The principal value of the tetration of the imaginary unit i^i^i in ONE minute! Check out the i-th root of i: • i-th root of i / blackpenredpen #blackpenredpen #math #complexnumbers
The infinite power tower i^i^i^i... has a fixed point. I spent a couple of minutes this afternoon working it out. Now you've done several videos on the Lambert W function this could be another one.
There is no "i" in the exponent. Wolframalpha says it is single-valued. Why is this? And why is i^i^i^i multi-valued but i^i^i single valued? Maybe wolframalpha is wrong?
swiss knight Because we are saying i^(i^i), not (i^i)^i. These are completely different. For example, what is 3^(2^3)? That would be 3^8 = 6561. If I changed it to (3^2)^3, we get 9^3 = 729. When we don’t include parentheses, we assume the former, not the latter; therefore, you cannot simply multiply the exponents.
I'm in 9 (14 year old) Don't judge , I tried to solve it , my answer was coming -i i^i^i = i^i² ( using the exponent laws) Now , since i² is just -1 . it comes down to i^-1. Again using using exponent laws it is = 1/i and now rationalising this would give i/-1 which is just *-i* . *Now my question to blackpenredpen is what is -i equal to ..I don't know if it's solvable coz it is complex but maybe you could try*
Dear mr BlackpenRedpen I have a question for you. One of my favourite graphs is (X^2 + 2x -8)^2 + (y^2 +2y -8)^2 = 81 I wondered if it is possible to find the area that is enclosed by this graph (it’s shape is two diagonally overlapping ovals). I am not aware of any rules that make it impossible to calculate this so I would appreciate if you would reply or maybe do a video on it :)
@@camdamcool6125 Correct me if I am wrong, but I think that's not quite the answer. Common interpretation of i^i^i^i is i^(i^(i^i))), while (e^-pi/2)^(e^-pi/2) is (i^i)^(i^i).
I got the answer as -i. Please verify. A={(i)^(i)}^ (i) (A)^(1/i) = (i) ^ (i)....... (eq.1) Now, 1/i = i/ (i*i) ....... Multiplying denominator and numerator by i. i= (-1) ^(1/2). After squaring both sides:- i^2= i*i= -1. So (1/i) = (- i). Using this in eq. 1, we get:- (A) ^ (-i) = (i) ^(i).. or, (1/A) ^(i) = (i) ^(i)...... Comparing both sides we get:- (1/A) = i Or, A=(1/i). And as already derived above, after multiplying denominator and numerator by i, we get:- A = -i. Which I guess is the simplified answer and it took less than 30 seconds😅.
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Did I make it?
blackpenredpen yes good job
Sonic the hedgehog is impressed : D
blackpenredpen barely, well done
We want to explain the million dollar problems, the Clay Institute, can you?
Your videos raise my mood so much, even when I'm already happy. It's wonderful to see someone appreciate the beauty of mathematics that much!
Next week: 61 seconds in ONE minute.
Your comment has 61 likes! (I'm not one of them, unfortunately)
@@pronounjow and now it has 69 lol!
And now it doesn't... 420, here we come!
I want to like this comment but it has 420 sooooo....
Next week: π is irrational (1min uncut)
Next week: π is transcendental (1min uncut)
Next week: Poincare conjecture in 1 min(uncut)
Next week: proving the Riemann hypothesis (1 min uncut)
Next week: proving the Collatz Conjecture (1 min uncut)
Next comment:something about Fermat last theorem and how the proof is too short for a 1 minute video...blablabla
You're deriving so fast you could get speeding ticket.
😂😂😂😂
Lmao. And he's switching colors too, which makes it more impressive
Lmao best one so far
Did this inspire the Lambertghini?
As long as he doesn't drink and derive it's perfectly legal.
The way he just cut himself at the end
Da E
I didn’t cut the video. I pressed the stop button on my laptop.
@@blackpenredpen
😂😂
Sonic: I am speed
blackpenredpen: Hold my pen
There too many they keep falling off Sonic's hand 😂
BlackpenBluepenRedpen and legendary purple pen
Or in this case: "Give me my markers".
the red one or the black?
Teacher: you have one minute to finish the exam
Me:
Hahaha
😂😂
Did I make it?
Not only did he solve it under 1 minute, he switched pens while doing that.
The marker switching is on point today 👌
Oracle thanks!!! And I stay true to black pen red pen
One thing I love about you is how genuine you are. Never change !
Thank you!! : )
The infinite power tower i^i^i^i... has a fixed point. I spent a couple of minutes this afternoon working it out. Now you've done several videos on the Lambert W function this could be another one.
@@sasmitvaidya3594 what
@@sasmitvaidya3594 i don't think that would be obvious to everyone.
@@boomerboxer3574 Yes, most folk are oblivious! to such fun stuff. ;-)
Crossover:
100 integrals in 1 minute?
Physically impossible. But 1 hour would be interesting
@@Aditya_V_R it was a joke boomer
Nicola Barbaro that doesn’t even make sense
@@markq722 ok boomer
@Popa Ionut ok boomer
wtf did just happend
quick maffs
General solution:
i^i^i = e^[(2m+1/2)π * (e^(2n-1/2)π)]
Where {m, n}∈Z
Okay, I know, sprint run right? Then let m=n=1.
There is no "i" in the exponent.
Wolframalpha says it is single-valued. Why is this? And why is i^i^i^i multi-valued but i^i^i single valued? Maybe wolframalpha is wrong?
The proof of Fermat's Last Theorem in 1 minute.
I have a most elegant proof that can be explained in under 1 minute.
But this comment box is too small to contain it.
@@SuperYtc1
Ah, the true spirit of Fermat.
Why isn't it 1/i?
Because i^i^i = i^i*i = i^(-1) = 1/i
Right?
(1/i = i/-1= -i)
swiss knight Because we are saying i^(i^i), not (i^i)^i.
These are completely different. For example, what is 3^(2^3)? That would be 3^8 = 6561. If I changed it to (3^2)^3, we get 9^3 = 729. When we don’t include parentheses, we assume the former, not the latter; therefore, you cannot simply multiply the exponents.
@@FACH-nr3jz thank you kind stranger
😏Yeah but we can make it 30 sec by keeping playback speed 2x.
Protip!
Yes, you did!
Incredible job, BPRP!
Love ur videos)
FrozenArtStudio thank you!!!
THIS MAN GOING LIGHT SPEED
Just the speed of the sound at the moment and thanks!!
Please do i^i^i^i^i^i^... (infinite cascade) in one minute!
Just write it as i^x, where x is i^i, and express it like e^(ln(i^x))=e^(i^i ln(i)), since we know what i^i and ln(i) are, the rest is easy peasy.
Next: Solving String theory in pi minutes.
+1
S P E E D
We can also use logarithm for more simplification🌟🌟
We want more 1 minute challenge
We love it
I just watched this video in x2 and I have travelled to the past.
Eminem: im the fastest rapper!
BPRP: *hold my pens*
blackpenredpen make a vid on, find the coordinates of the intersection between y=sin(1/x^x) and y=cos(1/x^x) pls.
The way he stopped the recording at the end to see if he could make it or not😂😂. Really appreciate it, it must jad took several re-attempts!
“No one can possibly solve this in a minute.”
BPRP: 🙋🏾♂️
What about i^i^ (infinite times?)
Does it will have solution?
Why didn't you raise i^i = e^(-pi/2) to the i-th power, just multiply the powers? You would get e^i(-pi/2) = -i
Elx_Dude Because a^b^c=a^(b^c), not (a^b)^c.
Usain bolt : i am the fastest man alive
Blackpenredpen : hold my pen ... wait, how am i suppose to write on board ?.
This is why I like math. You can flex on people with very fast minecraft enchanting symbols.
integral (sinx)^2/(x^2) from 0 to infinity
=0.947158991 + i(0.32076445), in case you wanted to know the values
Could you also have done (i^i)^i?
Would haven been e^-pi/2e^ipi/2
Bruh wtf I thought my playback speed was 2x when the video started 💀
I'm in 9 (14 year old) Don't judge , I tried to solve it , my answer was coming -i
i^i^i = i^i² ( using the exponent laws)
Now , since i² is just -1 .
it comes down to i^-1.
Again using using exponent laws it is = 1/i and now rationalising this would give i/-1 which is just *-i* .
*Now my question to blackpenredpen is what is -i equal to ..I don't know if it's solvable coz it is complex but maybe you could try*
nah, i^2 is -1, not i^i. Look at 3^2 and 3^3, they are 2 different things. That is why ur method does not work.
2021:
"proving N=PN speedrun"
I love these 1 minute explanations even though I usually have no idea what he says at the end
This is clearly just a slanted -i.
Well done
That handwriting at the end though.
Plz explain calculus 3 section 16,15
Wait, isn’t this just i^-1, e.g -i? I suppose we can do it this way, but simply multiplying the i in the powers should suffice, or no?
That’s quite straightforward once you write (exp(a))^b=exp(ab).
He even switches the pens
Exynouz : ) yea
love your 1m chalange
I am not sure it is precise. You computed i^(i^i) which is not equal to (i^i)^i=(e^(-\pi/2))^i = -i.
May you comment?
well, general convention is that a^b^c is a^(b^c)
Yes
Lowkey this actually made sense to me
Please, plot the series a(1)=i, a(n+1)=a(n)^i in the complex plane.
blackpenredpen 一位用生命拍片的TH-camr
Kobe Cheung wow what a comment!!
famous last words: did I make it
How about the proof that √2 is irrational in less than √2 minutes?
I want i^i^i^i in 1 minute next time!!!
Dear mr BlackpenRedpen I have a question for you.
One of my favourite graphs is (X^2 + 2x -8)^2 + (y^2 +2y -8)^2 = 81
I wondered if it is possible to find the area that is enclosed by this graph (it’s shape is two diagonally overlapping ovals). I am not aware of any rules that make it impossible to calculate this so I would appreciate if you would reply or maybe do a video on it :)
that was quick
For the next video, you should try to evaluate or take the derivative of i↑↑i.
1. That's not a function of x, so you can't differentiate it.
2. The mathematics to evaluate that hasn't been invented yet.
@@oledakaajel oh :(
I was also going to say differentiate x↑↑x but you're right. The math hasn't been invented yet.
(i^i)^i = i ^ (i * i) = i ^ (-1) = 1 / i = -i (checked by Matlab)
I think that it is much easier form of result.. and also faster.
Peter Krammer I thought that as well, but it seems it is i^(i^i) what is meant here, which is different. The last power goes first.
@@rolflangius1119 Ouch.. OK, you have right... (It is signal for me, that I am really tired)
Bprp: 1 minute
TH-cam: 1:01
why do you use the big ball shaped mic? wouldn't it be easier to buy a mic that clips on to your shirt so you have 2 hands?
In other simple words (i^i)^i = -i
But i^i^i = i^*(i^2) = i^(-1) = -1 / i = 1/sqrt(-1).
multiply top and bottom by i/i => i/i^2 = i/(-1) = -i.
How about that?
YAY! GOOD JOB may you please explain the integral of ln(i+x)?
Next the integral of cbrt tanx dx in one minute 😂
Lol it will take me one minute to just write the answer down. Hahaha
@@blackpenredpen Well, you could use a bunch of variables and let them equal to arctan, ln aso. to shorten the things you need to write out 😂
best minute spent today
LOL, next up i^i^i^i in 1 minute!
Just (e^-pi/2)^(e^-pi/2) lol
@@camdamcool6125 Correct me if I am wrong, but I think that's not quite the answer. Common interpretation of i^i^i^i is i^(i^(i^i))), while (e^-pi/2)^(e^-pi/2) is (i^i)^(i^i).
Blackinem - Math god
20 seconds in my head is hurting
Thanks Euler
i just know that you were originally gonna not do the principal value initially, but then you decided you had to in order to save time
what a result at end !!! amazing what we can do in complex numbers
bprp gets i^i^i=.947+.321i I get i =exp(i pi/2) so i^i=exp( i*i pi/2)=exp(-pi/2) so (i^i)^i=[exp(-pi/2)]^i =exp(-i pi/2) = -i where did I go wrong?
i to the i to the i in |i| minute.
That was an intense sports
Great Job!!
Next: Twin prime conjecture in under one minute
What about i^i^i^........... ?
(Infinite power)
@Q Q still computing....
Why could it not be e^-iπ/2? Since i^i is e^-π/2 when you do this to the power of i it should just multiply the power should it not?
Maths make people crazy
Next week: i^i^i^i^i^i^i^i.... 1min uncut
Next video: Holding a microphone for 100 hours nonstop while doing math
You made it alright!
Can you prove Ramanujan summation with help of differentiation it's a challenge for u..
RIP blue pen
😭😭the blue pen died in the making of this video
Does i^i^i=(i^i)^i or i^(i^i)?
(i^i)^i=(e^(-pi/2))^i=e^(-i pi/2)=-i.
I think this was unlisted, but why. And you did it 👌👍
I thought I had speed times 2
I had to check my speed setting too, but it was already set to "normal"...
you should do "proving that p=np (1 minute uncut)"
I love this
This is content I subbed for
Wow nice timing
Brain.Exe stopped working
No one:
Absolutely no one:
Blackpenredpen: *i am speed*
I got the answer as -i. Please verify.
A={(i)^(i)}^ (i)
(A)^(1/i) = (i) ^ (i)....... (eq.1)
Now, 1/i = i/ (i*i) ....... Multiplying denominator and numerator by i.
i= (-1) ^(1/2). After squaring both sides:-
i^2= i*i= -1.
So (1/i) = (- i).
Using this in eq. 1, we get:-
(A) ^ (-i) = (i) ^(i)..
or, (1/A) ^(i) = (i) ^(i)......
Comparing both sides we get:-
(1/A) = i
Or, A=(1/i).
And as already derived above, after multiplying denominator and numerator by i, we get:-
A = -i. Which I guess is the simplified answer and it took less than 30 seconds😅.
Could be i^i^i=-i?
If you would have gone for (e^(-pi/2))^i you would have ended with -i as result.
OMG that was impressive :>
If 3w4 = 3^3^3^3
And 8w9 = 8^8^8^8^8^8^8^8^8
And 4w4 = 4^4^4^4
What is iwi
And most importantly, owo.
What's this?