Home | | Maths 12th Std | Properties of Modulus of a complex number

# Properties of Modulus of a complex number

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.

## Triangle inequality

For any two complex numbers z1 and z2, we have |z1 + z2| Ōēż |z1| + |z2|.

Proof ŌćÆ  |z1 + z2|2 Ōēż  (|z1| + |z2|)2

ŌćÆ  |z1 + z2| Ōēż |z1| + |z2|

## Geometrical interpretation

Now consider the triangle shown in figure with vertices O, z1  or z2 , and z1 + z2. We know from geometry that the length of the side of the triangle corresponding to the vector  z1 + z2 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:

|z1 + z2 + z3 + ŌĆ”. + zn | Ōēż |z1| + |z2| + |z3| + ŌĆ” + |zn| for n = 2,3,ŌĆ”.

## The distance between the two points z1and z2in complex plane is | z1ŌłÆ z2 |

If z1 = x1 + iy1 and z2 = x2 + iy2 , then

| z1 - z2| = | ( x1 - x2 ) + ( y1 - y2 )i|

= ŌłÜ [( x1 - x2 )2 + ( y1 - y2 )2] ### Remark

The distance between the two points z1 and z2 in complex plane is | z1 - z2 |

If we consider origin, z1 and z2 as vertices of a triangle, by the similar argument we have |z1 - z2| Ōēż |z1| + |z2|

| |z1| - |z2| | Ōēż | z1 + z2|  Ōēż  |z1| + |z2| and

| |z1| - |z2| | Ōēż | z1 - z2|  Ōēż  |z1| + |z2|

## Modulus of the product is equal to product of the moduli.

For any two complex numbers z1 and z2, we have |z1 z2| = |z1| |z2

### Proof Note:

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

|z1 z2 z3 ŌĆ”.. zn| = |z1| |z2| |z3| ŌĆ” ŌĆ” |zn|

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.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 2 : Complex Numbers : Properties of Modulus of a complex number |