## Complex Numbers and Quadratic Equations

# Algebra of Complex Numbers

- Addition for 2 complex number z
_{1}= a_{1}+ ib_{1}and z_{2}= a_{2}+ ib_{2}, their sum is defined as z = z_{1}+ z_{2}= (a_{1}+ a_{2}) + i(b_{1}+ b_{2}) - Subtraction for two complex numbers z
_{1}= a_{1}+ ib_{1}and z_{2}= a_{2}+ ib_{2}, the subtraction of z_{2}from z_{1}is defined as z_{1}– z_{2}= (a_{1}− a_{2}) + i(b_{1}− b_{2}) - Multiplication of two complex numbers z
_{1}= a + ib and z_{2}= c + id is defined as z_{1}.z_{2}= (ac − bd) + i(ad + bc) - Division of complex numbers z
_{1}= a + ib and z_{2}= c + id where z_{2}≠ 0 then \frac{z_{1}}{z_{2}} = \frac{a + ib}{c + id} - Multiplicative inverse of non-zero complex number z = a + ib is defined as \tt z^{-1} = \frac{1}{z} = \frac{1}{a + ib}.

### View the Topic in this video From 19:58 To 52:48

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**1. Addition of Complex Numbers**

Let z_{1} = (x_{1} + iy_{1}) and z_{2} = (x_{2} + iy_{2}) be any two complex number, then their sum defined as

z_{1}+z_{2}=(x_{1}+iy_{1})+(x_{2}+iy_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})

**Properties of Addition**

a) **Commutative** z_{1 }+ z_{2 }= z_{2 }+ z_{1}

b) **Associative** (z_{1 }+ z_{2}) + z_{3 }= z_{1 }+ (z_{2 }+ z_{3})

c) **Additive Identity** z + 0 = z = 0 + z

Here, 0 is additive identity.

**Subtraction of Complex Numbers**

Let z_{1 }= (x_{1 }+ iy_{1}) and z_{2 }= (x_{2 }+ iy_{2}) be any two complex numbers, then their difference is defined as

z_{1 }− z_{2 }= (x_{1} − x_{2}) + i (y_{1} − y_{2})

**Multiplication of Complex Numbers**

Let z_{1} = (x_{1} + iy_{1}) and z_{2} = (x_{2} + iy_{2}) be any two complex numbers, then their multiplication is defined as

z_{1} z_{2} = (x_{1} + iy_{1}) (x_{2} + iy_{2}) = (x_{1}x_{2} − y_{1} y_{2}) + i (x_{1}y_{2 }+ x_{2 }y_{1})

**Properties of Multiplication**

a) **Commutative** z_{1} z_{2} = z_{2} z_{1}

b) **Associative** (z_{1} z_{2}) z_{3} = z_{1} (z_{2} z_{3})

c) **Multiplicative Identity** z.1 = z = 1.z

Here, 1 is multiplicative identity of an element z.

d) **Multiplicative Inverse** Every non-zero complex number z there exists a complex number z_{1} such that z.z_{1} = 1 = z_{1} . z

e) **Distributive Law**

i) z_{1} (z_{2} + z_{3}) = z_{1}z_{2} + z_{1} z_{3}

ii) (z_{2} + z_{3}) z_{1} = z_{2}z_{1} + z_{3} z_{1}

**Division of Complex Numbers**

Let z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} be any two complex numbers, then their division is defined as

\tt \frac{z_{1}}{z_{2}}=\frac{\left(x_{1}x_{2}+y_{1}y_{2}\right)+i\left(x_{2}y_{1}-x_{1}y_{2}\right)}{x_2^2+y_2^2}(where z_{2} ≠ 0).

**2. Integral Power of lota** (i)

\tt i=\sqrt{-1},i^{2}=-1,i^{3}=-i,i^{4}=1

So, \tt i^{4n+1}=i,i^{4n+2}=-1,i^{4n+3}=-i,i^{4n+4}=i^{4n}=1

In other words, \tt \begin{cases}i^{n}=\left(-1\right)^{n/2},if \ n \ is \ an \ even \ integer & \\ \tt i^{n}=\left(-1\right)^{\frac{n-1}{2}}.i,if \ n \ is \ an \ odd \ integer & \end{cases}