why can't we take any number divided by zero as a constant something like we do in the case of complex numbers

Note by Ajmal Ima
4 years, 10 months ago

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You can't do it because multiplication by $$0$$ destroys information. That is, for $$z = x \times y$$, $$y \neq 0$$, we can recover $$x$$ via $$\frac {z} {y}$$. However, when $$y = 0$$, $$0 = x \times y$$ for all $$x$$, so there's no way to recover $$x$$ through a division by $$y$$--the information about $$x$$ has been destroyed. Anthropomorphically speaking, the division operator "doesn't know" what $$x$$ was--$$x$$ could have been any number.

This isn't true of things like $$i$$, since there's a reversal operation to recover what $$i$$ "used to be"--namely, squaring it.

It's possible you could put together a system of math where division by zero is defined in cases where you know the value of $$x$$, since then you know how to reverse $$x \times 0$$. That is, $$z = x \times 0$$, and $$\frac {z} {0} = x$$. I leave that as an exercise for the reader. :)

- 4 years, 10 months ago

That is a clear and charitable answer, Christopher. Thanks for doing this.

Staff - 4 years, 10 months ago

Thanks;

- 4 years, 10 months ago

Umm ... $$\dfrac{z}{0}$$ is undefined (not a constant) for $$z \in \mathbb{C}$$.

- 4 years, 10 months ago