**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
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

for
.

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

Department of Mathematics