What is the square root of i^2

AI Thread Summary
The discussion centers on determining the value of \sqrt{i^2}. It is established that \sqrt{i^2} equals \sqrt{-1}, which is defined as i. The principal branch of the square root function is emphasized, indicating that the square root is typically taken to be the positive root. The conversation also highlights the importance of correctly identifying the argument of complex numbers when calculating their roots. Ultimately, the conclusion is that \sqrt{i^2} is indeed equal to i.
murshid_islam
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what does \sqrt{i^2} equal to? is \sqrt{i^2} = i or \sqrt{i^2} = \pm i?
 
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im guessing its plus&minus
 
The square root is usually taken to be the principal branch of the function. I.e. the root whose argument lies in [0,pi), so you would conventionally obtain i.

In general the n'th root of a complex number is taken to be the principal root unless otherwise stated: the root with argument in the interval [0,2pi/n)
 
I gave you the usual definition of the square root function in your other thread. You can use that to figure out what it is.
 
Data said:
I gave you the usual definition of the square root function in your other thread. You can use that to figure out what it is.
you said, \sqrt{z} = z^{1/2} = \sqrt{r}e^{i\theta/2}.

Data said:
So let's try out this definition. If z=-1, then we write z = e^{i\pi}, and we get \sqrt{z} = e^{i\pi / 2} = i. As you might expect.
but if z = x + iy, then r = \sqrt{x^2 + y^2} and \theta = \tan^{-1}\left( y \over x \right)

if z = -1 = -1 + 0.i, then x = -1, y = 0.

r = \sqrt{\left(-1\right)^2} = \sqrt{1} = 1

\theta = \tan^{-1}\left( {0} \over {-1} \right) = 0

\sqrt{z} = \sqrt{-1} = \sqrt{r}e^{i\frac{\theta}{2}} = \sqrt{1}e^{0} = 1

but where did i go wrong?
 
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Because arctan only returns values in the region -pi/2 to pi/2, so it is wrong to say that theta is arctan(y/x) alone. You must also use your common sense to see that the argument is actually pi, not zero. It is correct to say that tan(theta)=y/x, but that is *not* the same as theta =arctan(y/x) at all.
 
yeah, i can see that \theta = \pi from the position of z = -1 in the argand plane. so,

\sqrt{z} = \sqrt{-1} = \sqrt{r}e^{i\frac{\theta}{2}} = \sqrt{1}e^{i\frac{\pi}{2}} = 1.(\cos{\frac{\pi}{2}} + i.\sin{\frac{\pi}{2}}) = i

am i right?
 
murshid_islam said:
what does \sqrt{i^2} equal to? is \sqrt{i^2} = i or \sqrt{i^2} = \pm i?


This is my assumption:

\sqrt{i^{2}} = \sqrt{-1} = i

i^{2} is equal to -1 and square root of -1 is defined as i.
 
murshid_islam said:
you said, \sqrt{z} = z^{1/2} = \sqrt{r}e^{i\theta/2}.but if z = x + iy, then r = \sqrt{x^2 + y^2} and \theta = \tan^{-1}\left( y \over x \right)

That expression for r works for all z, but the one for \theta only works for x>0, assuming you're using the usual branch of arctan. In general finding the polar representation of a complex number involves solving

\sin{\theta} = \frac{y}{r}
and
\cos{\theta} = \frac{x}{r}

simultaneously (taking \arctan (y/x) makes the possible solutions obvious though!).
 
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