Properties of Modulus of a complex number: Let us prove some of the properties.

**Properties of Modulus of a complex number**

Let us prove some of the properties.

For any two complex numbers z_{1} and z_{2}, we have |z_{1} + z_{2}| â‰¤ |z_{1}| + |z_{2}|.

**Proof**

â‡’ |z_{1} + z_{2}|^{2} â‰¤ (|z_{1}| + |z_{2}|)^{2}

â‡’ |z_{1} + z_{2}| â‰¤ |z_{1}| + |z_{2}|

Now consider the triangle shown in figure with vertices *O*, *z*_{1} or *z*_{2} , and *z*_{1} + *z*_{2}. We know from geometry
that the length of the side of the triangle corresponding to the vector *z*_{1} + *z*_{2} cannot be greater than
the sum of the lengths of the remaining two sides. This is the reason for calling the
property as "Triangle Inequality".

It can be generalized by means of mathematical induction to finite number of terms:

|z_{1} + z_{2 }+ z_{3 }+ â€¦. + z_{n }| â‰¤ |z_{1}| + |z_{2}| + |z_{3}| + â€¦ + |z_{n}| for *n* = 2,3,â€¦.

If *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} , then

| z_{1} - z_{2}| = | ( *x*_{1} - *x*_{2} ) + ( y_{1} - y_{2} )*i*|

= âˆš [( *x*_{1} - *x*_{2} )^{2} + ( y_{1} - y_{2} )^{2}]

The distance between the two points *z*_{1} and *z*_{2} in complex plane is | *z*_{1} - *z*_{2} |

If we consider origin, *z*_{1} and *z*_{2} as vertices of a
triangle, by the similar argument we have

|z_{1} - z_{2}| â‰¤ |z_{1}| + |z_{2}|

| |z_{1}| - |z_{2}| | â‰¤ | z_{1} + z_{2}| â‰¤ |z_{1}| + |z_{2}| and

| |z_{1}| - |z_{2}| | â‰¤ | z_{1} - z_{2}| â‰¤ |z_{1}| + |z_{2}|

For any two complex numbers *z*_{1} and *z*_{2}, we have |z_{1 }z_{2}| = |z_{1}| |z_{2}|

**Note:**

It can be generalized by means of mathematical induction to any
finite number of terms:

|z_{1 }z_{2} z_{3} â€¦.. z_{n}| = |z_{1}| |z_{2}| |z_{3}| â€¦ â€¦ |z_{n}|

That is the modulus value of a product of complex numbers is equal
to the product of the moduli of complex numbers.

Similarly we can prove the other properties of modulus of a
complex number.

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12th Mathematics : UNIT 2 : Complex Numbers : Properties of Modulus of a complex number |

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