Question

if z1 and z2 are complex numbers prove that conjugate of (Z1 -Z2)= conjugate of Z1-conjugate of Z2​

Answer 1

SOLUTION

GIVEN

$\sf{z_1 \: and \: z_2 \: are \: complex \: numbers }$

TO PROVE

$\sf{ \overline{z_1 - z_2} =\overline{z_1} - \overline{z_2} }$

EVALUATION

Let

$\sf{ z_1 = a + ib \: \: \: and \: \: z_2 = x + iy }$

Then

$\sf{ \overline{z_1} = a - ib \: \: \: and \: \: \: \overline{ z_2 }= x - iy }$

Now

LHS

$\sf{ = \overline{z_1 - z_2} }$

$\sf{ = \overline{(a + ib) - (x + iy)} }$

$\sf{ = \overline{(a - x)+ i(b - y) } }$

$\sf{ = (a - x) - i(b - y) }$

$\sf{ = (a -ib) - (x - iy) }$

$\sf{ =\overline{z_1} - \overline{z_2} }$

= RHS

Hence proved

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