Question

Let z1 = x + iy and z2 = a + ib:Suppose z1 = z2: Which of the following is/are true?
A. x2

Answer 1

\[z_1=x+iy,~z_2=a+ib\]
If \(z_1=z_2\), then \(x=a\) and \(y=b\). The first can't always be true. Consider \(z_1=z_2=1\). Then \(x^2a^2=x^4=1^4=1\), but \(b^2y^2=y^4=0\).

The second is also not true. If \(z_1=z_2\), then the angle/argument (in terms of polar coordinates) must be the same, meaning the arguments may only differ by a multiply of \(2\pi\), not 2 (unless you consider \(2\pi\) to be a mutliple of 2).