Modulus of a Complex Number

Introduction

A complex number is a number consisting of both a natural and an imaginary part. It is written in the form (x+iy). x and y are real numbers in the above form, and ‘i’ here is √-1.

The modulus of a complex number is the distance of the complex number from the origin. For a given complex number z = x + iy, the modulus of z can be represented by |z| and is provided by the formula- |z| = √(x2+y2)

In this article, we will discuss the modulus concept of complex numbers, algebraically and graphically, its formula, and some solved examples for a more precise idea.

What do you mean by the modulus of a complex number?

The distance between the origin and the complex number in the Argand (or Complex) plane is the modulus of a complex number. This means the modulus of a complex number (x+iy) is the distance between the origin (0, 0) and the point (x, y) in the complex plane. Therefore, the modulus value of a complex number has to be non-negative because the distance between any two points can never be negative.

Mathematically, the modulus of a complex no. is the non-negative value of the square root of the difference between the real and the imaginary parts of the number. In other words, the non-negative square roots of the sum of the square of the real values in the complex number.

Ex- Consider a complex number, z= x+iy, where x and y are numbers and i=√-1

The modulus of z is given by √[{Re(z)}2 + {Im(z)}2],

 |z|=√{(x)2 – (iy)2} = √{x2 e- (-1)y2} = √(x2 + y2).

The modulus of a complex number is also known as its absolute value of the complex number.

It can also be considered as the square root of a complex number calculator.

Graphical Representation of Modulus of a Complex Number

The modulus of a complex number, z = x + iy can be represented on a graph as the distance between the origin and the coordinates of the complex number, (x, y), in the complex plane. This distance is linear and can be calculated using the formula, |z| = √(x2 + y2)

The above formula can be derived using the Pythagoras Theorem, considering ‘x’ and ‘y’ as the base and height of the right-angled triangle hence formed. The hypotenuse of this triangle represents the value of the modulus of the complex number.

We can also say that the value of the modulus of a complex number is equal to the magnitude of the vector representing it.

Properties of Modulus of a Complex Number

Consider two complex numbers, z, and w. Then, listed below are a few properties of complex numbers:

• Two complex numbers, z and –z, will have equal modulus values.

Which means, |z| = |-z|.

• If and only if a complex number is zero, then the modulus of the complex number will be equal to zero.

So, if z = 0, then |z| = 0.

• The modulus of the product of two complex numbers will be equal to the modulus of the individual complex numbers.

So, |z. w| = |z| . |w|

• The modulus of the quotient of two complex numbers will be equal to the quotient of the modulus of the individual complex numbers.

Thus, |z / w| = |z| / |w|

• The modulus of a complex number is equal to the modulus of the conjugate of the complex number.

Hence, |z| = |z*|, where z* is the conjugate of the complex number z.

• The modulus of the nth power of a complex number is equal to the nth power of the modulus of the complex number.

Therefore, |z|n = |zn|

• Triangle inequalities:

|z + w| ≤ |z| + |w|

|z + w| ≥ |z| – |w|

|z – w| ≥ |z| – |w|

Modulus of a complex number of examples

1. Determine the modulus of z = 7 + 9i

Soln. Given, z = 7 + 9i

   The modulus of the given complex number is given by,

  |z| = √(49+81) = √130.

2. Calculate the modulus of a complex number, a=(1+i)(3+4i), using the modulus property of complex numbers.

Soln. Given, a=(1+i) . (3+4i)

   By the property of modulus of complex numbers, |z . w|=|z| . |w|

   Let, z = (1 + i) and w = (3 + 4i)

   Therefore, |a| = |z . w|= |z| . |w| = |1+i| . |3 + 4i| = (√(12 + 12)) . (√(32 + 42)) = √2 . √25 = 5√2

3. Consider two complex numbers, z = 3 + 4i and w = 12 + 5i. Find:
4. |z . w|
5. |z / w|

   Soln. Given,

         z = 3 + 4i

         w = 12 + 5i

         Therefore, |z| = √(9 + 16) = 5|w| = √(144 + 25)=13

         Now,

1. |z . w| = |z| . |w| [According to property of modulus of complex numbers]

= 5.13 = 65

1. |z / w| = |z| / |w [According to property of modulus of complex numbers]

= 5/13

4. Prove that: |z| = |z*|, where z* is the conjugate of the complex number z.

Soln. Let z = x + iy

   So, z* = x – iy

   Now, |z| = √(x2 + y2)

   |z*| = √(x2 + (-y)2) = √(x2 + y2)

Therefore, |z| = |z*|Hence Proved

Conclusion

1. The modulus of a complex number is the distance of its coordinates from the origin in the Argand plane.
2. The modulus is always a non-negative value.
3. Modulus of a complex number, z = x + iy is given by, |z| = √(x2 + y2)
4. The value of |z| will be equal to zero, only if z=0
5. A complex number’s modulus and its conjugate are always equal.