An imaginary number involves the letter i, representing the square root of negative 1.
So, the square root of -4 is ±2i.
Raising a number to an imaginary power, such as 2^i or 2^(3+4i) for example, involves logarithms and Euler's formula.
Euler's formula is
e^(ix) = cos x + i sin x
The right side of Euler's formula is a form of a complex number, with cos x being the real part and i sin x being the imaginary part, with x in radians.
If you want to raise a number to the power of i, you consider the i as 1 times i.
That is, the x is 1 radian.
To use Euler's formula, you need to recall that e and the natural log (ln) are inverses.
So
2^i
is the same as e^(ln(2^i))
Using log properties, we can put the i in front of the log...
2^i = e^(i ln(2))
Then by Euler's formula...
2^i = cos (ln(2)) + i sin (ln(2))
The natural log of 2 is just a constant number.
Being sure your calculator is set to radians, you can take the cosine and sine of ln2.
You get the complex number...
0.769 + i (0.639)
We have evaluated a number (2) raised to an imaginary power and found it to be a complex number with a real and imaginary part. But we no longer have an imaginary number in the exponent.
Once you get this figured out, you will be "math
student number -e^(-i(pi)) "
That is... Math Student #1 !
-e^(-i(pi)) evaluates to 1
