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so, typically to find the square root of a complex number, you can just cut its angle in half. for example, sqrt(i) is halfway between 1 and i on the complex unit circle. you also have to take the square root of the number's magnitude, but 1, i, -1, and -i all have a magnitude of 1.
in other words, sqrt(-1) is half the angle between 1 and -1, so you land on i. sqrt(i) is half of that, so you land on (sqrt(2)/2 + sqrt(2)/2*i). Thinking about it this way makes it easy to picture the answer in your head.
The problem is theres two possible values for every square root. For example sqrt(4) could either be 2 or -2, since you can square either of those and get back to 4. So there's something called the "principal root", which basically means when we take the square root, we always take the positive one since thats the principal value.
But the question is, how do you extend this to complex numbers? For example sqrt(i) could either be (sqrt(2)/2 + sqrt(2)/2*i) in quadrant 1 or it could be (-sqrt(2)/2 - sqrt(2)/2*i) in quadrant 3. So which one do we consider the "principal value"? In this case, the answer is to pick the number in the first quadrant. In general, I THINK the rule is: pick the number with the angle between (-pi/2, pi/2].
So for sqrt(-i), you cant just half the angle, cuz the number you get wouldnt be the "principal value" (though it would work). Instead you take that angle and go halfway around the circle to get to the one in quadrant 4. Geometrically, that means you'd land on (sqrt(2)/2 - sqrt(2)/2*i).
Anyway, thats how to think about complex square roots geometrically. For cube roots you'd need to take 1/3 of the angle, and you'd end up with 3 potential values, etc. If you want to solve it algebraically instead of using geometric intuition, you can do what bprp did in his video.
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so, typically to find the square root of a complex number, you can just cut its angle in half. for example, sqrt(i) is halfway between 1 and i on the complex unit circle. you also have to take the square root of the number's magnitude, but 1, i, -1, and -i all have a magnitude of 1.
in other words, sqrt(-1) is half the angle between 1 and -1, so you land on i. sqrt(i) is half of that, so you land on (sqrt(2)/2 + sqrt(2)/2*i). Thinking about it this way makes it easy to picture the answer in your head.
The problem is theres two possible values for every square root. For example sqrt(4) could either be 2 or -2, since you can square either of those and get back to 4. So there's something called the "principal root", which basically means when we take the square root, we always take the positive one since thats the principal value.
But the question is, how do you extend this to complex numbers? For example sqrt(i) could either be (sqrt(2)/2 + sqrt(2)/2*i) in quadrant 1 or it could be (-sqrt(2)/2 - sqrt(2)/2*i) in quadrant 3. So which one do we consider the "principal value"? In this case, the answer is to pick the number in the first quadrant. In general, I THINK the rule is: pick the number with the angle between (-pi/2, pi/2].
So for sqrt(-i), you cant just half the angle, cuz the number you get wouldnt be the "principal value" (though it would work). Instead you take that angle and go halfway around the circle to get to the one in quadrant 4. Geometrically, that means you'd land on (sqrt(2)/2 - sqrt(2)/2*i).
Anyway, thats how to think about complex square roots geometrically. For cube roots you'd need to take 1/3 of the angle, and you'd end up with 3 potential values, etc. If you want to solve it algebraically instead of using geometric intuition, you can do what bprp did in his video.
It's just (1-i)/sqrt(2) :)