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As the title suggests, why are some algebraic equations displayed as inverses instead of equalling the direct variable we're trying to solve?

For example, consider the d-spacing formulas used in crystallography where the variable of interest is d.

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Why not just make all the equations equal to d not 1/d^2? If I want to use the equations I have to rearrange them anyway. is there a visual aspect to the equations I'm not seeing? I've seen a lot of other equations be put in inverse form too but I'm not too sure what the benefit is.

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    $\begingroup$ I have the impression that the author might have wanted to emphasize that all these cases expressed for 1/d^2 correspond to quadratic forms in the Miller indices h,k,l. $\endgroup$
    – alphanum
    Commented May 17, 2022 at 8:17
  • $\begingroup$ Maybe $1/d^2$ has some physical meaning in crystallography? $\endgroup$ Commented May 17, 2022 at 8:41

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Because the expressions for $1/d^2$ are overall simpler than for $d^2$. While the expressions for $d^2$ would be no different in complexity in the first, fourth, and seventh cases, they would be more complex in the other four cases: either introducing double-decker fractions or longer polynomial expressions if the numerators and denominators are multiplied up to clear the extra layer of fractions.

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  • $\begingroup$ Really? I don't see how they're simpler... why not just leave it as d instead of d^2 or 1/d^2? $\endgroup$
    – Hendrix13
    Commented May 17, 2022 at 8:58
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    $\begingroup$ $$\frac1a=\frac1b+\frac1c$$ takes fewer lines than $$a=\dfrac1{\dfrac1b+\dfrac1c}.$$ Also, there is nothing to stop you putting a square-root sign over the whole expression, if you so wish. Similarly, you could write Pythagoras' theorem as $c=\sqrt{a^2+b^2}$ instead of $c^2=a^2+b^2$ (etc.). $\endgroup$ Commented May 17, 2022 at 11:38
  • $\begingroup$ Great point, I see what you mean now, thanks! :) $\endgroup$
    – Hendrix13
    Commented Nov 12, 2023 at 4:59

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