@Another Random Cuber Depends if you're making a legitimate calculation or just back of the envelope. For sure back of the envelope engineers do exactly this, because really you're just looking for order of magnitude. On a real calculation the engineer will use the exact value + 20% for safety, plus an extra 20% for safety on the final result of the calculation!

😂👌

Well, that looks like an approximation that would cause a lot of errors

@@hatasesasd9631 , its joke

... and 30 is "close enough" to 42.

LOL. I have heard so many different nicknames for my mic and "thermal detonator" is so funny!

In case the proof doesn't work.

Because 50 thousand, no less

OOOOH its a mike

Exactly Pi hi exactly Tau /2

japeking1 mysteries of life....it’s enjoyable to see a thought a human mind solve a non trivial question.....logically....its bueatiful actually ..like the Mona Lisa painting...

Is the blank space... is ok I guess...

Lies again? EPA EPI

I'm seeing it at 1, four years later

Beautiful!!!

Jonathan Ma loved it. I must congratulate you on this beautiful proof

Amazing friend I have a doubt how x^(1÷x) can't be zero. If we put x=0 we get 0^infinity and we know 0^(any number) is zero. 😕😕

sonu shaw x^infinity doesn't have an answer.

sonu shaw Watch this video. It doesn't explain your question exactly, but the thinking process is in the correct place: kzfaq.info/get/bejne/eLiCos98udKqZoU.html

Very nice!!

Essence of Abstract analysis.

I was going to write same thing bro

Goddamnit

It was a piece of cake for sure!

r/iamverrysmart

SpeedyMicherGD r/whooosh

@@castroploiin r/ihavereddit

Yay!!

I am glad to hear! thank you

Engineers be like

Funny thing: in astronomy this is actually correct, you're never concered about what's behind the decimal symbol but rather about the magnitude of the result. So 3*3=pi*pi=10

@@alexbarac 10?

@@alexbarac also, its 3^3, not 3*3. 3*3=9, 3^3=27

@@methatis3013 Yeah, it's an accepted approximation. In many cases you tend to get out of the equations pi*pi and to simplify the calculations, you just aproximate it to 10.

e = 2 = 3 = pi

TheMichus Wow!! Ok!!!!! Maybe Bc of the music??

@@blackpenredpen I think this might be the case

TheMichus : )

alkankondo89 thank you for finding my video enjoyable. I also have another playlist called the "math for fun" you can check that out too

@@blackpenredpen Sir how would you even have known that the maximum value of x^(1/x) is e^(1/e)? It is clear when we take the derivative but honestly I don’t see how anyone would even have thought of doing this in the first place to solve this problem.

@@captainhd9741 we have 2 statements as e^(1/e) and π^(1/π) and we want to find which one is bigger. İn those statements, different values are e and π so we just write x for them. Than we can make the graph of the x^(1/x) function and see the results. Or try to prove as he did in the video. We wrote x for them mainly because we don't want to calculate the result, rather we want to prove that one is bigger than the other. Also if we have a proper function, it's sometimes easier to find the extremum points rather then calculating.

Nice pfp

Sad Prof.

It's a JEE ADVANCED Question.

Project Overturn aka RareBeeph apple pie^e is bigger

apple

That doesn't make any sense in this world , how can u say that 2.7^3.1 >3.2^2.8 without calculator

@@theimprudentman7272 just simply calculate it in the head !!!

You were a computer? How did you become a human?

Joel c.l what if he is a cat, not a hooman?

Mark Zuckerberg was an android, so you are in good company.

r/Not_cheating

Now are you human?

No need to take double log, take single log and one no. Will be negative other will be positive , the positive one will be greater

My pleasure!!

as long as x > 0

To expand on zyrohnmng's comment: Need to avoid using negative numbers in such an example, because a negative number raised to a non-integer power is highly problematic, and almost always, non-real & multi-valued. And complex numbers do not have ordering.

That, and during his proof, he made the assumption that x was positive on multiple occasions for the very reason you just gave.

True -- I should have said ALL *non-negative* x except x=e, and not used -420000 as an example! (Note to +zyrohnmng: the relation holds ok for x=0)

The proof doesn't work for x=0 since the function and its derivative is undefined for x=0. You can check the case separately.

Where c=1, mind you.

Thank you! The work will be very similar. The ones with base e will win. I think I can work out e^phi vs. phi^e for fun : )

blackpenredpen thank you for you reply Sir, but why are you so late in reply? I've commented on you "sine of 18 degree" video about a week earlier, I asked a question in that comment, but you havn't replied me,

مشكور جدا حل ممتاز

Muhammad Hussain Sarhandi lol

@@divyoroy9056 what happened?

yes, and it was generalized in the video using the function x^(1/x). a^b > b^a iff e = a > b

very impressive!... but this is just devine inspiration. I find his systematic approach a lot more useful.

thats true

Equation isn't correct for x=0 ?

≥

Hi sitangshu sekhar bhattacharjee

How’re u doing now?

we notice that x^(1/x) falls faster from its maximum for x < e than for x > e, i.e., x^(1/x) is flatter right of e in case (3a) above, i.e., when x < e < y, can we show x^y < y^x when the x distance to the max (e) is bigger than the y distance to e, i.e., e-x > y-e, i.e.: 3a: if x

Indranath Mukherjee thanks!!!

Yah but it's hard to predict between (1.23)^e and e^1.23? By intuition .. Calculus is crazy....it was created with intuition and it give solutions to problems which are not at all intuitive....

This is why you can't assume anything in math! Appearances are often deceiving

@Sulayman Hussain the reason why implicit differentiation works is not that hard. It basically works because of the chain rule, keep in the back of your mind that y is a function of x ;)

Dy/dx (y) = d/dy(y) x (dy/dx) = 1 dy/dx = dy/dx You have always been implicit differentiating and you never realised. Every time you differentiate a y=___ equation. The LHS goes to dy/dx because of this reason.

Have a look at the essence of calculus series on the channel 3blue1brown

@Sulayman Hussain what you are saying is true. The fact that we can simplify differentials is not well defined in cal1 and cal2. With teachers telling us to pretend differentials as fractions. We don't get any concrete proof why until real analysis.

The Epic Gamer I hate that ad too...

Brianna Baker waaaay harder? Please make it waaaaay easier for us. Apple sauce.

Okay I'll be honest, I forgot that using the exponential rule doesn't work for functions of base x, which is where I thought the majority of the optimization would come from. Still, it can be done without implicit derivation though apart from the first step this is fundamentally the same process you took. I guess I just hate doing implicit derivation unless it is absolutely necessary. Anyways here's the steps for those interested: imgur.com/a/sghux

-_-...... I was expecting something waaaaaay easier.......

easier solution: e^pi>pi^e ln(e^pi)>ln(pi^e) pi>e*ln(pi) pi-e*ln(pi)>0 let f(x)=x-e*ln(x). we want to prove f(pi)>0: f'(x) = 1-e/x so for (x > e) we have f'(x)>0 so we have f(x) increasing and f(e)=0 which means f(pi)>0. which means e^pi>pi^e

blackpenredpen... since you were expecting a derivative waaaaay easier, here it is: derivatives are just rules you memorise... like the product rule, similarly there is a power-exponent rule. For the product u*v the rule says "derivate u*v keeping u constant, plus derivate u*v keeping v constant". To derivate u^v "derivate u^v keeping u constant (exponential) plus derivate u^v keeping v constant (power)". Therefore d(u^v)/dx= u^v*Ln(u)*dv/dx + v*u^(v-1)*du/dx Using this rule (which is analogous to the multiplication rule) you can write the derivative in just one step. :-)

well I think it's obvious since 1/e and 1/pi are positive they are both greater then 1

Destroctive Blade neither 1/pi nor 1/e are greater than 1.

of course they aren't but 1/pi and 1/e are positive so e^(1/e)and pi^(1/pi) are greater then 1

this is what you want to prove, right? My initial comment was aiming to the fact that the proof was incomplete, that's all.

yeah sure I just said that it is obvious

Man, 3^3 is 27.😅

E is approximately 2.71828 whereas pi is approximately 3.14159

S.H. Kaifee and if you’re an engineer you round to the closest whole number and both e and π is 3 to their closest while numbers

Ohhh Getting smarter every day I didn't know that fact about engineers Thanks @@lucifersdevilishdetails.

Evil MrMuffinz thank you!!

Thank you~! = )

Same this guy is amazing. Does mathematical hocus pocus to make hard problems easy.

+blackpenredpen was there a context like this in Stewart's Calculus?

Hmm, I am not sure..

+blackpenredpen it's okay! Anyway I hope you could do Calc III videos. Just a suggestion though.

Simon Ruszczak The problem is, the sizes do matter. If you, for example, have 3^2 vs 2^3, 3^2 is greater. This is the simplest example without using zeroes and ones (which would naturally mess things up), and it goes against the rule.

wouldn't 3^2 being bigger show that π^e be bigger ( pi~3 and e~2.6), which was proved otherwise?

e^pi < pi^epi, so what? Where do you go from there?

ln(pi) > 1

The natural log of pi is greater than 1 tho.

gotta love physicists

quik maffs

Mr. American man that’s a bad estimate 😂

but e^pi is greater

Wow same method bro, but wrong estimation, it supposed to be e = 2.5 π = 3

That was e'asy pieasy

yea, you can do that so long as you leave the dy/dx at the end.

Yes since you know that y is a function of x you evaluate it as such. By using dy/dx

Yes. It's called *implicit differentiation* because the derivative of y is derived implicitly.

Peter Bočan by using chain rule, we have: (d/dx)(ln y) = (d/dy)(dy/dx)(ln y) = (d(ln y)/dy)(dy/dx) = (1/y)(dy/dx) isn't it?

ENGINEER DETECTED

that wouldn't have worked, since 2^3>(pi-3)

La Tortue PGM LOL That's exactly how I guessed my answer (I got it right using a slightly better estimate). And I am an engineer.

well, I did it for 2.7^3.1 and 3.1^2.7 and it worked

btw mathematical mathematics memes is the best fb group ever

Thank you!

JiveDadson but I have more KZfaq views tho! ：）

JiveDadson Fucking rekt

Matheus Cardoso Thanks. I have been a teacher for many years already.

You can prove that pi lies between 3 and 4, by bounding the number with a hexagon on the inside of a circle with a diameter of 1 unit, and a square on the outside. The distance across opposite vertices of the hexagon equals the diameter of the circle. The side length of the square equals the diameter of the circle. The square's perimeter is 4 units, and the hexagon's perimeter is 3 units. This means pi is between 3 and 4. You can prove that e is between 2 and 3, by using the infinite series expansion that defines e e = sum of 1/k! from k=0 to infinity = 1/0! + 1/1! +1/2! +1/3! etc The first two terms are positive 1, and when added together equal 2. Since the remaining terms are all positive, this gives us 2 as the lower bound. To prove that e 2*2 for 3, 2*3*4 for e < 2*2*2 for 3. This means every subsequent term after 1/1! will be less in the series expansion for e, and therefore add up to less than 3. This shows that e is

Sorry... I am not teaching that anytime soon.