[math] sqrt{-1} = (-1)^{frac{1}{2}} = (-1)^{frac{2}{4}} = sqrt[4]{(-1)^{2}} = sqrt[4]{1} = 1 [/math]

[math] \sqrt{-1} = (-1)^{\frac{1}{2}} = (-1)^{\frac{2}{4}} = \sqrt[4]{(-1)^{2}} = \sqrt[4]{1} = 1 [/math]

Other urls found in this thread:

en.wikipedia.org/wiki/Exponential_sheaf_sequence
youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF
twitter.com/NSFWRedditVideo

...

not how it works

The square root is defined in terms of the complex logarithm which can only be defined locally
en.wikipedia.org/wiki/Exponential_sheaf_sequence
>The exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the logarithm of g in a neighborhood of P.

FUCK

imaginary numbers is when i stopped believing in math...

"imagine" is when i stopped listening to John Lennon...

I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.
Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

What the fuck did you just fucking say about complex numbers, you little bitch? I’ll have you know I stopped caring about math when I was introduced to the concept of imaginary numbers, and I’ve been involved in numerous secret raids on Al-Gebra, and I have over 300 crocks of shit. I am trained in equations that can only be solved by inventing numbers that can't exist and I’m the top math deity in the entire US academic forces. You are nothing to me but fucking wrong. I will wipe you the fuck out with math the flaws of which have never been seen before on this Earth, mark my fucking words. You think you can get away with saying that shit to me over the Internet? Think again, fucker. As we speak I am contacting my secret network of algebra solutions across the USA and your IP is being traced right now so you better say "the correct answer is whatever the correct answer is", maggot. The math that says the pathetic little thing transcribed to words. You’re fucking dead, kid. I can be anywhere, anytime, and I can mark you wrong in over seven hundred ways, and that’s just if you write it down in english instead of ancient math runes. Not only am I extensively trained in unarmed combat, but I have access to the entire arsenal of the United States Logical Math Corps and I will use numbers that never lie to their full extent to wipe your miserable ass off the face of the continent, you little shit. If only you could have known what unholy flaws your little “clever” human construct was about to bring down upon you, maybe you would have held your fucking tongue. But you couldn’t, you didn’t, and now you’re paying the price, you goddamn idiot. I will shit complex numbers all over you and you will drown in it. You’re fucking dead, kiddo.

translate to brainletish pls

In complex analysis, a lot of inverse functions (square root, logarithm, etc.) have multiple possible answers.
Because of this you can do various forms of fuckery unless you're consistent about which one you use.

There is no global square root function for the complex numbers. Any complex number has two square roots sitting on opposite sides of the complex plane. For positive numbers we can just choose the positive square root, but there is no way of choosing one square root for each complex number in a smooth way.

youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF
This is made for people like you. If you're impatient skip to the 12th video.

Yeah except you're a fucking nigger.
I hope you're trolling, but you might actually think you're cleaver. Kill yourself anyways just to be safe.

I'm seeing some complicated answers. The dumbass answer is that you aren't allowed to
[eqn] \sqrt{-a}\sqrt{-b} = \sqrt{\test{someshit}} [/eqn]

\test was suppose to be \text

Someone tell me why this is wrong

>he thinks real numbers exist

"You aren't allowed to..." is another way of saying "the reasoning is incorrect". But then you miss out on a broader context where you see what fails, exactly, and in which cases things like [math]\sqrt{xy}=\sqrt{x}\sqrt{y}[/math] can be made to work, (which is not just when x and y are positive real numbers). The theory of Riemann surfaces connects complex numbers to more abstract concepts in topology and geometry.

It doesn't work that way. You have to show why it's right, which you generally do by referring to earlier results.

so basically the rule [math] (\sqrt[x]{y})^z = \sqrt[x]{y^2}[/math] doesn't hold over the field of complex numbers?

oops, replace that [math]^2[/math] with a [math]^z[/math]

You do realize the 4th root of 1 is also i right

and also neither can [math]\sqrt{(ab)} = \sqrt{a} \sqrt{b}[/math]

If we consider the field of positive real numbers then a number [math]a[/math] will only have one root so obviously

[math]\sqrt{a} = \sqrt{a}[/math]

but if we apply the same reasoning over the field over the field of real numbers then we get this result:

[math]\sqrt{1} = 1, \sqrt{1} = -1 \therefore 1=-1[/math]

right? no.

[math]\sqrt{1} = 1, -1 [/math]

likewise, even though:

[math]\sqrt[4]{1}^2 = 1,\sqrt[4]{1}^2 = -1 [/math]

[math]\sqrt{-1}\ne1[/math]

[math]\sqrt[4]{1} = 1, (-1), i, (-i)[/math]
[math]\sqrt[4]{1}^2 = 1, (-1)^2, i^2, (-i)^2=1,-1 [/math]

take this with a grain of salt, I'm a high school dropout brainlet and I'm just making this up as I go along but I think it explains why OP's conclusion is wrong

also what I said hereis true too I think in the sense that that is where his reasoning went wrong.

I think more generally what we can say is that just because [math]f(a)=x[/math] and [math]f(b)=x[/math], [math]b \ne a[/math]
meaning that just because applying the same operation to two different numbers yields the same result, those numbers are not necessarily the same, even if in one field of numbers they are always the same the same rule may not hold for another field of numbers. But I don't know.

damn, now i realise how retarded it was of me to type all of that out, all I had to do was point out that the 4th root of 1 has 4 answers

[math] \sqrt{-1} = (-1)^{\frac{1}{2}} = (-1)^{\frac{2}{4}} = \sqrt[4]{(-1)^{2}} = \sqrt[4]{1} = 1,-1,i,-i [/math]

The composition to product rule for exponents doesn't apply to complex numbers. Much like the power of product is equal to product of powers, this is only something that works for real numbers, not complex numbers.

[math] \displaystyle
\left (-1 \right )^{ \frac{2}{4}} \neq \sqrt[4]{ \left (-1 \right )^{2}}
[/math]

this

Not that guy, but that's one of the best series I've ever seen on youtube, thanks m8

Well, I mean, i is 1 in the imaginary plane

Anyway, -1 * -1 * -1 * -1 = 1.

np. I asume you already know 3blue1brown.

so is this the point where math stops being a system of universal rules and starts becoming like chemistry where all their "principles" have loads of annoying exceptions?

No.