- Complex numbers are the numbers of the form x+iy where $ i=\sqrt{(-1)}$ and x and y are real numbers.
**Definition:-**Complex numbers are defined as an ordered pair of real numbers like (x,y) where

$z=(x,y)=x+iy$- Both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.
- We also write that as Re(z) =x and Im(z) =y
- Two complex Number z
_{1}= a + ib and z_{2}= c + id are equal if a = c and b = d.

z

z

z

Therefore , Real part of both the complex number are added and similarly imaginary part of both the complex numbers are added

- Addition of Complex numbers also follows the closure, commutative,associative Laws.I will leave the proof of them to you.
- We also have additive identity (0) for Complex numbers.
- Additive inverse also exists for the complex number for z=x+iy, Additive inverse = -x-iy such that z+(-z) =0

Here the real part are subtracted and imaginary part is also subtracted

=x

Multiplication is similar to multiplication of algebraic expression and we just need put out i

- Multiplication of Complex numbers also follows the closure, commutative,associative Laws.I will leave the proof of them to you.
- We also have Multiplicative identity (0) for Complex numbers.
- Multiplicative inverse also exists for the complex number .Details are given below

z

=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$

To divide complex number by another complex number, first write quotient as a fraction. Then reduce the denominator complex number to multiplicative Inverse and then simple multiplication applies

Find the value of

$\frac {1+i}{1+2i}$

The multiplicative inverse of (1+2i) is given as

$=(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$

$=(\frac{1}{5}-i \frac{2}{5})$

So $\frac {1+i}{1+2i}$

$=(1+i)(\frac{1}{5}-i \frac{2}{5})$

=$\frac {1}{5} -i \frac {2}{5}+ i \frac {1}{5}+\frac {2}{5}$

=$ \frac {=i}{5} + \frac {3}{5}$

Find the value of (1+3i)(2-5i)

(1+3i)(2-5i) = 2-5i+ 6i-15i

Now i

Therefore,

=17+i

**Notes**- What is complex numbers
- Algebra Of complex Numbers
- Conjugate of Complex Numbers
- Modulus of complex numbers
- Argand Plane
- Polar Representation of the complex number
- Rotation of Complex Number
- Identities for Complex Numbers
- Eulers formula and De moivre's theorem
- Cube Root of unity
- Complex roots of Quadratic equations

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Class 11 Maths Class 11 Physics Class 11 Chemistry Class 11 Biology

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