well-defined, and extremely useful and important.

[skun]

Let's have a moment to appreciate how great the continuous functions are!

[miloweising]

Good video but from a bare bones stand point it is perfectly fine as a definition of exponents (with positive base) to use limits of rational exponents. This is what many real analysis books do. It is much easier to show that this gives a well defined notion of exponents than it is to develop Riemann integrals and the theory of Taylor series. For calculus students though this is a good explanation because they already have intuition for series and integrals but may have not seen the pieces come together like this before.

[willlowtree]

i was just thinking about the actual definition of logarithms recently!!! brilliant video, great delivery and very simple at the same time<3

[hovedgadegaming]

I learned surprisingly a lot during these short 5 minutes.