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What is the notation for showing that equations are equivalent by rearranging terms?

For example, for the arc length formula:

$$s=r\theta$$

I sometimes solve for $r$ and write it as

$$r=\frac{s}{\theta}$$

When I show my work, I usually write this relationship like this:

$$s=r\theta\Rightarrow r=\frac{s}{\theta}$$

Is this the correct way to write it or is there a better way? Up to this point, my teachers haven't cared, but I know in college and later in high school it will definitely matter.

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  • $\begingroup$ More context is preferable (i.e. are you doing a proof or a computation?). In general, however, this is fine. If it's a bit more involved, I write "Rearranging," and put the equation on the next line. $\endgroup$ – user217285 Apr 18 '16 at 17:30
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    $\begingroup$ What you have written says that the truth of the expression on the left implies the truth of the expression on the right. It is correct. If you want equivalence you can use the double arrow that shows how the truth of either implies the truth of the other. $\endgroup$ – John Douma Apr 18 '16 at 17:32
  • $\begingroup$ @Nitin just a computation at the moment, but I am looking for a general rule. $\endgroup$ – GamrCorps Apr 18 '16 at 17:32
  • $\begingroup$ @JohnDouma that is a good explanation. Thanks. Can you post that as an answer? $\endgroup$ – GamrCorps Apr 18 '16 at 17:34
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    $\begingroup$ There is really nothing wrong with just following Nitin's advice above. However, you should exercise some caution when rearranging equations. Example: your new expression is valid only for all non zero values of $\theta$, but your original expression can handle a zero angle. $\endgroup$ – Xoque55 Apr 18 '16 at 17:35
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Per request by GamrCorps, I post my original comment here.

What you have written says that the truth of the expression on the left implies the truth of the expression on the right. It is correct. If you want equivalence you can use the double arrow that shows how the truth of either implies the truth of the other.

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