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Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

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.

 

Property 

Triangle inequality

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

Proof


⇒  |z1 + z2|2 ≤  (|z1| + |z2|)2

⇒  |z1 + z2| ≤ |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 + …. + zn | ≤ |z1| + |z2| + |z3| + … + |zn| for n = 2,3,….

 

Property

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

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

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

= √ [( 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| ≤ |z1| + |z2|

| |z1| - |z2| | ≤ | z1 + z2|  ≤  |z1| + |z2| and

| |z1| - |z2| | ≤ | z1 - z2|  ≤  |z1| + |z2|

 

Property 

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 ….. zn| = |z1| |z2| |z3| … … |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.

 

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

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12th Mathematics : UNIT 2 : Complex Numbers


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