Modulus of Complex Number
The modulus of a complex number is the distance of the complex number from the origin in the argand plane. If z = x + iy is a complex number where x and y are real and i = √1, then the nonnegative value √(x^{2} + y^{2}) is called the modulus of complex number (z = x + iy). The modulus of a complex number is also called the absolute value of the complex number.
In this article, we will understand the concept of modulus of complex numbers algebraically as well as graphically along with its formula and some solved examples for a better understanding.
1.  What is Modulus of Complex Number? 
2.  Modulus of Complex Number Formula 
3.  Modulus of Complex Number Using Graph 
4.  Properties of Modulus of Complex Number 
5.  FAQs on Modulus of Complex Number 
What is Modulus of Complex Number?
The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number. If z is a complex number, then the modulus of the complex number z is given by, √{[Re(z)]^{2} + [Im(z)]^{2}} and it is denoted by z. The modulus of complex number z = a + ib is the distance between the origin (0, 0) and the point (a, b) in the complex plane. Since the modulus of a complex number is the distance, its value is always nonnegative.
Modulus of Complex Number Formula
The modulus of a complex number z = x + iy, denoted by z, is given by the formula z = √(x^{2} + y^{2}), where x is the real part and y is the imaginary part of the complex number z. The modulus of complex number z can also be calculated using the conjugate of z. Since \(z.\bar{z} = (x+iy)(xiy) = x^2+y^2\), we have \(z.\bar{z} = z^2\) ⇒ \(z = \sqrt{z.\bar{z}}\).
Modulus of Complex Number Using Graph
When a complex number z is plotted on a graph, the distance between the coordinates of the complex number and the origin on a complex plane is called the modulus of the complex number. The distance of the complex number represented as a point in the argand plane (a, b) is called the modulus of the complex number. This distance is a linear distance from the origin (0, 0) to the point (a, b), and is measured as r = √(a^{2} + b^{2}).
Further, this can be understood as modulus of a complex number can be derived from the Pythagoras theorem, where the modulus is represented by the hypotenuse, the real part is the base, and the imaginary part is the altitude of the rightangled triangle. The modulus of a complex number a + bi is the same thing as the magnitude (or length) of the vector representing a + bi.
Properties of Modulus of Complex Number
Given below are some of the properties of a complex number. If z and w are two complex numbers, then we have:
 The modulus of complex numbers z and z are equal, that is, z = z
 The modulus of a complex number is 0 if and only if the complex number is zero, that is, z = 0 iff z = 0
 The modulus of the product of complex numbers is equal to the product of the modulus of the complex numbers, that is, z.w = z.w
 The modulus of the quotient of two complex numbers is equal to the quotient of the modulus of the complex numbers, that is, z/w = z/w
 The modulus of a complex number is equal to the modulus of the conjugate of the complex number, that is, \(z = \bar{z}\)
 The modulus of the nth power of a complex number is equal to the nth power of the modulus of the complex number, that is, z^{n} = z^{n}
Important Notes on Modulus of Complex Number
 The distance of the complex number represented as a point in the argand plane (a, b) from the origin (0, 0) is called the modulus of the complex number.
 The modulus of a complex number is 0 if and only if the complex number is zero.
 The modulus of a complex number a + bi is the same thing as the magnitude (or length) of the vector representing a + bi.
 The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number.
Related Topics on Modulus of Complex Number
Modulus of Complex Number Examples

Example 1: Determine the modulus of complex number z = 6 + 3i.
Solution: The modulus of complex number z = 6 + 3i is given by,
z = √((6)^{2} + (3)^{2})
= √(36 + 9)
= √45
= 3√5
Answer: z = 3√5

Example 2: Calculate the modulus of the complex number v = (1 + 2i)(2 + 3i) using the property of modulus of the complex number.
Solution: We have v = (1 + 2i)(2 + 3i). Let z = (1 + 2i), w = (2 + 3i).
We know that zw = zw, therefore we will determine the modulus of complex numbers z and w separately.
z = √((1)^{2} + (2)^{2}) = √(1 + 4) = √5
w = √((2)^{2} + (3)^{2}) = √(4 + 9) = √13
Modulus of complex number v, v = zw = zw = √5.√13 = √65
Answer: Modulus of complex number (1 + 2i)(2 + 3i) is √65.
FAQs on Modulus of Complex Number
What is Modulus of Complex Number?
The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number.
What Happens to the Modulus of a Complex Number when the Power of Complex Number Increases?
When the power of complex number increases, the modulus of the complex number also increases by the same power. This can be verified using the property of modulus of complex number: The modulus of the nth power of a complex number is equal to the nth power of the modulus of the complex number, that is, z^{n} = z^{n}
What is the Modulus of Complex Number z = x + iy?
The modulus of a complex number z = x + iy, denoted by z, is given by the formula z = √(x^{2} + y^{2}), where x is the real part and y is the imaginary part of the complex number z.
How Do You Find Modulus of Complex Number?
The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number. It can be calculated using the formula z = √(x^{2} + y^{2}).
What is the Modulus of Complex Number z = 1?
The modulus of complex number z = 1 is z = 1 as z = 1 = 1 + 0i. This implies the modulus of complex number z = 1 is √(1^{2} + 0^{2}) = 1
When is Modulus of Complex Number Equal to 0?
The modulus of a complex number is 0 if and only if the complex number is zero, that is, z = 0 iff z = 0.
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