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I'll occasionally write equalities within parentheses or sqrt signs to make my steps more compact.

E.g.:

$$ r = \frac{3}{4}\sqrt[4]{\frac{7}{9}\cdot\frac{16}{7}=\frac{16}{9}} = \frac{3}{4}\sqrt{\frac{4}{3}}=\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $$

I always assumed this was pretty clear, and an acceptable (if unusual) way to use equalities.
However, I was recently told that "you can't take the root of a truth value", which is certainly true.

Is my notation confusing? Does it classify as abuse of notation?

Is the below correct?

$$ r = \left(\frac{7}{9} \cdot \frac{16}{7} = \frac{16}{9}\right) \implies r = \top $$

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    $\begingroup$ Yuck! Please don't do this. Equations are often messy enough without extraneous calculation written internally! $\endgroup$ – Chappers Sep 15 '17 at 16:19
  • $\begingroup$ If you want to annotate them, write above or below the line, possibly with a curly brace to indicate which part is being altered. $\endgroup$ – Chappers Sep 15 '17 at 16:22
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    $\begingroup$ As a software developer in languages that do allow this, it seems pretty straightforward to me. It could potentially make a derivation easier to understand without adding a lot of extra text. As with a lot of notational questions, it ultimately comes down to your audience. A specific audience may well appreciate the brevity, whereas a general audience may be confused. On the point of a square root of a Boolean: If you define multiplication as the AND operation then taking the square root is trivial. $\endgroup$ – Χpẘ Sep 15 '17 at 16:23
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    $\begingroup$ On the plus side, it could help hold off the Rise of the Machines. Having to make sense of the fourth root of a boolean data type might prevent Skynet from becoming self aware. $\endgroup$ – Robert Soupe Sep 16 '17 at 17:47
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    $\begingroup$ @DanielWainfleet Seems fair. Your assessment led me to wonder, exactly what is abuse of notation? To my surprise there's actually a Wikipedia article. The article gives several examples where abuse is common, even saying that the correct notation quickly becomes pedantic. It also explains that abuse is strongly dependent on time and context. A cited contextual example is f: A → B as a partial function vs in category theory. MSE could be considered a context, and abuse could be defined as popular opinion among MSE users. $\endgroup$ – Χpẘ Sep 18 '17 at 14:14
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Your notation is very non-standard and will be confusing to most people. Many people use curly braces to denote intermediate results: $$ r = \frac{3}{4}\underbrace{\sqrt[4]{\frac{7}{9}\cdot\frac{16}{7}}}_{\sqrt[4]{\frac{16}{9}}} = \frac{3}{4}\sqrt{\frac{4}{3}}=\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} $$

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