equality of complex numbers: general case.

Question

Can someone help me to understand this definition (or proposition) for complex numbers equality, of the form $w=x+\xi y$. \begin{align*} &\xi\text{ is a complex number such that } \Im(\xi)\neq0.\\ &\forall (a,b,c,d)\in\mathbb{R^4}:\quad a+\xi b=c+\xi d\:\iff\:a=c \text{ and }b=d.\\ \end{align*} I know that Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. but this a special case for $\xi=i$.

Answer 1

Let $\xi=x+iy$, so:

$$a+\xi b=a+bx+byi$$

$$c+\xi d=c+dx+dyi$$

So you know that $a+b\xi=c+d\xi$ are equal if and only if $a+bx=c+dx$ and $by=dy$. Next you know that $y \neq 0$, so you can divide by $y$ and get $b=d$. Next from first equation and $b=d$ you have $a+bx=c+bx$, so $a=c$.