1.1. The problem with reality. In calculus, most of the functions that you
considered

had both domain and range contained in the real numbers R. The real numbers are
good for

most real-life purposes, and we can describe most everything in the world around
us using

real numbers. However, if we are interested in the algebra of polynomials, we
soon run into

a problem. Given a polynomial p(x) with real coe cients, such as p(x) = x^{2} - 1,
one of the

things we can do is factor it.

x^{2} - 1 = (x + 1)(x - 1)

into linear or rst order polynomials. If we factor a polynomial as

the values
are called the roots or zeroes of the polynomial, because p(c_{i}) = 0 for

each i = 1, . . . , n. Equivalently, they are the solutions of the polynomial
equation p(x) = 0.

The problem with the real numbers is when we consider a polynomial such as
p(x) = x^{2}+1.

It cannot be factored using real numbers, and the equation x^{2} + 1 = 0, or
equivalently

x^{2} = -1, has no real solutions. This is because a negative number has no real
square root.

More generally, if we consider quadratic polynomials of the form p(x) = ax^{2} + bx
+ c, recall

that the roots are given by the quadratic equation.

We have a problem if the discriminant b^{2} - 4ac is negative, since then we
cannot take the

square root, and the quadratic polynomial has no real roots.

To work around this problem, we introduce a new type of number, called the
imaginary

numbers which are the square roots of negative real numbers. In particular, if
we let

then all imaginary numbers are just real multiples of i. For instance, we
compute

and

Taking the real and imaginary numbers together, we obtain the complex numbers.

**Definition 1.1. **The complex numbers are numbers of the form.

Using the complex numbers, we can factor every quadratic polynomial using the
quadratic

formula. In fact, we can do much more.

**Theorem 1.2 **(Fundamental Theorem of Algebra). Every polynomial with
complex coefficients

(and in particular, every polynomial with real coefficients, since the real
numbers are

contained in the complex numbers with imaginary part 0) can be completely
factored as a

product of linear polynomials.

A number system satisfying this property, that every polynomial can be
completely fac-

tored into linear polynomials, is called algebraically complete.

**1.2. Complex arithmetic.** In order to understand the complex numbers,
we must describe

how the normal arithmetic operations of addition and multiplication work in C.
We follow

the normal rules of algebra, treating i as an unknown variable, but substituting
i^{2} = -1.

We have.

(a + bi) + (c + di) = (a + c) + (b + d)i

and

(a + bi) (c + di) = a c + bi c + a di + bi di = (ac - bd) + (bc + ad)i.

How do we subtract complex numbers? That's easy. We just add the negative (or
additive

inverse) of the complex number.

-(a + bi) = -a + (-b)i.

How do we divide complex numbers? That's rather harder. We must
first find the
reciprocal

(or multiplicative inverse) of a complex number. To do that, we introduce the
complex

conjugate.

if z = a + bi, then = a - bi,

obtained by flipping the sign of the imaginary part only. The complex conjugate
is useful

since multiplying a complex number by its conjugate gives us

which is a real number with no imaginary part. In fact
is a non-negative
real number,

and
only if both a = 0 and b = 0. We are now able to compute the reciprocal, as

Example 1.3. Here we divide

Note that we can also use the complex conjugate to extract the real and
imaginary parts

of a complex number, as follows.

**1.3. Geometric interpretation. **One of the most interesting features of the
complex num-

bers is that they can be viewed as points in the two dimensional xy plane by
identifying the

complex number z = a + bi with the point with coordinates (x, y) = (a, b). In
fact, they

are better viewed not just as points but rather as displacements or vectors. A
vector can be

thought of as an arrow starting at the origin (0, 0) and ending at a particular
point. The

advantage of thinking of complex numbers as vectors is that two vectors are
added by placing

their arrows end to end, and constructing a new arrow from the origin to the
destination.

(This is basically the two dimensional version of how you were taught to add
using a number

line.) This vector addition corresponds precisely to the addition of complex
numbers.

There is also a geometric interpretation of multiplication, although for that
we need to

use polar coordinates. Recall that the the polar coordinates
for a point in the plane

are given by

&bull,its modulus r is the distance from the origin, and we usually take r≥,0.

•,its angle θ, formed between the point (or, more precisely, its vector) and the
positive

real axis, measured counterclockwise, and we usually

Given a complex number z = a + bi, the modulus is given by

and the angle is given by

Using polar coordinates, a complex number can be written in the form

Given another complex number also written in polar coordinates,

their product is given by

where we observe that our formula for the product of two complex numbers
corresponds to

the angle addition formulae for sin and cos. So, taking the product of two
complex numbers

multiplies their moduli and adds their angles.

Using polar coordinates, we can also take powers of complex numbers. Given a
complex

number
,
then its nth power is

In addition, we can take fractional powers, or roots, of complex numbers. For
instance, a

square root of a complex number is given by

There is no problem taking
since r≥,0. We get the second square root (minus the first)

by replacing the original angle θ, with the equivalent
angle

In general, a nonzero complex number has a total of n nth roots, given by
starting with the

original angle θ,, as well as taking

1.4. Euler's formula. The mathematician Leonhard Euler discovered a beautiful identity.

We can show why this is true using Taylor series. We have

In particular, Euler noted his famous formula

Using this exponential notation, our polar coordinate formula for the product
of two complex

numbers becomes

which follows from the identity
Also, the formula for the powers of a complex

number is simply

which follows from the identity
In the particular case of n = -1, the reciprocal

of a complex number is

taking the reciprocal of the modulus and the negative of the angle.

Using this exponential notation, the complex conjugate of
Extracting

the real and complex parts of
we obtain the formulae.

**1.5. Exercises.
**(1) Write the complex fraction
in the form a + bi.

(2) In the complex numbers, the number -1 has two square roots, i. Compute
the

three cube roots of -1. Write each of them both in exponential form and in the
form

a + bi.

(3) Write the number 1 - i in exponential form, and compute (1
- i)^{4}.

(4) What can you say about the logarithm of a complex number? In other words,
given

a complex number z, what complex number or numbers w satisfy