okay as we all know square root of negative 1 is equal to i but have we ever thought about what is
square root of i okay as we all know when we have square root of i this right here is just equal to
1 over square root 2 plus 1 over square root of 2 i but have you ever thought about what if we have
square root of i plus square root of i plus square root of i and so on so on so but before
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it does converge and because everything's in the complex world so we'll just say it converges to a
complex number i'm just going to use a complex variable z you can put down a plus bi but
let me tell you this is better i have something better for you guys coming up so that's why
so to continue we can just look inside right here you know the usual strategy
this is actually the same as z because this right here actually keeps on going on forever
so we can take this equation and write it as square root of i plus z and that's equal to this e
and then from here we can just square both sides and move things around and we'll just
get z squared minus z minus i equals zero and we have a complex quadratic equation
because we have c being negative i and that's usually complex quadratic formula
but i know there's no complex quadratic formula anyway so z is equal to and let's do this in our
head negative b b is negative one so we just have one and then it's plus or minus but again we just
want the first answer right the principle square root so i don't want to look at the negative
and then we have the square root b squared is just going to be 1 and then minus 4
times 1 times negative i so it's plus 4 i and then all over 2.
so that's what we have ladies and gentlemen this right here is equal to 1 plus square root
of 1 plus 4i over 2. we are done of course not and the reason is because we have this
i inside of the square root you see back in 2017 when we had this i in the square root
we had to find it and we did of course we will also have to simplify this right here in 2022
so now we can just use this formula to figure out this part right here
and we see this is the same as saying one over two
plus and we have 1 over 2 times that square root part and we will just write it down right here
and i will put on the 1 and 4 in red because they are the a and b values respectively and we see
that b is equal to 4 which is positive that means this and that have the same sign and we'll just
take it past the sign because we want to get the principal square root for this right here
okay to continue we have one half here and then plus another one half and then we multiply by
here we go a and b right one and four and we'll just have the square root here first
a is 1 and then here we will have to do the square root of a squared plus b squared 1 squared
plus 4 squared is 17. so we have 17 here in that square root and over 2. and that's the real part
and then we add the second part it's pretty much the same thing except for we have this negative
here so let me put that down negative 8 which is 1 and then we add the same square root of 17
over 2 and then put on the i and that's pretty much it yeah and now we will just have to clean
things up a little bit and in fact i would like to rationalize the denominator here and here because
i see these two tools are in the square root but these are not i don't feel right so anyway
let's put down the one half right here first and then i will take this times that well
it's just one times that's where i'm just going to put a big square root up right here
but to rationalize i will multiply this by 2 and then the top by 2 so i will have 2 plus 2
square root of 17 all right and now the button will be just a regular two times that regular two
it will be a regular four of course and in fact this is the real part and then i will take the
one half multiplied with this so again same thing multiply this and that by 2 inside so we will have
the square root and we will have negative 2 and then plus 2 square root of 17. and then again
the bottom will be a regular 2 times that regular two we will have that regular four
and finally ladies and gentlemen i'm just going to put an i right here so that's the imaginary part
so ladies and gentlemen square root of i plus square root of the height plus square right and so
so is equal to this right here i'm not going to say a lot but you can see i think it's really
really cool yeah but of course this is just one answer out of infinitely many answers of
this technically speaking yeah and i know i owe you guys this formula so of course