sqrt(i+sqrt(i+sqrt(i+...))) = ?

Показать описание

It’s risk-free with Nord’s 30-day money-back guarantee! 👍 #ad

What is sqrt(i+sqrt(i+sqrt(i+...))) = ?, i.e. the infinite nested square root of the imaginary unit i. Note, we will just first the principal square root of this since this infinite nested square roots have infinitely many answers. We will first write sqrt(i+sqrt(i+sqrt(i+...))) as a complex number z. then we can convert this into a complex quadratic equation z^2-z-i=0. Then we can use the quadratic formula to solve for z. But we will end up needing to find out the square root of a complex number.

10% off with the code "WELCOME10"

----------------------------------------
***Thanks to ALL my lovely patrons for supporting my channel and believing in what I do***
AP-IP Ben Delo Marcelo Silva Ehud Ezra 3blue1brown Joseph DeStefano
Mark Mann Philippe Zivan Sussholz AlkanKondo89 Adam Quentin Colley
Gary Tugan Stephen Stofka Alex Dodge Gary Huntress Alison Hansel
Delton Ding Klemens Christopher Ursich buda Vincent Poirier Toma Kolev
Tibees Bob Maxell A.B.C Cristian Navarro Jan Bormans Galios Theorist
Robert Sundling Stuart Wurtman Nick S William O'Corrigan Ron Jensen
Patapom Daniel Kahn Lea Denise James Steven Ridgway Jason Bucata
Mirko Schultz xeioex Jean-Manuel Izaret Jason Clement robert huff
Julian Moik Hiu Fung Lam Ronald Bryant Jan Řehák Robert Toltowicz
Angel Marchev, Jr. Antonio Luiz Brandao SquadriWilliam Laderer Natasha Caron Yevonnael Andrew Angel Marchev Sam Padilla ScienceBro Ryan Bingham
Papa Fassi Hoang Nguyen Arun Iyengar Michael Miller Sandun Panthangi
Skorj Olafsen Riley Faison Rolf Waefler Andrew Jack Ingham P Dwag Jason Kevin Davis Franco Tejero Klasseh Khornate Richard Payne Witek Mozga Brandon Smith Jan Lukas Kiermeyer Ralph Sato Kischel Nair Carsten Milkau Keith Kevelson Christoph Hipp Witness Forest Roberts Abd-alijaleel Laraki Anthony Bruent-Bessette Samuel Gronwold Tyler Bennett christopher careta Troy R Katy Lap C Niltiac, Stealer of Souls Jon Daivd R meh Tom Noa Overloop Jude Khine R3factor Jasmine
----------------------------------------

Рекомендации по теме

Комментарии

It’s risk-free with Nord’s 30-day money-back guarantee! 👍 #ad

blackpenredpen

Ah so thats what the root of all imagination is. The neuroscientists will be pleased with this breakthrough.

teelo

By the way, around 2:26 where you replaced the nested roots with z, you need to make sure the function is bijective, or doesn't diverges at i. Or you will make a mistake like the infinite power tower =4 and you get 2=4

dqrksun

In college i was taught that y=sqrt(x) always referred to the prinicipal root, and if you want both roots you write y^2=x

nictamer

I won my college's integration bee last week due, in part, to your videos. Thanks for your help :)

joshascher

cool video! just at 4:08 it made me think of a video idea that you could make. a complex number calculation: z3 = z1^z2 where z1 = a+bi and z2 = c+di and where the objective is to explicitly express z3 in the form X + Yi. Small clue, personally to get there I went through ln(z3), then I had to explicitly express ln(a+bi) (complex logarithm which can also be a good video subject;-)) with a little trigonometry and using several times the identity of euler e^ia = cos(a) + i sin(a). Good day (and sorry for the mistakes I am French😅)

tifng

Good to see you again. I always enjoy your videos and though I do wish you could do more of them, especially number theory and matrices and modulo calculations (and other stuff that's over my pea brain), I'm just glad to see you whenever you do present a video. Thank you.

OCinTexile

I saw the equation at the thumbnail and I attempted to solve this algebraically. I'm proud of myself I got the right result. Good video as always.

anic

to compute the sq root of 1+4i, I would just go to polar coordinates.

icizay

I did the first steps just like you but for computing √(1+4i) I converted (1+4i) into a matrix form, diagonalazied it and computed the 1/2th power and then computed it back again

jakubpacua

But at 3:17, when you remove one root, what does it mean the "principle square root" for complex numbers ? Because there is no way to arrange the complex numbers like we do for the reals, so how do you choose wich one is the "principle square root" ?
It make sense for the reals because we have this notion of positive and negative, but this doesn't work for complex numbers

baptiste

studied complex numbers😂😂😂and just opened youtube and boom ...he arrives😂lol

anjanyadav

I just love how enthusiastic he is and he’s yet to dive into the equation.

Don’t ask me how I got here… algorithm at its best yet again 💪

RobinHunuki

"disregard the negative sign" yeah in an equation full of i lol

jamirimaj

Interestingly if you plug in a=1 and b=2 into the formula for the complex square root you get the golden ratio bacause then you get:
sqrt(phi)+(1/sqrt(phi))*i
even though wolframalpha spits out something completly different.... I am confused now xD

Melanie