## 1.1. Complex Numbers | ICA |

Natural numbers (* N*), integers (

N Z Q R

derives from simple counting numbers, but addition does not have an inverseNadds inverse numbers with respect to addition, but multiplication does not have an inverseZadds inverse numbers with respect to multiplication, but the limit process is not completeQ

Example 1.1.1: Properties of number systemsFind

- a natural number without an additive inverse
- an integer without a multiplicative inverse
- a sequence of rational numbers that converges but the limit is not rational

* R*, the set of real numbers, on the other hand, is (by definition)
complete
with respect to the limiting and has all properties that are useful in
daily life. We can add, subtract, multiply, and divide (albeit not by zero),
we can take limits of real numbers and get back other real numbers, and any
two real numbers are comparable (i.e. either they are identical or one is
smaller than the other). Such a system, incidentally, is called a

Definition 1.1.2: A FieldA field is a set together with two operations commonly denoted asF+and*, as well as two different special elements commonly denoted as0and1, that satisfies the following axioms:

- Both
+and*areassociative, i.e.a+(b+c)=(a+b)+canda*(b*c)=(a*b)*c- Both
+and*arecommutative, i.e.a+b=b+aanda*b=b*a- The
distributive lawholds, i.e.a*(b+c)=(a*b)+(a*c)0is theadditive identity, and1is themultiplicative identity, i.e. for allxwe havex+0=xandx*1=x- There are
additive and multiplicative inverses, i.e. for allxexistsysuch thatx+y=0and for all non-zeroaexistsbsuch thata*b=1

In other words, * R* is a field where in addition the limit process
works (the proper way of saying it is that

Example 1.1.3: Is Q a fieldIs a field?Q

So, what is wrong with * R* to - possibly - warrant yet
another number system? One problem is that:

not every polynomial equation is solvable inR

In other words:

- (1)
*x - 1 = 0*has one solution - (2)
*x*has two solutions^{2}- 1 = 0 - (3)
*x*has^{3}- 1 = 0*one*solution ? - (4)
*x*has^{2}+ 1 = 0solution ??**no**

We can certainly live with those facts, but they seem not logical (as Mr. Spock from the USS Enterprise would say): the third eqution should really have three solutions, not just 1, and the fourth equation should have two, not zero, solutions! The solution to the problem will come from a simple, yet ingenious definition:

Definition 1.1.4: The Imaginary UnitDefine a new symbol called the imaginary unit, or i, as:i =

This may not make sense, since traditionally we are not allowed to take
even roots of negative numbers. However, this is a *definition* so
henceforth the symbol *i* simply means: *square root of negative
one*. If you are worried, think about another, probably more familiar
symbol: . It *looks* familar, but
what does it really mean? Of course it simply means that it is that symbol
that by definition solves the equation *x ^{2}-2=0*, just as

What matters are the consequences and reprecussions that the introduction, or definition, of a new symbol has: does it result in any contradictions to known results? Does it yield new results that are compatible - but perhaps extend - existing theorems? Is it 'useful'?

Just as the symbol provides a reason of
extending the rational numbers to the real ones, the symbol
*i = * will cause us to extend
the real numbers to the *complex* numbers (a step that we will formally
do below).

Example 1.1.5: Properties ofiUse the definition of ito:

Using our new symbol we could define a new number system: informally we simply combine real numbers with our new symbol as:

z = x + i*y, wherex, yR

We can use the definition of *i* to add, subtract, multiply, and
divide such entities. For example:

The new number system could now consist of all symbols of the
form *x + i*y*, but we will use a more formalized approach to avoid
the somewhat strange symbol *i* (just in case someone still
objects to it). Just as is not used in the
definition of the real numbers, the symbol *i* should not be used
in the *formal* definition of the complex numbers.

Definition 1.1.6: Complex NumbersThe set of complex numbers is defined as the set of all pairsC(x,y),x, y, where forRz = (x,y)andw = (u, v)we define:Two complex numbers

z + w = (x,y) + (u,v) = (x+u, y+v)z * w = (x,y) * (u,v) = (x*u - y*v, x*v + y*u)zand_{1}= (x_{1},y_{1})zare called equal if_{2}= (x_{2},y_{2})bothx_{1}= x_{2}andy. We will frequently use the symbol_{1}= y_{2}0to mean(0,0).

This definition does not pull any strange symbols out of the hat (*i*
is nowhere to be seen), but it does look perhaps a little contrived: addition of
two tuples is defined as it should, but why is multiplication defined so strangely?
Why not simply define *(x,y) * (u,v) = (x u, y v)*?

Please note that two complex numbers being equal results in *two*
equations that need to be true *simultaneously*. You should also observe
that we have defined equality of two complex numbers, but not inequality. In
other words, we *can not* decide if one complex number is less or greater
than another! But first we want to find out what this definition has to do with
our new symbol *i*. Let's play with the definition to find out:

Example 1.1.7: Simple Complex Numbers

Now we can formalize the properties of our new set of numbers:

Theorem 1.1.8: Complex Numbers are a FieldThe set of complex numbers with addition and multiplication as defined above is a field with additive and multiplicative identitiesC(0,0)and(1,0). It extends the real numbersvia the isomorphismR(x,0) = x.We define the complex number

i = (0,1). With that definition we can write every complex number interchangebly asz = (x,y) = x + i*y = x + i y

Complex numbers are an extension of the reals, but not *all* properties
of real numbers extend to complex ones: the real numbers are *ordered*, i.e.
two real numbers are either equal or one is great than the other. Complex
numbers, however, are *not* ordered: two complex numbers are either equal or not,
but if they are not equal we *can not* decide which one is greater.

We now have two representations of the field of complex numbers:

- as pairs of real numbers
(x,y)with a 'funny' multiplication- as a sum of two real numbers
z = x + iyusing the symboliwith the property thati^{2}= -1

The first definition is the mathematically proper one, since it does not need a new symbol. However, the second definition is the easier one to work with in almost all situations, and we will hence think of complex numbers as:

## Complex Numbers

C= {z = x+iy, wherex,yandRi}^{2}= -1

Let's put our new numbers to work:

Example 1.1.9: Algebra with complex numbers

Now we can also remedy, at least partially, our original problem where polynomial equations of degree 2 sometimes have two, one, or no solution.

Now we have an elementary understanding of complex numbers, but to better work with them we need a helpful visual representation. That will be the topic of the next section.

Proposition 1.1.10: Two complex rootsShow that every equation zhas exactly two solutions in^{2}= c(unlessCc = 0). In other words, every complex (non-zero) numberchas exactly two complex square roots.