Algebra of Complex Numbers

In this Section, we shall develop the algebra of complex numbers.

Addition of two complex numbers

Let $z_{1}=a+ib$ and $z_{2}=c+id$ be any two complex numbers.Then the sum $z_{1}+z_{2}$ is defined as follows:

$z_{1}+z_{2}=(a+c)+i(b+d)$ which is also a complex number.

Example: Let $z_{1}=2+i3$ and $z_{2}=-6+i5$ . Hence $z_{1}+z_{2}=(2-6)+i(3+5)=-4+i8$

The addition of complex numbers satisfy the following properties:

Closure law

The sum of two complex numbers is a complex number. $z_{1}+z_{2}$ is a complex number for all complex numbers $z_{1}$ and $z_{2}$ .

Commutative law

For any two complex numbers $z_{1}$ and $z_{2}$ , $z_{1}+z_{2}=z_{2}+z_{1}$

Associative law

For any three complex numbers $z_{1},z_{2},z_{3}$ , $(z_{1}+z_{2})+z_{3}=z_{1}+(z_{2}+z_{3})$

Existence of additive identity

There exists the complex number $0+i0$ (denoted as $0$ ) called the additive identity or the zero complex number, such that for every complex number $z$ , $z+0=z$

Existence of additive inverse

To every complex number $z=a+ib$ , we have the complex number $-z=-a+i(-b)$ called the additive inverse or negative of $z$ . We observe that $z+(-z)=0$ (the additive identity).

Difference of two complex numbers

Given any two complex numbers $z_{1},z_{2}$ the difference $z_{1}-z_{2}$ is defined as follows

$z_{1}-z_{2}=z_{1}+(-z_{2})$

Example: $(6+3i)-(2-i)=(6+3i)+(-2+i)=4+4i$

Multiplication of two complex numbers

Let $z_{1}=a+ib$ and $z_{2}=c+id$ be any two complex numbers.Then the product $z_{1}z_{2}$ is defined as follows:

$z_{1}z_{2}=(a+ib)\times (c+id)$

$z_{1}z_{2}=a\times (c+id)+ib\times (c+id)$

$z_{1}z_{2}=ac+iad+ibc+i^{2}bd$

$z_{1}z_{2}=ac+i(ad+bc)+-1\times bd.......(i^{2}=-1)$

$z_{1}z_{2}=(ac-bd)+i(ad+bc)$

The multiplication of complex numbers satisfy the following properties:

Closure law

The product of two complex numbers is a complex number. $z_{1}z_{2}$ is a complex number for all complex numbers $z_{1}$ and $z_{2}$ .

Commutative law

For any two complex numbers $z_{1}$ and $z_{2}$ , $z_{1}z_{2}=z_{2}z_{1}$

Associative law

For any three complex numbers $z_{1},z_{2},z_{3}$ , $(z_{1}z_{2})z_{3}=z_{1}(z_{2}z_{3})$

Existence of multiplicative identity

There exists the complex number $1+i0$ (denoted as $1$ ) called the multiplicative identity, such that for every complex number $z$ , $z\times 1=z$

Existence of multiplicative inverse

For every non-zero complex number $z=a+ib$ $(a\neq 0,b\neq 0)$ , we have the complex number ${\frac {a}{a^{2}+b^{2}}}+i{\frac {-b}{a^{2}+b^{2}}}$ (denoted by ${\frac {1}{z}}$ or $z^{-1}$ ) called the multiplicative inverse of $z$ such that $z\times {\frac {1}{z}}=1$ (the multiplicative identity).