Given the complex number z show that i(z-z1) is a real number if z1 is the conjugate of z.
Let the complex number z = a + ib
z1 is the cojugate of z, z1 = a - ib
i*(z - z1)
=> i*(a + ib - a + ib)
but i^2 = -1
-2*b is a real number.
This shows 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.