Algebra of Complex Numbers

Algebraic Operations on Complex Numbers

The basic algebraic operations on complex numbers discussed here are:

  • Addition of Two Complex Numbers
  • Subtraction(Difference) of Two Complex Numbers
  • Multiplication of Two Complex Numbers
  • Division of Two Complex Numbers.

Addition of Two Complex Numbers

We know that a complex number is of the form z=a+ib where a and b are real numbers.

Let two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2

Then the addition of the complex numbers z1 and z2 is defined as,

z1+z2 =( a1+a2 )+i( b1+b2 )


The real part of the resulting complex number is the sum of the real part of each complex number and the imaginary part of the resulting complex number is equal to the sum of the imaginary part of each complex number.

That is, Re(z1+z2 )= Re( z1 )+Re( z1 )

Im( z1+z2 )=Im( z1)+Im(z2)

Example:

z1 = a+3i, z2 = 4+bi, z3 = 6+10i

Find the value of a and b if z3 = z1+z2

Solution:

By the definition of addition of two complex numbers,

Re(z3 ) = Re(z1 )+Re(z2 )

6 = a + 4

a = 6 – 4 = 2

Im(z3 ) = Im(z1 ) + Im(z2 )

10 = 3+b

b = 10-3 =7

Properties of Addition of Complex Numbers

Name of the Property

Description

Expression

Closure property

Addition of two complex numbers is a complex number

z1 + z2 = z

Commutative property

Order of addition of two complex numbers, does not change the result

z1 + z2 = z2 + z1

Associative property

Regrouping three complex numbers, while adding them, does not change the result

(z1+z2)+z3 = z1+(z2+z3)

Additive inverse property

If z = a+ib is a complex number, then its additive inverse will be -z = -a – ib

z+(-z) = 0

Additive identity

If a value added to complex number results in the same complex number, then it becomes the additive identity

(a+ib) + (0 + i0) = a + ib

Difference of Two Complex Numbers

Let the complex numbers z1 = a1+ib1 and z2 = a2+ib2, then the difference of z1 and z2, z1-z2 is defined as,

z1-z2 = (a1-a2)+i(b1-b2)

Re(z1-z2)=Re(z1)-Re(z2)

Im(z1-z2)=Im(z1)-ImRe(z2)

Example:

z1 =8+ai,z2=6+4i,z3 =2. Find the value of a if z3=z1-z2

Solution:

By the definition of difference of two complex numbers,

Im3=Im1-Im2

0 = a – 4

a = 4

Note: All real numbers are complex numbers with imaginary part as zero.

Multiplication of Two Complex Numbers

We know the expansion of (a+b)(c+d)=ac+ad+bc+bd

Similarly, Let the complex numbers z1 = a1+ib1 and z2 = a2+ib2

Then, the product of z1 and z2 is defined as:

z1 z2=(a1+ib1)(a2+ib2)

z1 z2 = a1 a2+a1 b2 i+b1 a2 i+b1 b2 i2

Since,  i2 = -1, therefore,

z1 z2 = (a1 a2 – b1 b2 ) + i(a1 b2 + a2 b1 )

Example:

z1=6-2i, z2=4+3i. Find z1 z2

Solution:

z1 z2 = (6-2i) (4+3i)

= 6 × 4 + 6 × 3i + (-2i) × 4 + (-2i)(3i)

= 24 + 18i – 8i – 6i2

= 24 + 10i + 6

= 30 + 10i

Multiplicative inverse of a complex number

For any non-zero complex number z=a+ib(a≠0 and b≠0) there exists another complex number z-1 or 1/z, which is known as the multiplicative inverse of z such that zz-1 = 1.

z = a+ib, then,

Example:

z = 3 + 4i

Solution The numerator of z-1 is conjugate of z, that is a – ib

Denominator of z-1 is sum of squares of the Real part and imaginary part of z

Here, z =  3 + 4i 

Properties of Multiplication of Complex Numbers

Name of the Property

Description

Expression

Closure property

Product of two complex number is a complex number only

z1 x z2 = z

Commutative property

Change of order of complex numbers, does not change the result of their product

z1.z2 = z2.z1

Associative property

Regrouping of complex numbers, does not change the result of their product

z1(z2.z3) = (z1.z2)z3

Distributive property

Multiplication of a complex number with the sum of two complex numbers is given by:

z1(z2+z3) = z1.z2 + z1.z3

Division of Complex Numbers

Let the complex number z1 = a1 + ib1 and z2 = a2 + ib2, then the quotient of z1/z2 is defined as, z1/z2= z1(z2-1)

Therefore, to find z1/z2, we have to multiply z1 with the multiplicative inverse of z2.

Now, let us discuss in detail about the division of complex numbers:

Let z1 = a1+ib1 and z2 = a2+ib2, then z1/z2 is given as:

z1/z2 = (a1+ib1)/(a2+ib2)

Hence, (a1+ib1)/(a2+ib2) = [(a1+ib1)(a2-ib2)]/[(a2+ib2)(a2-ib2)]

(a1+ib1)/(a2+ib2) = [(a1a2)-(a1b2i)+(a2b1i)+b1b2)]/[(a22+b22)]

(a1+ib1)/(a2+ib2) = [(a1a2)+(b1b2) +i(a2b1-a1b2)]/(a22+b22)

Hence,

Example:

If z1 = 2 + 3i and z2 = 1 + i, find  $\frac{z_1}{z_2}$z1z2