Can the power of two irrationals be rational?

Are there two irrational numbers, let’s say x and y, such that x to the y is rational? The answer to this question brings us through a fascinating journey about transcendental numbers and the Gelfond-Schneider theorem.
Additional Resources:
Where I found this theorem:
math.stackexchange.com/questi...
Music: www.purple-planet.com
Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
github.com/3b1b/manim

[@jHan]

This is a shorter video that I made to fill the gaps between my longer projects. I'd probably do one or two of these 2-4 minute "shorts" between every project, since there's so much math I want to share that unfortunately would take too long to devote an entire >10 minute video. Anyways, I hope you enjoyed, and again thank your so much for 1000 subscribers.

[@raifegeozay687]

0:00 obviusly yes, e^In2=2

[@morveman3564]

The hard part is to prove e and ln2 are irrational (e is quite easy with the right idea but I don’t see one for ln2 tho) but you’re right lol

[@KrasBadan]

This is amazing, I never knew that √2^√2 is transcendental.

[@Farzriyaz]

But (√2^√2)^√2 isn't.
1. Use one of the Rules of Exponents to manipulate (√2^√2)^√2 and turn it into √2^(√2*√2).
2. Multiply √2 by √2 to get 2 and turn √2^(√2*√2) into √2^2.
3. Square √2 and turn √2^2 into 2.
(√2^√2)^√2=2

[@electricboogaloo11]

small point on 2:16
Not sure I'd call 'i' irrational, sure it can't be expressed as a ratio of two integers but so is a margarita pizza, doesn't mean that it's an irrational number.
Irrationals are typically defined as reals that can't be expressed as a ratio of two integers.
Great video btw

[@jHan]

You are absolutely right! To clarify, I used the term "irrational" in the context of the Gelfond-Schneider theorem, where -i would be considered not rational. I realize that could be a bit confusing, so thanks for pointing it out!

[@seamusfinnerty5897]

a margarita pizza can absolutely be expressed as a ratio of two integers, 2pizza/9. of course we all know pizza is the only integer that does not lie on the number line or complex plane

[@canyoupoop]

​@@seamusfinnerty5897Pizza belongs to pizza box. Pizza box is also a better version of field, it's a hut

[@gm90rt24]

Continue in this channel mate you are a great explainer 💙we support you

[@leandrobacelar7557]

Excellent video, good job!!

[@darthvitiate7373]

nice video keep up the good work and you will massively grow

[@laxminarayanbhandari855]

You give me 3b1b vibes. Nice video. ❤️

[@darthvitiate7373]

ye with the Euler and the style of explaining with those drawings

[@culater]

Love it !

[@johnzinhoinhoinho]

Great content, please explain about Fourier series

[@boldandbrash2499]

So professional!

[@geraltofrivia9424]

Very nice

[@GeoffryGifari]

if x and y are transcendental, can xʸ be algebraic?

[@tipeg8841]

Yes e^ln(2) = 2

[@shreymehta4973]

an absolute legend

[@kliqx]

The content is obviously amazing, but - and I mean to say this as respectfully as I can - I feel the music is too... "upbeat", I guess, for it. The more chill, slower, quieter stuff that 3b1b has playing in the background fits it better imo.

[@peace7784]

Both are good tbh

[@shreymehta4973]

ayo youtube monetize this man

[@mahatmaniggandhi2898]

e^ln2

[@shreymehta4973]

newton who???

[@srinidhir2188]

Sir upload more questions
Plz...