Complex Numbers

Complex numbers are a new set of numbers other than real numbers. We know about natural numbers, integers, rational numbers and irrational numbers. All these numbers taken together form the set of real numbers.

The system of real numbers is complete in itself – every real number corresponds to a point in the real line and conversely. However, there is an important fact about real numbers which is,

The square of a real number is never negative.

Because of this fact, we are unable to solve an equation which demands a value whose square is negative. For example, an equation $x^2+4=0$ demands a value of $x$ whose square is negative. Hence, there is no solution for this equation in the system of real numbers.

To solve such equations, we have to extend the system of numbers. This is done by introducing a new set of numbers called complex numbers. It is believed that Cardon used complex numbers in 1546.

Now we shall proceed to define the complex numbers.

Simply, a complex number is defined as an ordered pair of real numbers.

If $a$ and $b$ are real numbers, then the ordered pair $(a,b)$ is a complex number. The first number, $a$, is called the real part and the second number, $b$, is called the imaginary part of the complex number. In line with this interesting terminology, the x-axis is often referred to as the real axis and the y-axis as the imaginary axis.

A complex number is usually denoted by a single letter such as $z$, $w$, etc. Now we shall give the complete definition of a complex number.

A number of the form $z=a+ib=(a,b)$ is called a complex number where $a$ is the real part of $z$ denoted by $\text{Re}(z)$ and $b$ is the imaginary part of $z$ denoted by $\text{Im}(z)$.

In $z=a+ib$, $\;i$ is called the imaginary unit.

Since a complex number $(a,b)$ is an ordered pair of real numbers $a$ and $b$, two complex numbers $(a,b)$ and $(c,d)$ are said to be equal if and only if $a=c$ and $b=d$ respectively.

Some Definitions Relating Complex Numbers

1. The sum of two complex numbers $z=(a,b)\;\;\text{and}\;\;w=(c,d)$ is defined to be a complex number $z+w$ such that $z+w=(a+c,b+d)$

2. The product of two complex numbers $z=(a,b)\;\;\text{and}\;\;w=(c,d)$ is defined to be a complex number $zw$ such that $zw=(ac-bd,ad+bc)$

3. If $z=(a,b)≠(0,0)$ be a complex number then the reciprocal of $z$ denoted by $\frac{1}{z}$ or $z^{-1}$ is defined as $z^{-1}=\left(\frac{a}{a^2+b^2},\frac{-b}{a^2+b^2}\right)$

This is called the multiplicative inverse of $z$ as $zz^{-1}=(1,0)$ Also, $\frac{w}{z}=wz^{-1}$ where, $w$ and $z$ are complex numbers.

Hence, if $z=(a,b)$ and $w=(c,d)$ be two complex numbers, then, $\frac{w}{z}=wz^{-1}=(c,d)\left(\frac{a}{a^2+b^2},\frac{-b}{a^2+b^2}\right)$ $=\left(\frac{ac+bd}{a^2+b^2},\frac{ad-bc}{a^2+b^2}\right)$