i-th root of i

Рет қаралды 561,944

6 жыл бұрын

We will solve the most complex math question, i.e. evaluating the principal value of the i-th root of i, namely i^(1/i). In fact, the answer is actually a real number!
Do you enjoy complex numbers and up for a challenge? If so, check out x^x=i • they never teach equat...
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#blackpenredpen #math #calculus #apcalculus

Пікірлер: 603

@blackpenredpen 9 ай бұрын

Do you enjoy complex numbers and up for a challenge? If so, check out x^x=i kzfaq.info/get/bejne/rKmUdst10rO9fGQ.html

@vincentkingsdale8334 7 ай бұрын

How is this practically applied to real life????

@Unlimit42 Ай бұрын

You can't solve subfactorial of i.

@pseudotenshi3431 3 жыл бұрын

The most surprising thing about this is I understood more than half of what he explained

@thekiwininjasmusic4928 2 жыл бұрын

I agree

@findystonerush9339 Жыл бұрын

I understood it all!

@poodle101 9 ай бұрын

thats because hes a good teacher

@RK-qi4tv 8 ай бұрын

Yeah and I'm terrible at maths.

@michaelliu8887 4 жыл бұрын

*Learns about imaginary numbers* KZfaq: What's 1/i? What's e^i? What's i^i? What's sqrt(i)? What's log(i)? What's i-th root of i? What's sin i? What's sin-1(2)? What's i factorial? What's integral of x^i?

@blackpenredpen 4 жыл бұрын

刘颢云 lolll

@Ramu-10 3 жыл бұрын

Ya got lucky... I haven't even learned about logarithms and have no clue what e is...

@papaganush420 3 жыл бұрын

@@Ramu-10 you gotta love when you figure out whats going on

@valeriolocatelli665 3 жыл бұрын

@@Ramu-10 it is euler's number, which approxamatly is 2.71.......

@nuzlock4481 3 жыл бұрын

@@Ramu-10 then watch this amazing video about "e" and it's derivation. kzfaq.info/get/bejne/o5h9fNN116ybdac.html

@hyperpsych6483 5 жыл бұрын

0:32 "I don't like to be on the bottom, I like to be on the top" Me: *scrolls down to comments to see if anyone made a joke about that*

@hendrik5919 5 жыл бұрын

A joke would be like what?

@TheLifeofJay1 5 жыл бұрын

Super Psych that is literally exactly what I just did 😂😂😂 you made my day

@paulgoogol2652 4 жыл бұрын

I assume the top would be the video section, while the bottom is the comment section.

@rogerkearns8094 4 жыл бұрын

In the video it's already meant to be a joke.

@kabochaVA 4 жыл бұрын

Same here... 0:32 "That's what she said!" xD

@dallasboringnews7157 4 жыл бұрын

"i" understand this a whole lot better now.

@kazifarhadulhaque686 4 жыл бұрын

😂😂😂

@randomavenger3048 4 жыл бұрын

lmao

@shalomirewole-ojo5335 3 жыл бұрын

😂

@bedbyashkhadagray3265 3 жыл бұрын

@Shri Hari's Animations it's not comedy it's komedi

@stephenfiore9960 3 жыл бұрын

“i” get it, even though the “roots” of the ha”i”r on my head are gray

@willy3506 6 жыл бұрын

Do the e-th root of i and the i-th root of e

@lyrimetacurl0 5 жыл бұрын

The second one is easy:- e^(i*pi) = -1 ith root of e = e^(1/i) So (ith root of e)^(pi*i^2) = -1 So (ith root of e)^-pi = -1 Flip over fraction So (ith root of e)^pi = -1 We know e^(i*pi) = -1 So (ith root of e) = e^i That's a distance of 1 radian around the unit circle So (ith root of e) = cos(1)+i*sin(1) Or about 0.54030 + 0.84147i

@anselmschueler 5 жыл бұрын

just go on wolfram|alpha

@camerongray7767 5 жыл бұрын

Lyri Metacurl Lol

@dnpendown3199 5 жыл бұрын

@@anselmschueler wtf

@anselmschueler 5 жыл бұрын

@@dnpendown3199 Or use Qalculate, the best desktop and command line calculator available! > root(i,e), root(e,i) [root(i, e), root(e, i)] = approx. [0.8376315 + 0.54623573i, 0.54030231 - 0.84147098i]

@austinlincoln3414 3 жыл бұрын

“I dont like to be on the top. I like to be on the bottom” - blackpenredpen

@arinroday302 3 жыл бұрын

lmao

@favourrege 3 жыл бұрын

Kinky

@obinnanwakwue5735 6 жыл бұрын

Wow, complex roots of imaginary numbers are real......math is so weird!

@ffggddss 6 жыл бұрын

Sometimes they are, but not usually.

@stranger0152 6 жыл бұрын

But bro real root of real numbers is also imaginary. For example square root of -1 😂

@gaeb-hd4lf 4 жыл бұрын

@Smash Boy Yeah, technically speaking Complex numbers are just an algebraic estructure made from real numbers (just like matrices for example) so it makes sense that if you apply a function to an "imaginary" number (a number on the Y axis of the complex plane) it gives you a "real" number (a number on the X axis of the complex plane).

@philiptymon3380 4 жыл бұрын

because “One of the miseries of life is that everybody names things a little bit wrong.” - Richard Feynman 1985.

@saswatsarangi6669 4 жыл бұрын

Purely imaginary in this case, imaginary root of imaginary...

@cangursoy 6 жыл бұрын

I just stumbled upon your videos while procrastinating. You're a really warm and enthusiastic talker, keep it up sir! :)

@Rose-vb4wk 6 жыл бұрын

Or you could just i = e^(iπ/2) ith root both sides i√i = e^(π/2)

@arthurmoiret6076 3 жыл бұрын

+2kπ ?

@dns911 3 жыл бұрын

@@arthurmoiret6076 That's the point when he said there are infinitely many solutions to this

@adityasharma2380 3 жыл бұрын

I did that too!!!!!

@diegocabrales 3 жыл бұрын

@@arthurmoiret6076 or -2kπ

@tracychacon1210 10 ай бұрын

This video is amazing. Thank you for the break down!

@trip_on_earth 6 жыл бұрын

I'm in love with mathematics now

@D1M1T 3 жыл бұрын

I was in love with math. Then they said “do specialists”

@Fematika 6 жыл бұрын

The reason why i^i = e^(-pi/2) is because i^i = e^(log(i)i) = e^((pi/2)i * i) = e^(-pi / 2). There are infinite answers because the log(z) has infinite answers; as it is defined as ln |z| + arg(z)i, where arg(z) is the angle (which can be, in this case, pi/2, 5pi/2, 7pi/2, ...).

@jco6345 6 жыл бұрын

but why log(i)=(pi/2)i ?

@angelmendez-rivera351 5 жыл бұрын

J CO Because log(-1) = iπ and -1 = i^2

@suryanshkhatri7929 3 жыл бұрын

This guy rekindles our faith in the fact that things that seem to be very tough, are actually very easy, we just need to take one right step, and everything else becomes so easy!

@parthiban5663 4 жыл бұрын

Your voice is a lullaby 😭❤️

@martinhughes2637 3 жыл бұрын

This is so fantastic and beautiful. Thank you for sharing!!

@rolexmarcelo3218 3 жыл бұрын

I'm even more amazed that you can formulate these math questions with answers.

@travishayes6037 6 жыл бұрын

As mentioned before, I love this channel. Found it randomly, but so glad I did. Currently a grad student taking complex analysis (as an elective, took it 2years ago). All of your complex videos are just supplementary/review for me, but they're still very satisfying to watch. Got any plans for singularities?

@blackpenredpen 6 жыл бұрын

Travis Hayes great idea! I will discuss this with peyam bc we are filming together this Friday

@retired5548 6 жыл бұрын

in a sense,all of his videos are complex

@retired5548 6 жыл бұрын

in two,actually. at least two

@normanbraslow7902 3 жыл бұрын

I don't believe a word of what you say! You doubtless are as flummoxed as the rest of us mortals. Quit showing off!

@SoumilSahu 6 жыл бұрын

black mic red shirt yay!

@dane1234abc1 3 жыл бұрын

You are the most brilliant math person I have ever come across. I am in awe.

@japeking1 6 жыл бұрын

These just make me so cheerful..... I have to limit myself to one a day ( and then I'll start again from the beginning because I'm still forgetting stuff like when I was a kid. )

@davidkippy101 6 жыл бұрын

You could actually just start with e^(ipi)=-1. Take the square root of both sides: e^(ipi/2)=i. Then take the ith root of both sides. e^(pi/2)= i^(1/i).

@kevinpaulsen6103 5 жыл бұрын

I love it when blackpenredpen features blackpenredpen.

@brandongammon6978 5 жыл бұрын

That shirt is crazy!! Also, I love your vids ! 😄

@RizzY_RUS 4 жыл бұрын

Drink game: Take a shot after he says the letter "i"

@skullzs1983 25 күн бұрын

You'd get drunk after the first 10 seconds.

@radward7173 2 жыл бұрын

damn I didn't know about imaginary numbers more than how to write them and how to show them on complex plain. yet after watching several other videos of yours I managed to get this one by myself.

@anirudhbhalekar 6 жыл бұрын

You're explanations are simple and crisp, really helping me learn mathematics better

@decaybeats Жыл бұрын

Your*

@thibautklinger5178 3 жыл бұрын

When I saw the video I took a few minutes to think about it for myself and then come back to see if I was right and I was. Thanks I like keeping my mind sharp with little math riddles like this one.

@devjain2319 3 жыл бұрын

e, pi, and i: The Zeus Poseidon and Hades of math

@user-zy2dw3zn8m 3 жыл бұрын

Do Aid correct name of gades?

@arf101088 5 ай бұрын

such an interesting edge case!

@danialbrown4417 4 жыл бұрын

Black pen blue pen would make it easier for my eyes to read. Thank you so much for your videos.

@Vince0 6 жыл бұрын

That ball that you always keep in your hand makes you look like an ood from DW xD

@knnedmeitei3323 5 жыл бұрын

Great work. Thank u

@isavenewspapers8890 2 жыл бұрын

Euler’s number and pi, name a more iconic duo

@dhyeypatel8399 6 жыл бұрын

That was the question that i had not been able to solve in my exam thanks

@markhamclorado8391 3 жыл бұрын

same😢

@Treegrower 6 жыл бұрын

You are a math god

@rishabhdhiman9422 6 жыл бұрын

And you are invisible

@manda3dprojects966 6 жыл бұрын

Invisible, you scared me

@lPlanetarizado 5 жыл бұрын

the heck is happenin here????

@epicawesomenessnucleartrin1585 5 жыл бұрын

Hangul Choseong Filler

@DankStuff101 4 жыл бұрын

I FINALLY FOUND YOU (only intellectuals understand)

@fountainovaphilosopher8112 6 жыл бұрын

Who thinks new BPRP is more interesting and better? *raises hand*

@blackpenredpen 6 жыл бұрын

Ognjen Kovačević thank you!!!

@maxhaibara8828 6 жыл бұрын

yay!!

@blackpenredpen 6 жыл бұрын

Ognjen Kovačević u know. It's weird that even tho I teach the same way in my classes. But I have been doing different things on YT nowadays and I have been enjoying it way more also.

@maxhaibara8828 6 жыл бұрын

blackpenredpen wait you're a teacher?! I thought you're just a super smart student!!

@camerongray7767 5 жыл бұрын

I don’t even know how to begin to do this

@somyaranjanpanda4620 5 жыл бұрын

Thank you sir🙏🙏🙏

@flamingpaper7751 6 жыл бұрын

What would an ith root(x)graph look like?

@harrisonoberg8085 2 жыл бұрын

i blinked and he got pi out of seemingly nowhere

@cleiven3533 5 жыл бұрын

Could one also write i^(1/i) as (e^(pi/2)i)^(1/i), because e^(pi/2) = i according to Euler's formula, allow the i and 1/i to cancel out, and take the remaining e^(pi/2) as a final answer?

@HanzCastroyearsago 2 жыл бұрын

“I dont like to be on the bottom, i like to be on the top” - pen guy

@NardiPaffon 2 жыл бұрын

"Let me do a REAL quick video for you" I see what you did there

@nisithranjanhazra9298 3 жыл бұрын

Thanks for your smile too ❤️

@amineelarif7001 4 жыл бұрын

The ad is trying to kill the youtuber 🤣

@mikeCavalle 2 жыл бұрын

nicely done.... the use of multi color to force focus reminds me of a math professor in the 1960's at the University of Maine.

@blackpenredpen 2 жыл бұрын

Thank you Mike!

@HelloWorld72459 3 жыл бұрын

'THATS IT' PROCEEDS TO VANISH IN THIN AIR

@raiturner2132 5 жыл бұрын

0:33 that’s what she said😂😂

@ErwinSalasErwin 3 жыл бұрын

Professor, I have a question, is it possible to have a real number to the power of a matrix?

@GrowlyBear917 3 жыл бұрын

I'm going to use this information when I file my taxes.

@Lily-zd6dx 6 жыл бұрын

"What we do now? We do our usual business" if math was that easyyy

@sudipmaity8390 6 жыл бұрын

cool.i wish my math teacher had an amazing personality lyk u.

@KalikiDoom 6 жыл бұрын

amazing!

@TuanLe-td1dg 4 жыл бұрын

That was so clean, damn

@czerbniak75 3 жыл бұрын

Tak na dobry sen. U mnie luzik. Dobranoc, kurde jeszcze jasno.

@EdKolis 9 ай бұрын

Imagine if imaginary roots of other imaginary things could be real. Like trees.

@squeet6831 5 жыл бұрын

0:33 of course you do. Your entire channel is about doing hard work. Lol

@shotasdg3679 6 жыл бұрын

Beautiful!

@notthefez3634 3 жыл бұрын

I just wasted 2 minutes being confused by a problem I never even knew existed and still don't understand

@WorriedtheCircusDemon 3 жыл бұрын

Damn, same, I clicked this in my recommended (I've always been horrible at math) I don't get how people in the comments get it so easily

@samuelwillowcreek8764 3 жыл бұрын

That moment when you try to checkmate a grandmaster and he just literally flips the board. Great stuff man, I didn't this was a question that bothered me since 3 minutes ago but thanks for the answer anyways.

@charles7623 6 жыл бұрын

thank you

@darkalpha6530 3 жыл бұрын

So you have this chanel too. Subscribed to this too ❤️❤️🔥🔥😂😂👍👍.

@gauravbhalerao7420 3 жыл бұрын

My mind is blown and my curiosity satisfied

@shibsankarnaskar357 3 жыл бұрын

Really impressive.. 💗💗

@S1nwar 6 жыл бұрын

so i kind of simplify here but does "complex number (some operation) complex number" return a real number almost all the time?

@krimbus1236 6 ай бұрын

Another way to do this is using De Moivre’s theorem, where i on the complex plane would be pi/2, so theta is pi/2/i, so e^i theta would simplify to e^pi/2

@billf7585 3 жыл бұрын

You can check its true, if you raise both sides to the power i, you get i = e^(i*pi/2) which is the complex polar form of a vector with argument pi/2 and magnitude 1, which is of course just i on the complex plane. The family of solutions are associated with all arguments that equal pi/2 mod 2pi, so any integer number of times around the units circle landing back at i.

@meerable Жыл бұрын

As far as I know, the transition from sqrt to power imposes restrictions on the argument. And here it is observed?

@ianboard3555 3 жыл бұрын

I've given this as an interview question (I'm an EE) - it's easy if you just work your way through it.

@BeIteshazzar 3 жыл бұрын

hush, hush... i to i.. too shy, shy..

@noodleplexium5953 3 жыл бұрын

The thumbnail has tortured me after a tough maths test. Stop

@nosirrahx 3 жыл бұрын

If I have learned anything from this channel it is that if you something strange to i, e and/or pi are going to pop out of the equation.

@nucderpuck 3 жыл бұрын

Always being on top is like having only real numbers. Broaden your horizon!

@SNOWgivemetheid 3 ай бұрын

I did some other method : Consider i^(1/i) and get i = (-1)^(1/2) and -1=e^iπ so finally : i=e^(iπ/2) so in the end we have i^(π/2) as the "i"s cancel out !

@112BALAGE112 6 жыл бұрын

There are other solutions as well: (e^(pi/2+2*n*pi))^i = i for integers n.

@fpgamer4566 3 жыл бұрын

That mic kinda look like a cartoon bomb

@marcusrosales3344 5 жыл бұрын

You could easily motivate this by analogy with a square root. Like square root of 4 can be found by finding a number a such that: a^2=4 ==>a=2. Similarly, the ith root of i means find an a such that: a^i=i ==> a=e^(pi/2)

@jeim376 4 жыл бұрын

The greatest crossover in history.

@plaina4628 4 жыл бұрын

wow i thought it will be complicated but then its just simple by knowing what is i^i and the rest are just very easy

@hamsack981 3 жыл бұрын

I don't even think about this kind of stuff. It's crazy that some dude was like, oh there's an i'th root of I, let's solve what it is...

@Dav753 7 ай бұрын

How or why can you plug in that e to the negative pi halves in the place of i to the i?

@joshtechindia 6 жыл бұрын

YAY! This was interesting!

@1234vedas 6 жыл бұрын

Emmanuel Franklin your name n your profile matches a lot with a malayali. Are you one?

@DylanSnider 2 жыл бұрын

Can you graph "i-th root of x = y" and another that's "2nd root of x = y"? Also, can absolute value prove i-th root is the same as 2nd root?

@TheMartnK 6 жыл бұрын

You should probably also update this version with e^(pi/2+2*pi*n)

@carlthorellstein53 3 жыл бұрын

but is it allowed to expand by i? I mean do we know for sure that i/i = 1?

@SwasthikVideos 4 жыл бұрын

Very good and nice videos with interesting content sir

@alexeypopov314 11 ай бұрын

Dude, you rule!

@fouadmeddah2790 2 жыл бұрын

Explaining this Our dude here became closer to being a pirate

@Avinashkumar-jj8yt 3 жыл бұрын

You should use collor Mike.. The Mike which you hold in your hand looks very awkward

@AtariDays80 3 жыл бұрын

I still find it odd that "i" with absolute magnitude of "1" can ever twist itself into something outside of that, as in > 1.

@danofyork 3 жыл бұрын

Based

@_DD_15 5 жыл бұрын

Why did you multiply by i over i? You could have substituted e to the i pi/2 as first step. And then cancel out the i and get the result in 2 steps. :)

@proislam1447 2 жыл бұрын

Use polarform z=re^ip and you get immediately =+-cos45°+-isin45° or am I missing something?

@markjackson6642 6 жыл бұрын

Amazing!!!

@Sma.Das. 6 жыл бұрын

Considering that on a Cartesian plane moving by pi is 180° so since it is pi/2 which is 90° therefore you can have the statement pi*(4n-3)/2 where n is any real number ;)