Complex NumbersWe start with the question of finding solutions to the followingequations

x2 = −1

Notice this equation has no real solution. However, if we define

i =√−1

then either i or −i satisfies the above equation. That is,

(±i)2 = (±√−1)2 = −1

Complex numbers

Definition (Complex Numbers)

Complex numbers are of the form a + bi , where a, b ∈ R, andi =√−1

Formal Definition of CThe complex number system C is created by the rules for additionand multiplication on R2, the set of ordered pairs of real numbers.For instance,

2 + i ⇒ (2, 1)

DefinitionThe complex number system is defined to be the set C of allordered pairs (x , y) of real numbers equipped with operations ofaddition and multiplication defined as follows:

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)

(x1, y1) · (x2, y2) = (x1x2 − y1y2, x1y2 + y1x2)

Algebraic propertiesThe operations of addition and multiplication on C satisfies thefollowing filed axioms.

1. The commutative laws for addition and multiplication: for allz1, z2 ∈ C, we have

z1 + z2 = z2 + z1, z1z2 = z2z1

2. The associativity laws for addition and multiplication: for allz1, z2, z3 ∈ C, we have

(z1 + z2) + z3 = z1 + (z2 + z3), (z1z2)z3 = z1(z2z3)

Algebraic properties (Cont’d)3. The distribution laws: for all z1, z2, z3 ∈ C, we have

z1(z2 + z3) = z1z2 + z1z3

4. The additive and multiplicative identity: the complex number(0, 0) is the additive identity and (1, 0) is the multiplicativeidentity.

5. For each complex number z , there is an associated additiveinverse.

6. For each complex number z 6= (0, 0), there is an associatedmultiplicative inverse.

Proof of property 6For each complex number z = (x , y) 6= (0, 0), there is anassociated multiplicative inverse z−1. Moreover,

z−1 =

(x

x2 + y2,−y

x2 + y2

)

Moduli and conjugates - Geometric representationRecall: A complex number is an ordered pair of real numbers! So,it can be thought of as a point in the Euclidean plane. Thus, if

z = x + iy

is a complex number,

I |z |, called the modulus or the absolute value of z , is theEuclidean distance of z to the origin. Moreover,|z | =

√x2 + y2.

I z , called the complex conjugate of z , is the reflection of zwith respect to the real axis. Moreover, z = x − iy .

Prove the following:Let z = (x , y) = x + iy ∈ C. Then,

1. |z | = |z | =√zz

2. Rez = z+z2

3. Imz = z−z2i

Prove the following:Let z1 = x1 + iy1 and z2 = x2 + iy2 be complex numbers. Then,

1. z1 + z2 = z1 + z2, and z1z2 = z1z22. |z1z2| = |z1||z2|3.∣∣∣ z1z2 ∣∣∣ = |z1|

|z2| , provided z2 6= 0.

Triangular InequalityThe modulus of the sum of two complex numbers has an upperbound. If z1 and z2 are two complex numbers, then

|z1 + z2| ≤ |z1|+ |z2|

When do we |z1 + z2| = |z1|+ |z2|?

Triangular Inequality: CorollaryIf z1 and z2 are two complex numbers, then

||z1| − |z2|| ≤ |z1 + z2|.Note: The immediate consequence is that |z1| − |z2| ≤ |z1 − z2|.

Triangular Inequality: GeneralizedIf z1, z2, · · · zn are complex numbers, then

|z1 + z2 + · · ·+ zn| ≤ |z1|+ |z2|+ · · ·+ |zn|