It may be easier to see using the exponential form for complex numbers. I'll use Q for theta.
iQ 1/n 1/n iQ/n
(re ) = r e is the principal nth root. The other n-1 roots, as Bob Bundy pointed
out, are equally spaced around the circle; that is, the increment to add to Q/n is 360/n. Keep adding
this increment to each succeeding angle until adding another increment would be a "wrap around."
i100 1/4 1/4 i100/4 i25 i(25+90) i(25+180) i(25+270)
Example: (16e ) produces 16 e = 2e , 2e , 2e , 2e
where the angles are in degrees for ease of typing.
The rectangular form for complex numbers is not as nice to work with for products, quotients and
powers and roots. The exponential form is not nice with sums and differences.
iC iD i(C+D) iC iD i(C-D) iC n n inC
(ae )(be ) = abe (ae )/(be ) = (a/b)e (ae ) = a e (Roots done above)
Integral powers can be done in rectangular form but are quite difficult if n is moderately large.
So we can switch to exponential form, do the power, and switch back to rectangular form.
Products and quotients are not so bad in rectangular form as usually taught in college algebra. It
is not worth the effort to switch to exponential to do products and quotients.
Finding roots in rectangular form? I've never seen it done.
Sums and differences in rectangular form are easy. (a+bi)+(c+di) = (a+c)+(b+d)i and
(a+bi)-(c+di) = (a-c)+(b-d)i.
I've never seen sums and differences done in exponential form.
So the more powerful the operation, the more we tend to use the exponential form for the
There are several forms that complex numbers can be written in.
r(cosQ+isinQ) or rcisQ for short; re ; (r,Q) ; a+bi ; (a,b)
Those with r and Q are just different "wrappings" for giving the r and Q. Each is useful for
interpreting in different settings. Those with a and b are different "wrappings" for giving
the a and b of the rectangular form. a+bi is more convenient for algebraic manipulations
and (a,b) is perhaps more convenient for graphing.
Taking square roots in the exponential form gives us 180 degrees for the increment. So we
can see why square roots of a positive numbers are two real numbers (Eg sqr(4)=2 or -2).
Also we can see why square root of a negative numbers are pure imaginary numbers
(Eg sqr(-4) = 2i or -2i).
i0 1/2 i0/2 i0 1/2 i(0+180) i180
sqr(4) = sqr(4e ) = 4 e = 2e = 2. Also 4 e = 2e =-2
i180 1/2 1/2 i90 i90 i270
The principal square root of -4 is (4e ) = 4 e = 2e = 2i. The other is 2e =-2i.
Have a great day!