1.Properties of Complex Numbers

Properties of Complex Numbers

The primary purpose of this article “Properties of Complex Numbers” is to define and study the set of complex numbers.The set of real numbers which is generator of complex numbers,is of fundamental importance to this study and we understand that the reader is well acquainted with the set of real numbers R as well as all their properties.

Also Read This Article:- Complex Numbers in Class 11

2.Primary Properties of Complex Numbers

(1.)Equality of Two Complex Numbers

\text{Two complex numbers } z_1 = (x_1, y_1) \text{ and } z_2 = (x_2, y_2) \text{ are said to be equal if } x_1 = x_2 \text{ and } y_1 = y_2.
Thus two complex numbers are equal if and only if the real part of one is equal to the real part of the other and the imaginary part of the one is equal to the imaginary part of the other.
(2.)Addition (Sum) and multiplication (product) of two complex numbers.
If and are any two complex numbers,then their sum and product are defined by the following relations:

\text{If } z_1 = (x_1, y_1) \text{ and } z_2 = (x_2, y_2) \text{ are any two complex numbers,} \\ \text{then their sum and product are defined by the following relations:} \\ z_1 + z_2 = (x_1 + x_2, y_1 + y_2) \quad (\text{Sum}) \\ z_1z_2 = (x_1x_2 - y_1y_2, x_1y_2 + y_1x_2) \quad (\text{Product})
(3.)Usual represention of a complex number z=(x,y) in the form

i^2 = (\sqrt{-1})^2 = -1,  i^3 = i \cdot i^2 = i(-1) = -i \\ i^4 = (i^2)^2 = (-1)^2 = 1,  i^5 = i^4 \cdot i = 1 \cdot i = i,  i^6 = -1
Keeping these result in view the following computation is correct
But the computation is wrong

3.Properties of the Addition of Complex Numbers

Properties of Complex Numbers

(1.)commutative of Addition in C
z_1 + z_2 = z_2 + z_1 \\ \text{proof: Let } z_1 = (x_1, y_1), z_2 = (x_2, y_2) \text{ where } x_1, y_1, x_2, y_2 \text{ are real numbers.} 
We have z_1 + z_2 = (x_1 + x_2, y_1 + y_2) [by definition of addition in C]
= (x_2 + x_1, y_2 + y_1) (addition of real numbers is commutative)
= (x_2 ,y_2)+ (x_1 ,y_1) \\  =z_2+z_1 \\ \Rightarrow z_1+z_2=z_2+z_1
for all complex numbers z_1 and z_2
(2.)Associativity of Addiction in C
(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \\ \text{proof: Let } z_1 = (x_1, y_1), z_2 = (x_2, y_2), z_3 = (x_3, y_3)
Where x_1, y_1, x_2,y_2,x_3,y_3 are real numbers
We have (z_1 + z_2) + z_3 = ((x_1 + x_2) + x_3, (y_1 + y_2) + y_3) \\ (x_1 + x_2, y_1 + y_2)+(x_3,y_3) [by def. of addition in C]
=\left( (x_1 + x_2)+x_3, (y_1 + y_2)+y_3 \right) (by def. of addition in C)
=(x_1 + (x_2 + x_3), y_1 + (y_2 + y_3)) [addition of real numbers is associative)
(x_1, y_1)+ \left( (x_2, y_2)+(x_3, y_3) \right)  (by def. of addition in C)
(x_1, y_1)+ \left( (x_2, y_2)+(x_3, y_3) \right) \\ =z_1 + (z_2 + z_3)
Hence (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)
(3.)Additive Identity
The complex number (0,0) or 0+i.0 is the additive identity,since for every complex number (x,y),we have
(x,y)+(0,0)=(x+0,y+0)=(0,0)+(x,y)
The complex number and the zero complex number and is simply written as 0.
A complex number x+iy is said to be a non-zero complex number if at least one of x and y not zero.
(4.)Additive Inverse
The complex number (-x,-y) is the additive inverse of the complex number (x,y) since
(x,y)+(-x,-y)=(x-x,y-y)=(0,0)=the additive identity
and also (-x,-y)+(x,y)=(0,0)
The complex number (-x,-y) is called the negative of the complex number (x,y) and we denote (-x,-y) by – (x,y)
Thus if z=(x,y) then -z=-(x,y)=(-x,-y)
(5.)Cancellation law for addition in C
If z_1, z_2, z_3 are any complex numbers then z_1 + z_3 = z_2 + z_3 \Rightarrow z_1 = z_2

4.Properties of the Multiplication of Complex Numbers

(1.)Commutativity of Multiplication in C
z_1 z_2 = z_2 z_1 \\ \text{proof: Let } z_1 = (x_1, y_1), z_2 = (x_2, y_2)
where x_1, y_1,x_2, y_2 are real numbers
We have z_1 z_2 =(x_1, y_1) (x_2, y_2) \\ =(x_1x_2 - y_1y_2, x_1y_2 + y_1x_2) (by def. of multiplication in C)
= (x_2x_1 - y_2y_1, x_2y_1 + y_2x_1)  , (as real numbers are commutative for addition and multiplication)
z_1=(x_1, y_1), z_2 = (x_2, y_2) (by def. of multiplication in C)
= z_2 z_1
Hence z_1 z_2 = z_2 z_1
(2.)Associativity of multiplication in C
(z_1 z_2) z_3 = z_1 (z_2 z_3)
proof: Let z_1=(x_1, y_1), z_2=(x_2, y_2), z_3=(x_3, y_3)
Where x_1, y_1, x_2, y_2, x_3, y_3 are real numbers \\
we have (z_1 z_2) z_3 = [(x_1, y_1) \cdot (x_2, y_2)] (x_3, y_3) \\ = (x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2)(x_3, y_3) \text{ by def. of} \\ \text{multiplication in } \mathbb{C} \\ = [(x_1 x_2 - y_1 y_2)x_3 - (x_1 y_2 + y_1 x_2)y_3, (x_1 x_2 - y_1 y_2)y_3 + (x_1 y_2 + y_1 x_2)x_3] \\ = (x_1 x_2 x_3 - y_1 y_2 x_3 - x_1 y_2 y_3 - y_1 x_2 y_3, x_1 x_2 y_3 - y_1 y_2 y_3 \\ - y_1 y_2 y_3 + x_1 y_2 x_3 + y_1 x_2 x_3) \\ \text{by distributive law for real numbers } \cdots (1) \\ \text{ Also } z_1 (z_2 z_3) = (x_1, y_1) \cdot [(x_2, y_2) (x_3, y_3)] \\ = (x_1, y_1) (x_2 x_3 - y_2 y_3, x_2 y_3 + y_2 x_3), \text{ by def.} \\ \text{of multiplication in } \mathbb{C} \\ = (x_1 [x_2 x_3 - y_2 y_3] - y_1 [x_2 y_3 + y_2 x_3], x_1 [x_2 y_3 + \\ y_2 x_3] + y_1 [x_2 x_3 - y_2 y_3]) \\ = (x_1 x_2 x_3 - x_1 y_2 y_3 - y_1 x_2 y_3 - y_1 y_2 x_3, x_1 x_2 y_3 + \\ x_1 y_2 x_3 + y_1 x_2 x_3 - y_1 y_2 y_3) \text{ by distributive} \\ \text{law for real numbers} \\ = (x_1 x_2 x_3 - x_1 y_2 y_3 - y_1 x_2 y_3 - y_1 y_2 x_3, x_1 x_2 y_3 + \\ y_1 y_2 y_3 + x_1 y_2 x_3 + y_1 x_2 x_3) \cdots (2)
by laws of real numbers
From (1) and (2), we have
(z_1 z_2) z_3 = z_1 (z_2 z_3)
(3.)Multiplicative Identity
The complex number (1,0) or 1+i.0 or simply 1 is the multiplicative identity since for every complex number (x,y) we have
(x,y)(1,0)=(x.1-y.0,x.0+y.1)=(x,y)=(1,0)(x,y)
(4.)Multiplicative Inverse
The Complex number (x,y) is called the multiplicative inverse of the complex number (a,b) if (x,y)(a,b)=(1,0) or simply 1
We have (x, y)(a, b) = (1, 0) \\
\Rightarrow (xa - yb, xb + ya) = (1, 0) \\ \Rightarrow xa - yb = 1 \text{ and } xb + ya = 0
The equation xa-yb=1 and xb+ya=0 give
x =\frac{a}{a^2+b^2}, y = \frac{-b}{a^2+b^2}
Provided a^2 + b^2 \neq 0 which implies that a and b are not both zero i.e.,(a,b) is a non-zero complex number.
Thus every non-zero complex number possesses multiplicative inverse and the multiplication inverse of the complex number is the complex number
\left(\frac{a}{a^2+b^2}, \frac{-b}{a^2+b^2}\right)
If z is a non-zero complex number (a, b) \neq (0, 0) ,the multiplicative inverse of z is denoted by \frac{1}{z} \text{ or } z^{-1}
(5.)Cancellation law for multiplication in C
\text{If } z_1, z_2, z_3 \in \mathbb{C} \text{ and } z_3 \neq 0 \\ \text{then } z_1 z_3 = z_2 z_3 \Rightarrow z_1 = z_2
(6.)Multiplicative Distributes Addition in C
z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3 \\ proof: z_1(z_2 + z_3) \\ \text{Let} z_1 = (x_1, y_1), z_2 = (x_2, y_2), z_3 = (x_3, y_3) \text{where } x_1, y_1, x_2, y_2, x_3, y_3 \text{ are real numbers. we have} \\ z_1(z_2 + z_3) = (x_1, y_1)[(x_2, y_2) + (x_3, y_3)] \\ = (x_1, y_1)(x_2 + x_3, y_2 + y_3) \text{ by def. of addition in } \mathbb{C} \\ = (x_1[x_2 + x_3] - y_1[y_2 + y_3], x_1[y_2 + y_3] + y_1[x_2 + x_3]) \text{by def. of multiplication in } \mathbb{C} \\ = (x_1x_2 + x_1x_3 - y_1y_2 - y_1y_3, x_1y_2 + x_1y_3 + y_1x_2 + y_1x_3) \text{by distributive law of real numbers} \\ = ([x_1x_2 - y_1y_2] + [x_1x_3 - y_1y_3], [x_1y_2 + y_1x_2] \\ + [x_1y_3 + y_1x_3]), \text{ by laws of real numbers} \\ = (x_1x_2 - y_1y_2, x_1y_2 + y_1x_2) + (x_1x_3 - y_1y_3, x_1y_3 + y_1x_3) \text{by def. of addition in } \mathbb{C} \\ = (x_1, y_1)(x_2, y_2) + (x_1, y_1)(x_3, y_3) \text{ by def. of multiplication in } \mathbb{C} \\ = z_1z_2 + z_1z_3 \\ \text{Hence } z_1(z_2 + z_3) = z_1z_2 + z_1z_3

Also Read This Article:- Complex Numbers in Complex Analysis

5.Difference of Two Complex Numbers

With the above theory,one can understand the Properties of Complex Numbers.

6.Division in C

Properties of Complex Numbers

(x,y) (c,d)=(a,b)
We have (x,y) (c,d)=(a,b)
(xy-yd,xd+yc)=(a,b)
xy-yd=a and xd+yc=b
The equation xy-yd=a and xd+yc=b give
x = \frac{ac + bd}{c^2 + d^2}, y = \frac{bc - ad}{c^2 + d^2}
Provided c^2 + d^2 \neq 0 which implies that c and d are not both zero
Thus division,except (0,0),is always possible in the set of complex numbers.
If z_1  and  z_2  are two complex numbers such that z_2 \neq 0 then the quotient of the complex numbers z_1  and  z_2 is defined by the relation
\frac{z_1}{z_2} = z_1 \frac{1}{z_2} = z_1(z_2)^{-1}

With the above theory,one can understand the Properties of Complex Numbers.

7.Some Properties of Conjugate Complex Numbers

With the above theory,one can understand the Properties of Complex Numbers.

Also Read This Article:- Complex number

8.Properties of Absolute Values of Complex Numbers

Properties of Complex Numbers

With the above theory,one can understand the Properties of Complex Numbers.
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9.Frequently Asked Questions Related to Properties of Complex Numbers

\begin{array}{|c|} \hline \text{विद्यार्थियों के एक समूह में से कुल संख्या के} \\ \text{ घनमूल का} \frac{1}{2} \text{विद्यार्थी पढ़ रहे हैं।} \\ \text{शेष सात विद्यार्थी खेल रहे हैं।} \\ \text{बताओ कुल कितने विद्यार्थी हैं?} \\ \text{(Out of a group of students, } \\  \frac{1}{2} \text{of the cube root of the total} \\ \text{ number of students are studying.} \\ \text{The remaining ten students are playing.} \\ \text{How many students are there?)} \\ \text{दिनांक 01.07.2026 के प्रश्न का उत्तर:} \\ \text{ Answer to Question Dated 01.07.2026:} \\ 2(lb + bh + lh) = 24, \text{ and } 4(l+b+h )= 6 \\ \Rightarrow (l+b+h)^2=6^2\\ \Rightarrow l^2 + b^2 + h^2 +2(lb + bh + lh)=36 \\ \Rightarrow l^2 + b^2 + h^2 +22=36 \\ \Rightarrow l^2 + b^2 + h^2=36-22=14 \\ \sqrt{l^2 + b^2 + h^2} = \sqrt{14} \\ \hline  \end{array}
This article has been prepared by **Satyam Coaching Centre** on the **Satyam Mathematics** blog.”*