Are they irrational? Transcendental? | Epic Math Time

Showing that a number is transcendental can be difficult. While π and e have a deep connection involving exponentiation, other combinations of them, like π + e, are not as well understood.

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Definitions:

Algebraic number: A number that is a root of some polynomial with integer coefficients. Examples include 42 (a root of x - 42), √2 (a root of x² - 2), φ (The Golden Ratio, a root of x² - x - 1).

All rational numbers r = p/q are algebraic, because they are always the root of the equation qx - p. Irrational numbers can also be algebraic, such as the example of √2 above.

Transcendental number: A number that is not algebraic. Examples include π, e, cos(1), etc.

The usual transcendental numbers that one encounters cannot be expressed in terms of finitely many operations on integers. Practically speaking, this is why they often get their own symbol and name, like π, as any other way to express them (such as an infinite sum of rational numbers) is cumbersome.

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[adrianparism]

1) x²-(e+π)x+eπ is a polynomial with roots e and π. If e+π and eπ were both rational, e and π would be too.
2) As e^(iπ) is rational different from 0 and 1, and -i is irrational and algebraic, (e^(iπ))^(-i)=e^π is transcendental.
3) Not in the top of my head. I'll work on it.

[PapaFlammy]

I really like the animations on the numbers at the beginning! :D

[Schieber]

What's also neat is that the argument showing that either πe or π + e must be transcendental seems to work for any two transcendental numbers, not just π and e!

[jackhandma]

A truly marvelous proof that can fit on a small margin.

[judgejudy]

I love the content you produce! Stumbling on your channel feels like I’ve found a goldmine. Entertaining and informative. Awesome work! Thank you.

[michaeljburt]

Love the math as per usual. But can I just comment that the presentation of everything here is so cool. The music is cool, the game-style practice problems are also so cool. Kid me would've eaten this up (who am I kidding, adult me is eating this up)

[ultrio]

While it is difficult trying to figure out whether πe is transcendental, it is trivial to prove πe is, in fact, delicious

[TheMathCoach]

Great video, really liked the pacing and I can only imagine all the work you put in to make the animations! Thank you for taking the time and making such great math content :)

[rc]

Excellent video, high quality production Sir!

[altuber_athlete]

The name of this channel really fits the videos. They are epic

[aashsyed]

is the channel member at 0:12 clicked the membership button?

[conorbrennan]

You make excellent videos and are well thought out!
May I ask what equipment is it you use to draw on this invisible wall?

[petevenuti]

Backward writing, is that one of your talents or did it take years of practice?

[NikolajKuntner]

I like the color coding in red, when a proposition is concluded to be in contradiction to the axioms in our framework.

[mathOgenius]

Cool thing is learned today (a+b)^2 when a and b are irrational works the same lol 🤣🤣

[jikaikas]

3:17 Me an engineering student : Witchcraft

[robsbackyardastrophotograp]

Happy Festivus!

For the second exercise, I used the cyclic nature of exponents of i, namely i^4=1 and i^5=i etc. e^π is nothing but (e^iπ)^(i^3). Since i^3 is -i, we can rewrite as (e^iπ)^-i. From Euler's identity, e^iπ=-1 and is thus algebraic. -i is irrational under the definition provided in context of the Gelfond-Schneider Theorem. Thus we are raising an algebraic number to an irrational power, which is transcendental under Gelfond-Schneider.

Is there some principal value type gimmick in the background for which using cycles of powers of i in this context "breaks down" somewhere?

[MyAce]

Meanwhile uncomputable numbers out here making up the majority of R, but we barely know of the existence of any and by definition can't approximate any of them

[nicolasdellano]

Damn, everyday u are getting more epic

[Ryan-gqji]

Hey dude, you think you can make a video for the proof of the Taylor series?