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
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,
Euler's formula. In order to relate the trigonometric form to the exponential form , we can (at least superficially) introduce Euler's formula
Calculations of complex numbers. Let and . Then we can compute
Roots of . Recalling that the cosine and sine functions are periodic with period , we obtain