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Which is notation is stylistically preferred?

$$\Delta U = mg \frac{R^2h}{R^2 + R h} \approx mg \frac{R^2 h}{R^2} = mgh$$

or

$$\Delta U = mg \frac{R^2h}{R^2 + R h} \approx mg \frac{R^2 h}{R^2} \approx mgh$$

And does the rule change if you are using line breaks?

Logically, it seems like the second notation implies that you are making an additional approximation, so I am inclined to say that the first notation is correct.

But if a reader is just skimming the formulas without looking at the derivation, they might mistakenly think that an approximation is being used at all.

I am trying to find an example, but it seems like most of the books I can find have simply dodged the issue by adding a discussion in between the formulas. Perhaps there is a style guide somewhere that advises what to do in situations like this.

Max
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    The first one is correct, provided $R\not=0$. – morrowmh Jun 08 '20 at 07:09
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    I think „we find $\Delta U \approx mgh$ since [first version]“ is an option. Or just use an align to make the $\approx$ more prominent. – Jonas Linssen Jun 08 '20 at 07:27
  • They're both correct, since (strictly speaking) $a \approx b$ includes $a = b,$ so you probably want to ask which is more precisely correct (or some such wording). I would use the first one, as it conveys more information; indeed, it does so without any increase in words or symbolic clutter. If the approximation $\Delta U \approx mgh$ is very important in what follows, you could preface this with something like "For later use we note that $\Delta U \approx mgh,$ which follows from ..." – Dave L. Renfro Jun 08 '20 at 07:40

2 Answers2

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The first one is correct, in exactly the same way that you would write e.g. $$e^{\frac{1}{x-1}}\geq1+\frac{1}{x-1}=\frac{x}{x-1}.$$

Especially Lime
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Logically, it seems like the second notation implies that you are making an additional approximation, so I am inclined to say that the first notation is correct.

I agree with this reasoning.

But if a reader is just skimming the formulas without looking at the derivation, they might mistakenly think that an approximation is being used at all.

If you are worried about this, you might add an extra line containing $$ \implies \Delta U \approx mgh $$ or the like.

Vincent
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