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A YouTube video (https://www.youtube.com/watch?v=6SzZ_jAHasE) asked what was the answer to this type of fraction:

$$\dfrac {1+1+1+1+1+1}{\dfrac {1+1+1+1+1}{\dfrac {1+1+1+1}{\dfrac{1+1+1}{\dfrac{1+1}{1}}}}}\\$$

I had thought that this was equivalent to division, and hence:

$$6 / 5 / 4 / 3 / 2 / 1 = \frac{1}{20}$$

evaluating from left to right.

However, the solution appeared to start the evaluation of the fraction at the bottom.

Is there a defined rule for this? (Note: "no" would be an acceptable answer).

rghome
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    I would say that the notation used there is ambiguous (probably intentionally so), and that this isn't really a question about mathematics, but an annoying "guess the intentions of the author" kind of puzzle. – Xander Henderson Sep 19 '22 at 20:42
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    For what it is worth, the size of the fonts indicates that this should probably be understood as $6/(5/(4/(3/(2/1))))$. – Xander Henderson Sep 19 '22 at 20:43
  • I added a note to indicate that the font size isn't relevant. The formatter did that automatically. If I knew how to make them all the same size I would. – rghome Sep 19 '22 at 20:45
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    If it's a YouTube video, then anything is possible and you could make an argument for anything, but I think most mathematicians would go by "bigger fraction lines first". But if the font rendered as the same size, then I'd go top to bottom and hence get 1/20. But this is the sort of thing which mathematicians don't fuss over, and if there's potential for ambiguity, many would put brackets, because math is about communication and getting the ideas across clearly (sometimes even at the expense of brevity). – peek-a-boo Sep 19 '22 at 20:48
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    The point of notation is to clearly communicate an idea. The notation used here does not achieve this. No answer is correct (or, equivalently, any answer is correct). This isn't really a question about mathematics. Again, it is about guessing the intent of the author. – Xander Henderson Sep 19 '22 at 20:48
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    Minute 1:13 shows the intention with four rows was $\dfrac{4 \cdot 2}3$ i.e. $\dfrac{4}{\left(\dfrac{3}{\left(\frac{2}{1}\right)}\right)}$. So does the eventual conclusion $a_{2k}\approx \sqrt{\pi k}$ – Henry Sep 19 '22 at 20:57
  • If it is definitely ambiguous, then that answers my question. I am a little surprised as the precedence of common operators is well defined (PEMDAS). It hadn't occurred to me that fractions would be anything other than stacked division operators. As far as the intent of the author goes - calculating from the top isn't interesting and calculating from the bottom (spoiler) has pi in the answer, so he went with the interesting option. – rghome Sep 19 '22 at 20:58
  • Your fraction is \frac {1+1+1+1+1+1}{\frac {1+1+1+1+1}{\frac {1+1+1+1}{\frac{1+1+1}{\frac{1+1}{1}}}}}. Even if you didn't choose the font size, you did choose the curly brackets. – peterwhy Sep 19 '22 at 21:02
  • I used "dfrac" to force it to use display-style sizing throughout. – Akiva Weinberger Sep 19 '22 at 21:22
  • @AkivaWeinberger Thanks - that looks much better. – rghome Sep 19 '22 at 21:25
  • The size of the vinculum is still the determining factor. Consider $$\dfrac{1+1+1}{\dfrac{1+1}{1}} \qquad\text{vs}\qquad \dfrac{\dfrac{1+1+1}{1+1}}{1}. $$ The one on the left is $3/(2/1)$, while the one on the right is ambiguous, but would probably be interpreted as $(3/2)/1$ (that is how it is typeset). – Xander Henderson Sep 19 '22 at 21:35
  • @rghome: "The precedence of common operators is" not "well defined"; PEMDAS is a common convention, but not used universally. Another common convention is PE(MD)(AS), where the parenthesized groups have equal precedence, and are read left-to-right; thus (for example) $(8/4)\cdot2=8/4\cdot2\neq8/(4\cdot2)$. – Jacob Manaker Sep 20 '22 at 05:13

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The size of the horizontal lines distinguish this: $$\dfrac a{\dfrac bc}\text{ refers to }a/(b/c).$$ $$\dfrac{\dfrac ab}c\text{ refers to }(a/b)/c.$$ That said, you should never write like this.

Akiva Weinberger
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  • So the smaller the line, the higher the priority? (I can see that this wouldn't be a system to rely on). – rghome Sep 19 '22 at 21:13
  • I suppose that's a consequence, so yes. But the way to think about it is just that the horizontal line is drawn wider than its numerator and its denominator, so that in $\dfrac a{\dfrac bc}$, you cannot read it as $\dfrac ab$ over $c$ because the former is wider than the line. (LaTeX also automatically does weird things with the vertical spacing, which helps I suppose) – Akiva Weinberger Sep 19 '22 at 21:18
  • @rghome I wonder about the direction of causation: if LaTeX does it like this because we write like this, or if we write like this because LaTeX does it like this – Akiva Weinberger Sep 19 '22 at 21:43