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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 $\xi=i$.

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  • $\begingroup$ Does $a,b,c,d\in\Bbb C$? $\endgroup$ Commented Aug 9, 2014 at 13:20
  • $\begingroup$ what is the $z$ ? $\endgroup$
    – idm
    Commented Aug 9, 2014 at 13:21
  • $\begingroup$ @TheGreatSeo no in $\mathbb{R^4}$ $\endgroup$
    – user161440
    Commented Aug 9, 2014 at 13:22
  • $\begingroup$ @idm where is z? $\endgroup$
    – user161440
    Commented Aug 9, 2014 at 13:23
  • $\begingroup$ @idm ah okay sorry $\endgroup$
    – user161440
    Commented Aug 9, 2014 at 13:24

1 Answer 1

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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 $a=c$.

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