Question

Prove that modules of the ratio of any two
conjugate numbers is unity​

Answer 1

SOLUTION

TO PROVE

The modules of the ratio of any two conjugate numbers is unity

CONCEPT TO BE IMPLEMENTED

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

EVALUATION

Let us consider the complex number as

z = a + ib

Then conjugate of z = a - ib

So the modules of the ratio of two conjugate numbers

$\displaystyle \sf{ = \bigg| \frac{a + ib}{a - ib} \bigg| }$

$\displaystyle \sf{ = \frac{| a + ib| }{| a - ib| } }$

$\displaystyle \sf{ = \frac{ \sqrt{ {a}^{2} + {b}^{2} } }{ \sqrt{ {a}^{2} + {b}^{2} }} }$

$= 1$

Hence proved

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