Euler's formula

Complex numbers. By setting the imaginary unit '' which satisfies  ,'' we can introduce complex numbers of the form , where and are real numbers.

Sum, product, conjugate and absolute value. Let and . Then the sum '' and the product '' of the two complex numbers are defined respectively by

and

The conjugate  and the absolute value  of  are defined respectively by

and

Complex plane. Each complex number  uniquely determines the point  at  in the coordinate plane, which is referred as complex plane. In the complex plain, the -axis and the -axis are repectively called the real axis and the imaginary axis.

Trigonometric forms for complex numbers. Let be the length of , and let be the angle from the real axis to . (Recall Lecture summary No.13; if ; if .) Then we have , and therefore,

 (3-1)

Euler's formula. In order to relate the trigonometric form  to the exponential form  , we can (at least superficially) introduce Euler's formula

 (3-2)

where denotes the base for natural logarithm. Combining (3-1) and (3-2) together, we obtain .

Calculations of complex numbers. Let and . Then we can compute

which coincides with the formal'' calculation of exponents via (3-2). Since , we obtain the following result (De Moivre's Theorem):

Roots of . Recalling that the cosine and sine functions are periodic with period , we obtain

 (3-3)

Observing when in (3-3), we can find all the solutions to , known as th roots of unity'' as follows:

for .

When , all the solutions to  (that is, th roots of ) are given by

for .

Department of Mathematics