The Modulus of a Complex number defines the distance of the complex number from the origin of the plane. Suppose, z = x + iy is a complex number. Here x and y are mentioned as real, and i = √-1. So, the non-negative value √(x² + y²) is the modulus of complex numbers (z = x + iy). We can also call the Modulus of complex numbers the absolute value of the complex number. Thus, here we will discuss the properties of modulus Complex numbers with examples and formulas. We also learn how to find the Modulus of a complex number. 

Know More About the Modulus of Complex Numbers

We can say the Modulus of Complex numbers is the square root of the real part of the sum of squares and the imaginary part of Complex numbers. Suppose z is a complex number, then the Modulus of the complex number is denoted by |z|. Therefore z = a + ib is the distance between the origin (0, 0) and the point (a, b) in the complex plane. It should be noted that the value of the Modulus of complex numbers is non-negative. 

The formula of Modulus of Complex Numbers

We can write the Modulus of a complex number as |z|. And it’s formula is |z| = √(x² + y²). Here, x is considered the real and the ‘y’ imaginary parts of the complex number z. Thus, the formula for the Modulus of the complex number is:-

Modulus of a complex number 

 ‘z= a + ib’ is

|z|= √ {[Re(z)]² + [Im(z)]²}

    = √ a² + b²

Properties of Modulus of Complex Numbers

Here we discuss some properties of the Modulus of complex numbers. It will help to learn how to find the Modulus of a complex number. 

1 . The Modulus of complex numbers is equal in positive and negative. It means |z| = |-z|

2. If the complex number is zero, then the Modulus of the complex number is also zero. It means z = 0 then |z| = 0.
3. It should be noted that the Modulus of a complex number is the same value as the Modulus of the conjugate of the complex number. 
4. The nth power of the modulus of a complex number is the same value as the nth power of the complex number. 
5. We can say that the Modulus of the multiply of complex numbers is the same value as the multiply of the Modulus of complex numbers. It means |z.w| = |z|.|w|
6. The Modulus of the result of the division of two complex numbers is the same value as the division of the Modulus of the complex numbers. It means, |z/w| = |z|/|w|

Some Important Points Related To Modulus Of Complex Numbers

1 . The distance of the complex number set as a point in the argand plane (a, b) from the origin (0, 0) is called the Modulus of the complex number.

2. If we get the complex number zero, then we also get the Modulus of the complex number is zero. 
3. The Modulus of complex numbers is the square root of the real part of the sum of squares and the imaginary part of Complex numbers. 

Some Examples of Modulus of Complex Number

Here we learn how to find the Modulus of a complex number. 

1 . Prove |z1 z2| = |z1||z2|

Here, z1 = a + ib 

and z2 = c + id, 

 |z1 z2| = |(a + ib)(c + id)|

⇒ |ac + iad + ibc + i²bd|

⇒ |ac + iad + ibc – bd|

⇒ |ac – bd + i(ad + bc)|

⇒ (ac – bd)²+ (ad+bc)²

⇒ (ac)²+ (bd)² – 2abcd + (ad)² + (bc)² + 2abcd

⇒ a² c² + b² d² + a² d²+ b² c²

⇒ a² c² + b2 c² + b2 d² + a2 d²

⇒ (a² + b²)c² + (b2 + a²)d²

⇒ (a² + b²) (c² + d²)

⇒ |z1||z2| proved. 

2. Solve the modulus complex number z = -6 + 3i.

|z| = √((-6)² + (3)²)

= √(36 + 9)

= √45

= 3√5

3. Prove |z1 / z2 | = (|z1|) / (|z2|).

|z1| |1/z2|

⇒ |z1| 1/(|z2|)

⇒ (|z1 |) / (|z2|).

Conclusion

From the above discussion, we clearly understand the Modulus of complex numbers. It is the square root of the real part of the sum of squares and the imaginary part of Complex numbers. Also, we know the formula and the properties of the Modulus of complex numbers. We knew that the Modulus of complex numbers is equal in both positive and negative. And by the solved example, we learn how to find the Modulus of a complex number. Thus, it would be best if you solved more example questions related to the Modulus of complex numbers. It will help you to understand the topic clearly.