Imaginary to the power of fractional number I am sorry if it's stupid, but I don't know how to calculate imaginary number "$i$" power to fractional number. I know:

$$\begin{aligned}
i^2 &amp;= -1\\
i^3 &amp;= \;\;\;i \\
i^4 &amp;= -1
\end{aligned}$$

and so on. I know that multiplying a complex number by "$i^n$" rotates the magnitude vector of that complex number, like: $i(a+bi)$ rotates for $90$ degrees, $i^2(a+bi)$ rotates it for $180$ degrees, to the power of $3$ rotates for $270$ degrees... and so on. And I can even know how to calculate it:

$$\begin{aligned}
    i(a+bi) &amp;= -b + ai\\
    i^2(a+bi) &amp;= -a + bi
\end{aligned}$$

and so on. But my problem is, I know multiplying a complex number by $i^{0.5}$ rotates for $45$ degrees, but I have no idea how to calculate it, and represent as complex:

$$
i^{0.5} (a + bi) = ?????
$$

Could anyone explain it in simply words for simply guy like me? Or give some link with explanation? Thanks in advance for any help.

PS. Of course I know it's something like:

$$
i^{0.5}(a + bi) = (0.707\ldots * a) + (0.707\ldots * b)i
$$

But have no idea how it happened :)
(Slight correction in your question: $i^3 = -i$, and $i^4 = i^0 = 1$.)

Maybe polar representation will help.

If you have a complex number, you can express it either as real and imaginary parts $(a+bi)$ or as magnitude and angle in the complex plane ($re^{i\theta}$, with $r$ being how long it is, and $\theta$ being what angle it makes with the positive real (positive $x$) axis).

So your number is either represented as $0 + 1i$, or $1e^{i\pi/2}$.

Taking the square root of the latter expression directly give you one value: $\sqrt{1}e^{i\pi/4}$. (There's another value, $1e^{i5\pi/4}$, which comes by applying De Moivre's theorem.) So this square root of $i$ makes an angle of $\pi/4$ (or $45^{\circ}$) in the complex plane, and has a length of $1$. (The other is in the third quadrant.)

In real-imaginary form, this is $\sqrt{2}/2 + i\sqrt{2}/2$, which is what you wrote at the end. You can convert from polar to this form by the formula $r\cos \theta + i r \sin \theta$.

You can then use this expression to get a value for $i^{0.5}(a+bi)$ with the FOIL method.

Hope this helps!
You are right, is not as easy than a power by an integer. You must to use the De Moivre's formula