Answer to Question #135380 in Algebra for Yolanda

Question

let z1=x+iy and z2=a+ib.suppose z1=z2. Which one of the following is/are true? Explain
a)x=a and y=b
b)the arguments of z1 and z2 differ by a multiple of 2pi

Answer 1

a)

Definition of equality of two complex numbers:

Let $z_1 = x +iy, z_2 = a + ib$

$z_1 = z_2 \leftrightarrow Rez_1 = Rez_2, Imz_1 = Imz_2$

In our case:

$Rez_1 = x, Rez_2 = a \implies x =a$

$Imz_1 = y, Imz_2 = b \implies y = b$

So this following is true.

b)

The argument of complex number is angle between X axis and vector representing the complex number. Also each nonzero complex number has an infinite number of arguments that differ by a multiple of $2\pi k$, where $k$ - is any integer. Since z1 = z2, therefore this is the same number, it means their arguments differ by 2pi and second following is true.