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 $a,b,c,d\in\Bbb C$? $\endgroup$– The Great SeoAug 9, 2014 at 13:20
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$\begingroup$ what is the $z$ ? $\endgroup$– idmAug 9, 2014 at 13:21
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$\begingroup$ @TheGreatSeo no in $\mathbb{R^4}$ $\endgroup$– user161440Aug 9, 2014 at 13:22
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$\begingroup$ @idm where is z? $\endgroup$– user161440Aug 9, 2014 at 13:23
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$\begingroup$ @idm ah okay sorry $\endgroup$– user161440Aug 9, 2014 at 13:24
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1 Answer
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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 $a=c$.
