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Complex Numbers|Algebra of Complex Number





Complex Number

  • 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 z1 = a + ib and z2 = c + id are equal if a = c and b = d.
Example
z1= 2+4i
z1= 8-4i



Algebra of Complex Number

Just like real numbers, Addition ,subtraction,multiplication ,division is also performed on Complex numbers

Addition of complex numbers

Let z1=x1+iy1 and z2=x2+iy2 then
z1+z2=(x1+x2)+i(y1+y2)
Therefore , Real part of both the complex number are added and similarly imaginary part of both the complex numbers are added
Important Notes
  • 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

Subtraction of complex numbers

z1-z2=(x1-x2)+i(y1-y2)
Here the real part are subtracted and imaginary part is also subtracted

Mulitiplication of complex numbers

(z1.z2)=(x1+iy1).(x2+iy2)
=x1x2 -y1y2 + i(x1y2 + x2y1)
Multiplication is similar to multiplication of algebraic expression and we just need put out i2=-1 wherever it comes
Important Notes
  • 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

Multiplicative Inverse

for $z=a+ib$
z-1 is given by
=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$

Division of Complex Numbers


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
Example -1
Find the value of
$\frac {1+i}{1+2i}$
Solution:
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)(1/5-2i/5)
=$\frac {1}{5} -i \frac {2}{5}+ i \frac {1}{5}+\frac {2}{5}$
=$ \frac {=i}{5} + \frac {3}{5}$
Example -2
Find the value of (1+3i)(2-5i)
Solution:
(1+3i)(2-5i) = 2-5i+ 6i-15i2
Now i2=-1
Therefore,
=17+i

Quiz Time

Question 1 The multiplicative inverse of complex number $(1+i)$ is .
A) $\frac {1-i}{2}$
B) $\frac {1-i}{\sqrt {2}}$
C) ${1-i}$
D) None of the above
Question 2 Which of the following is incorrect statement ? A) Addition of complex number follows closure, commutative,associative Laws
B) Multiplication of complex number follows closure, commutative,associative Laws
C) The additive inverse of complex number is $z$ is $-z$
D) None of the above
Question 3 if 4x + i(3x – y) = 3 + i (– 6),the values of x and y are
A) 1/4, 33/4
B) 3/4, 1/4
C) 3/4,33/4
D)1/4,1/4
Question 4 What is the value $(33+ 45i) + (11-41i) - (1-i)$
A) $43 -5i$
B) $-43 -5i$
C) $43 +5i$
D) $5i -43$
Question 5 if $z_1= 1+ 3i$ and $z_2= 2-4i$,then the $Re(z_1z_2)$ then
A) -10
B) 14
C)10
D)-14
Question 6The value of $\frac {6+3i}{2-i}$ is
A) $\frac {1}{2} (-9+12i)$
B) $\frac {1}{2} (9+12i)$
C) $\frac {1}{5} (9-12i)$
D) $\frac {1}{5} (9+12i)$


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