e^ Pi vs Pi^e: which is bigger? II A Pre Pi-day battle. (visual proof; calculus)

Thanks!

This is a short, animated visual proof demonstrating which of the values e^pi or pi^e is larger (where pi is the circle constant and e is Euler's constant) . #math​ #inequality #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #calculus #mathshorts​ #mathvideo​ #mtbos​ #euler #pi #piday

For two visual proofs of a more general version of this inequality, see
and

To learn more about animating with manim, check out:
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Music in this video:
Meeting Again by Emily A. Sprague

[mathflipped]

Great neat proof! Well done, as always.

[RigorousIgor]

This is downright perfect! Thnak you for such a powerful visual and a straightforward derivation. My own epxerience with that fact was a bit vague: I was introduced to the result via the function x^(1/x), which is a weird function with its maximum occurring at x=e. Therefore, for any x, x^(1/x)<=e^{1/e), and if we rewrite e^pi and pi^e as having the identical exponents, we would get [e^(1/e)]^(e*pi) and [pi^(1/pi)]^(e*pi), from which we see that both expressions assume the functional form of x^(1/x), and hence e^pi>pi^e. But your result is much more intuitive, as most people get introduced to the function 1/x and the fact that it is a derivative of ln(x).

[bikashchakraborty]

I feel happy to see this! And many thanks for considering this one!

[asparkdeity]

Never seen a proof this simple, simply amazing!

[Ninja]

This proof was a lot easier to understand than the usual approach I’ve seen. Although, you should drop the equal sign in you inequality since the area under the curve is strictly less than the rectangle due to the decreasing trend.

That approach uses the function x^(1/x) and showing its maximum is at x=e using differentiation. Thus e^(1/e)>pi^(1/pi), which means e^pi>pi^e taking both sides to the power of e*pi.

[RafaelCouto]

Really nice to spice some calculus on the vid. In my head I imediatelly thought:
Pi > e then e^Pi must be bigger since that's the case for all numbers ...
there's power in the powers 😅but great vid as always!

[thegamesuniverse]

Amazing, great explanation. Thank you. ❤

[ceromat]

This version is different: April 27, 2024

[MichaelRothwell]

Amazing proof! This proof can easily be adapted to show e^x>x^e for any x>0, x≠e, equivalent to showing x^(1/x) has greatest value at x=e. First time I've seen a maximum found using integration!

[davidbrisbane]

e^x > 1 + x, where x > 0.
So, e^(π/e - 1) > 1 + (π/e - 1) = π/e, as π > e
So [e^(π/e)]/e > π/e
So e^(π/e) > π
So, e^π > π^e.

[likeaduck]

Wow wonderful !! my teacher proved it using monotonicity of x^1/x but i loved to see a new method and this is easier .

[tanishqbarsaiya]

I used aprox technique where I took e =2 and π=3 and it will give its most approximate answer and It will give a intusion that it will goes this way

[SunilMeena-doxn]

I used maxima of a function to solve it.
Assume e^π > π^e.
Means (log(e)/e)>(log(π)/π).
So we just to analyze logx/x.
It's derivative is (1-log(x))/(x^2).
It's +ve till x=e, then -ve.
So it's value at e is highest.
Hence proved.

[HareKrsnaRama]

Amazing proof, you are best, but in last we have an inequality, assuming pi and e are constant, is pi^e smaller than e^pi or they are equal?

[eddiechan]

can proof x^y > y^x if y>x similarily ?

[indarapusridhar]

But how can we say equality is not possible

[MultiDruba]

How don't understand how comes the equal to symbol in the inequality, as when the area under the curve is strictly less than the rectangle .

[sandanth]

A simple explanation:
pi=3.141...
euler's number=2.717...

If compared to one other "which is bigger" scenario, 2³ or 3²
pi is closer to 3
e is *almost* close to 3, but it's still in the range of 2 and 3

In the 2³ or 3² scenario, 3² is bigger, while 2³ is just 1 away from being the same

In assumption, pi^e is bigger than e^pi
Problem Solved.