What is the square root of the imaginary number i?

Question

How do you find it?

Answer 1

(1+i)/sqrt(2).

Notice that i= cos(90)+isin(90).
Then sqrt(i)=(cos(90) +isin(90))^1/2.
By De-Moivre's theorem, this reduces to
cos(45)+isin(45)=1/sqrt(2)+i/sqrt(2)= (1+i)/sqrt(2).

And indeed, ((1+i)^2)/2 = (1+i^2+2i)/2=i.
Similarly, the other root would of course be -(1+i)/sqrt(2)

Similarly, you can find the third,fourth,fifth,3/7th ,....roots of i.

****This is De-Moivre's Theorem:
(Cosx+isinx)^n=cos(nx)+isin(nx), for any real number n.
Note:
cosx + isnix is actually given a special name "cisx".It can be shown that cisx=e^ix.
similarly,you can try finding the ith power of i. Oooooo...complex numbers!!!!!!

Answer 2

The square root of the imaginary number i is:

(1 + i)/√2

You can verify this by squaring it to see if you get i.

Think of a complex number as a two dimensional number. The real numbers run along the horizontal axis. The imaginary numbers run along the vertical axis. This is similar to the x and y axes. You can plot a point such as 3 + 2i on the two dimensional plane just as you would plot P(x,y) = (3,2).

Any point you plot has a magnitude and direction, which can be measured like the angle theta by going counter-clockwise from the positive x axis.

The magnitude for P(3,2) = √(3^2 + 2^2) = √13

For the point in question, i, the magnitude is 1. Since the square root of 1 is 1, the magnitude of √i will also be 1. Now for the angle. The angle of i is pi/2. The angle of √i is half of that or (1/2)pi/2 = pi/4. Hence √i = (1 + i)/√2.

Answer 3

well the imaginary number i is equal to the square root of -1 so it would be -1 to the 1/4th power

Answer 4

OK.

0.5* (sqrt(2) + i *sqrt(2))

Solution:
Ask: What complex number (a+bi) multiplied by itself is i?


So, (a+bi)*(a+bi) = i = 0 + i

a^2 - b^2 = 0 (the real part), so a^2= b^2

2abi = 1*i (the imaginary part), so, 2ab=1

Now solve the two equations in two variables:

2ab=1, so (2ab)^2 = 4a^2b^2 = 1

4 a^4 = 1 ===> a = (sqrt2)/2 ===> b=(sqrt2)/2

Substitute into a + bi

Done.

Answer 5

You can find the answer at the below address.

Answer 6

the answer is one.

Answer 7

hmmmmmmm....... negative one to the one fourth?