# Why are algebraic equations sometimes displayed as inverses instead of solving for the variable in question?

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.

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.

• 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. Commented May 17, 2022 at 8:17
• Maybe $1/d^2$ has some physical meaning in crystallography? Commented May 17, 2022 at 8:41

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.
• $$\frac1a=\frac1b+\frac1c$$ 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.). Commented May 17, 2022 at 11:38