Algebra of Complex Numbers

What is Complex Number?

Complex Number is an algebraic expression including the factor  i = √-1. These numbers have two parts, one is called as the real part and is denoted by Re(z) and other is called as the Imaginary Part. Imaginary part is denoted by Im(z) for the complex number represented by ‘z’.

Either of the part, real part or imaginary part, can be positive, negative, integer, fraction, decimal, rational, irrational or even zero. If only real part of any complex number ‘z’ is zero, i.e. Re(z) = 0, then these types of complex numbers are termed as ‘Purely Imaginary Number’. While if only imaginary part of any complex number ‘z’ is zero, that is. Im(z) = 0, then these are called as ‘Purely Real Numbers”.

Complex Number in its Cartesian form is expressed as z = a + ib or z = Re(z) + iIm(z). 

For Example, for a complex number, z = 2 + 3i, a = Re(z) = 2  and b = Im(z) = 3.

Complex Numbers form a super set of all the different types of numbers.
 

What are Imaginary Numbers in Math?

Imaginary Numbers are the real numbers multiplied with the imaginary unit ‘i’. ‘i’ (or ‘j’ in some books) in math is used to denote the imaginary part of any complex number. It helps us to clearly distinguish the real and imaginary part of any complex number. Moreover, i is just not to distinguish but also has got some value.

i = √-1

Main application of complex numbers is in the field of electronics. In electronics, already the letter ‘i’ is reserved for current and thus they started using ‘j’ in place of i for the imaginary part.

Explain Algebra of Complex Numbers?

Let’s understand the different algebras of complex number one by one below.
 

Equality of Complex Numbers

Two Complex numbers z₁ and z₂ are equal iff, 

Condition 1) Re (z₁) = Re (z₂)

Condition 2) Im (Z₁) = Im(z₂)

So If, z₁ = x + 3i and z₂ = -2 + y_i are equal, then as per above conditions,

Re(z₁) = x and Re(z₂) = -2, so x = -2

And Similarly

Im(z₁) = 3 and Im(z₂) = y, so y = 3
 

Addition of Complex Numbers

Let z₁ = a + ib and  z₂ = c + id, then the sum of this two complex numbers that is z₁+ z₂ calculated as:

z₁+ z₂ = (a + ib) + (c + id)

=(a + c) + i(b + d)

Therefore, 

z₁ + z₂ = Re (z₁+ z₂) + Im(z₁+ z₂)

Addition of complex numbers can be another complex number.

Example

Let z₁= -1 + 4i and z₂ = 8 + 2i,

Then z₁+ z₂ = (-1 + 8) + i(4 + 2) =7+ i6

Addition of complex numbers satisfy the following properties

• Closure Law: The sum of two complex numbers is another complex number, that is. if z₁ + z₂ where  z₁ and  z₂ are complex numbers, then z will also be a complex number

• Commutative Law: As per commutative law, for any two complex numbers z₁ and  z_2, z₁ + z₂ = z₂ + z_1.

• Associative Law: For any three complex numbers say ​(z₁+ z₂ )+ z₃ = z₁+ (z₂+ z₃).

• Existence of Additive Identity: Additive identity also called as zero complex number is denoted as 0 (or 0 + i0), such that, for every complex number z, z + 0 = z.

• Existence of Additive Inverse: Additive inverse or negative of any complex number z, is a complex number whose both real and imaginary parts have the opposite sign. It is represented by –z and z + (-z) = 0
   

Difference of two Complex Numbers

Let  z₁= a + ib and  z₂ = c + id, then the difference of this two complex numbers that is. z₁ -  z₂ is calculated as:

z₁- z₂= (a + ib) - (c + id)

= (a – c) + i (b – d)

Therefore, 

z₁ - z₂ = Re(z₁ - z₂ ) + Im(z₁ - z₂)

Difference of complex numbers can be another complex number.

Example

Let z₁= -1 + 4i and z₂ = 8 + 2i,

Then z₁- z₂ =(-1 -8) + i(4 – 2) = -9 + i2

Difference of two complex numbers also satisfies the same properties as the addition of the two follows.
 

Multiplication of two Complex Numbers

Let z₁= a + ib and z₂ = c + id, then the multiplication of this two complex numbers that is. z₁× z₂  is calculated as:

z₁× z₂ = (a + ib) ×(c + id)

z₁×z₂= (ac – bd) + i(ad + bc)

Therefore,

z₁ × z₂ = [Re(z₁) Re(z₂) – Im(z₁) Im(z₂)]+ i[Re(Z₁) Im(z₂) + Im(z₂) Re(z₂)]

Example

Let z₁= -1 + 4i and z₂ = 8 + 2i,

Then, z₁ × z₂ = (-8 -8) + i(-2 + 32) =-16 + i30

If k is any constant, then

kz = k(a + ib) = ka + ikb

Also, if k₁ and k₂ are any real constant, then

k(z₁ + z₂ )= kz₁+ kz₂

k₁ (k₂ z)=(k₁ k₂ )z

(k₁+ k₂ )z=k₁ z+k₂z

Multiplication of two complex numbers also posses few properties, let’s list them all here below:

• Closure Law: The product of any two complex numbers is another complex number, that is. if z = z_1- z₂ where z₁ and z₂ are complex numbers, then z will also be a complex number

• Commutative Law: As per commutative law, for any two complex numbers z₁ and z₂, z₁ – z₂ = z₂ z₁.

• Associative Law: For any three complex numbers say z₁, z₂ and z_3. (z₁ z₂ ) z₃ = z₁ (z₂ z₃).

• Multiplicative Identity: Multiplicative Identity  is denoted as 1 (or 1 + i0), such that, for every complex number z, z .1 = z.

• Multiplicative Inverse: For any non- zero complex number z,1/z or z^-1 is called as ssthe multiplicative inverse as z,1/z = 1 If z = x + iy, then

• Distributive Law:  For any three complex numbers z₁, z₂ and z₃ we have

z₁ (z₂+ z₃ )= z₁ z₂+ z₁ z₃

(z₁+ z₂ ) z₃ = z₁ z₃ + z₂ z₃

Division of two Complex Numbers

Let z₁ = a + ib and  z₂ = c + id, then the division of this two complex numbers that is z₁/z₂ is calculated as:

On Rationalization:

Example

Let   z₁ = -1 + 4i and z₂ = 8 + 2i,

Power of i

Since i = √-1 or  i = -1 which means i can be assumed as the solution of the equation x² + 1 = 0 .i is called as Iota in complex numbers.

We can further formulate as,

 i² = -1

i³ = i² * i = -i

i⁴ = i² * i² =1

So we can say now, i⁴ⁿ where n is any positive integer.

i⁵ = i⁴ * i =1 *i = i

i⁶ =i⁴ * i² = i² = -1

Also, 

Also note that  i + i² + i³ + i⁴ = 0 or i⁴ⁿ⁺¹ + i⁴ⁿ⁺² + i⁴ⁿ⁺³ = 0 for any integer n.
 

Square root of a Complex Number

Let  z₁ = (a + ib) then the square root of a complex number z₁, that is √z₁ can be calculated as follows:

Assume, √z₁ = x + iy

that is √(a + ib) = x + iy, Now squaring both the sides,

On simplification we get

(a + ib) = (x²- y² )+ 2xyi,

Now comparing both sides real and imaginary parts, we get

a = (x²- y² ) and b = 2xy 

Now using the below identity:

(x²- y²) = (x² + y²) - 4xy, find the value of x² + y², 

 And then finally find the values of x² and y²,we get

On further simplification, get the value of x and y by taking square root both sides,

Finally we get,

Example

Let’s find the square root of 8 – 6i.

Assume, √(8 – 6i) = x + iy On squaring and simplifying, we get

x²- y² = 8 and 2xy = – 6

And finally we get, √(z₁) = x+ iy = ± (3 – i)
 

What does the asterisk in Complex Numbers mean?

Asterisk (symbolically *) in complex number means the complex conjugate of any complex number.

Let z₁ = x + iy is any complex number, then its complex conjugate is represent by

We can also define the complex conjugate of any complex number as the complex number with same real part and same magnitude of imaginary part but with opposite sign as of given complex number.

Refer the below table to understand it more clearly

Also note few important properties of conjugate:

More Readings

Algebra of Complex Numbers