Question

is the square root of a complex number ALWAYS going to be negative? please explain?

Answer 1

I think he means sqrt(5i+n) or just sqrt(ni)

Answer 2

like any imaginary number. like 9 or 64.

Answer 3

@wio

Answer 4

I have sqrt (3+2i) = 2+i which is not a negative one. So that the sentence is not right, right?

Answer 5

First of all, what is the square root of a complex number?
Will it complex, or always real?

If it is complex, then how do you define "negative"?

Answer 6

Instead of square root, do you mean the square?

Answer 7

i mean like √9i is -3. correct? so is this true for all numbers like that

Answer 8

square root. I go backward from (2+i)^2 to make sure that the square root is positive
after simplifying, I have 3+2i. Therefore, sqrt (3+2i) = 2+i

Answer 9

Nope. The square root of 9i is not -3. If you square -3 you get 9.

Answer 10

@wio the term "square root" indicates to +/- something, right? so that his "ALWAYS" confused me a lot

Answer 11

Suppose we have a complex number: a+bi
We square it:
(a+bi)(a+bi) = a^2 + 2abi -b^2 = 9i

Or another way to look at it is:
(a^2-b^2) + (2ab)i = (0) + (9)i

This leaves us with the equations:
a^2-b^2 = 0
2ab = 9

Answer 12

From the first equation we know that |a| = |b|, which is to say a = +-b
Plugging that into the second equation gives us:
2a(a) = 9 -> 2a^2 = 9 -> |a| = sqrt(4.5)
2a(-a) = 9 -> -2a^2 = 9 -> |a| = sqrt(-4.5)
This second equation is invalid, since a and b are real numbers.

Answer 13

Thus the square root of 9i is going to have multiple answers:
sqrt(4.5) + sqrt(4.5)i
sqrt(4.5) - sqrt(4.5)i
-sqrt(4.5) + sqrt(4.5)i
-sqrt(4.5) - sqrt(4.5)i

And we wouldn't really say that they are positive or negative as this concept doesn't apply to complex numbers. Does this make sense?

Answer 14

Here you can verify that [ sqrt(4.5) + sqrt(4.5)i ]^2 = 9i
http://www.wolframalpha.com/input/?i=%5Bsqrt%284.5%29+%2B+sqrt%284.5%29i%5D%5E2

Answer 15

Your silence concerns me...

Answer 16

me or the Asker?

Answer 17

Both

Answer 18

To me, this sentence is not correct because of the counterexample I gave out above. However, to my classes, whenever I have that kind of question, the probability of this sentence is correct is 90%. So I tag you to have a logic for what I thought.

Answer 19

Your counter example is valid. It seems that the asker wasn't even sure ho to find the square root of a complex number though.

Answer 20

so not every complex number has a negative square root.