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Complex Numbers | Algebra of Complex Number





Complex Number

  • Complex numbers are the numbers of the form x+iy where i=(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 in the equation
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
=(aa2+b2)+i(ba2+b2)

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
1+i1+2i
Solution:
The multiplicative inverse of (1+2i) is given as
=(aa2+b2)+i(ba2+b2)
=(15i25)
So 1+i1+2i
=(1+i)(15i25)
=15i25+i15+25
==i5+35

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) 1i2
B) 1i2
C) 1i
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(3xy)=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)+(1141i)(1i)
A) 435i
B) 435i
C) 43+5i
D) 5i43
Question 5
if z1=1+3i and z2=24i,then the Re(z1z2) then
A) -10
B) 14
C)10
D)-14
Question 6
The value of 6+3i2i is
A) 12(9+12i)
B) 12(9+12i)
C) 15(912i)
D) 15(9+12i)


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