Question

If z1 and z2 are 2 complex numbers, such that z1 + z2 is a real number, then

Answer 1

Answer:

Let z1, z2 be two complex numbers such that z1+z2 and z1z2 both are real, then (1) z1 = -z2 (2) z1 = bar z2 (3) z1 = -bar z2 (4) z1 = z2. Solution: ... z1z2 is real => (a+ib)(c+id) is real. ie imaginary part is zero.

Step-by-step explanation:

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Answer 2

z1 and z2 are such that their complex part is the same number but with opposite sign.

Explanation,

General representation of a complex number is $z = a + ib$

Let the complex numbers be,

    $z_{1} = a + ib\\z_{2} = p + iq$

Adding both the numbers,

            $z_{1} + z_{2} = a +p + ib + iq\\z_{1} + z_{2} = (a +p) + i(b + q)$

Now according to question, the sum is real then,

              $b + q =0\\b = - q$

Both the complex numbers can be written as

                   $z_{1} = a + ib\\z_{2} = p - ib$

And complex part of both the numbers just differ in a negative sign.