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ALGEBRA OF COMPLEX NUMBERS

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Introduction

algebraofcomplexnumbers1
A number of the form z=a + ib, i=√-1, where ‘a’ and ‘b’ are real numbers is called a complex number.  Here ‘a’ is the real part and ‘b’ is the imaginary part of ‘z’.

  • The real part is denoted by Re z and imaginary part by Im z.
  • If the real part is zero then is is said to be purely imaginary and if the imaginary part is zero then it is said to be purely real.
  • All real numbers are complex numbers with imaginary part zero.

For example:   5= 5 + i0
                     -3 =-3 + i0
Two complex numbers z1=a + ib and z2=c + id are equal if a=c and b=d
If two complex numbers are equal then their real parts are equal and imaginary parts are also equal.
Example: If 4x+ i(4y-x) = 8+ i6, then find ‘x’ and ‘y’
Solution:  Since the complex numbers are equal, we have
                 4x=8 and 4y-x=6
                   4x=8
                     x=2
               4y-x =6 becomes 4y-2 =6

                 4y=6+2=8
                   y=2
Hence x=2 and y=2

Algebra of Complex Numbers

In algebra of complex numbers we deal with addition of two complex numbers, subtraction of two complex numbers,  multiplication of two complex numbers , division of two complex numbers and the properties satisfied by them.

Addition of two complex numbers


Let z1= a + ib and z2= c + id be any two complex numbers, then the sum z1+z2 = (a+c) + i(b+d), which is again a complex number.  While adding two complex numbers, we have to add real parts together ad imaginary parts together.
Properties satisfied by addition of complex numbers.

  • Closure Property:  The sum of two complex numbers is a complex number.
  • Commutative Property:  For any two complex numbers z1 and z2, z1+z2= z2+z1
  • Associative Property:  For any three complex numbers z1, z2 and z3, z1+(z2+z3) = (z1+z2)+z3
  • Existence of additive identity:  For every complex number ‘z’ there exists a complex number 0+i0 (denoted as 0) such that z+0=z, which is called the additive identity.
  • Existence of additive inverse:  To every complex number       z=a +ib, we can find a complex number –z = (-a)+(-ib)               [ denoted as –z] called the additive inverse such that z + (-z) =0.


Difference of two complex numbers

Given two complex numbers z1 and z2, the difference z1 –z2 is defined as follows, z1 – z2 = z1 + (-z2)
All the above mentioned properties are satisfied here also.
Example: Evaluate (1-i)-(-1+i6)
Solution: (1-i) – (-1+i6) = (1-i) + (1-i6)
                           = [1+1] + [-i+-i6]
                           = 2 + (-i7)
                           = 2 + -7i
                           = 2 – 7i

Multiplication of two complex numbers

Let z1=a + ib and z2=c + id be any two complex numbers. Then, the product z1z2 is defined as follows,
z1z2 = (ac-bd) + i(ad+bc)
Multiplication of complex numbers satisfies the following properties.

  • Closure Property:  The product of two complex numbers is again a complex number.
  • Commutative Property:  For any two complex numbers z1 and z2, z1z2= z2z1
  • Associative Property: For any three complex numbers z1, z2 and z3, (z1z2)z3 = z1(z2z3)
  • Existence of multiplicative identity: To every complex number z, there exists 1_i0 (denoted as 1), called the multiplicative identity such that z.1 = 1.z =z.
  • Existence of multiplicative inverse:  For every non-zero complex number z we have 1/z or z-1, called the multiplicative inverse of z such that z.1/z=1
  • Distributive Law:  For any three complex numbers z1, z2, z3

a)    z1(z2+z3) = z1z2 + z1z3
b)    (z1+z2)z3 = z1z3 + z2z3

Division of two complex numbers


Given any two complex numbers z1 and z2, where z2≠0, the quotient z1/z2 is defined by z1. 1/z2
Here we multiply by the conjugate of the denominator to get the answer.
algebraofcomplexnumbers2
Example: Find the multiplicative inverse of √5+3i
Solution: The multiplicative inverse of √5 + 3i is  algebraofcomplexnumbers3
       algebraofcomplexnumbers4
Hence the multiplicative inverse of √5 + 3i is (√5/14) + i(3/14)

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