Given the complex number z show that i(z-z1) is a real number if z1 is the conjugate of z.

We'll consider the complex number z = a + bi and it's
conjugate z1 = a - bi.

To prove that i(z - z1) is a real
number, we'll replace z and z1 by the rectangular
forms:

i(z - z1) = i(a + bi - a +
bi)

We'll eliminate like terms inside
brackets:

i(z - z1) = i(2bi) =
2b*i^2

We'll keep in mind that i^2 = -1 and we'll evaluate
the result:

i(z - z1) = 2b*(-1) =
-2b

We notice that the result has no
imaginary part, therefore it is a real number: i(z - z1) = 
-2b.