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We know, $$3 \frac12=3+\frac12$$

Then, if we have

$$3 \frac12 ÷ 3 \frac12$$

It means:

$a)\,\frac72 ÷ \frac72=1$ or

$b)\, 3+\frac12 ÷ 3+\frac12 =\frac{7}2$ or

$c)\, 3 + 1 ÷ 2 ÷ 3 + 1 ÷ 2=\frac{11}3$

Which one is true?

Sorry, maybe it appears on another question, but i want to make it sure between b) and c).

user516076
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    @TMO As their title says, they're using mixed fraction notation. Of course that notation is quite annoying, but they're using it correctly. – Noah Schweber Jun 24 '20 at 23:58
  • Ah, did not read the title. Thanks for pointing that out Noah. – Con Jun 24 '20 at 23:58

1 Answers1

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This is correctly read as $${3{1\over 2}\over 3{1\over 2}}=1,$$ essentially because juxtaposition always binds tighter than explicitly-written infix operations like $\div$.

However, I would rather not read it at all! Not only is the notation a bit ambiguous, but mixed-fraction notation is generally pretty unwieldy for arithmetic operations - when looking at anything more complicated than addition of fractions with the same denominator, so-called "improper fractions" are better than mixed fractions.

Noah Schweber
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  • Thanks. Can i ask something related to this? If i have $$3\sqrt3 ÷ 3\sqrt3$$ is it $1$ or $3$. I doubt, that's a notation.. – user516076 Jun 25 '20 at 00:02
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    @user516076 I would again read that as $1$, for the same reason: juxtaposition binds more tightly than explicitly-written infix operations. Of course in this case juxtaposition denotes multiplication rather than addition, but the "notational emphasis" is the same. – Noah Schweber Jun 25 '20 at 00:03
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    (Incidentally, one really good reason to avoid mixed-fraction notation is that juxtaposition is almost universally used to denote multiplication, and so mixed-fraction notation introduces potential ambiguities.) – Noah Schweber Jun 25 '20 at 00:04
  • @user516076 What you have in $3\sqrt{3}÷3\sqrt{3}$ is a sucession of a multiplication, a division and a multiplication. Unless otherwise specified, division is explicitly between $\sqrt{3}$ and $3$ and this expression, because it is too ambiguous, had better be rewritten as : $3\times \frac{\sqrt{3}}{3}\times\sqrt{3}$. While this changes nothing for the $3$ on the left of the expression, it radically changes how the second $\sqrt{3}$ should be placed. Except if you have: $(3\sqrt{3})÷(3\sqrt{3})$. Then this should of course be taken to mean : $ \frac{3\times\sqrt{3}}{3\times\sqrt{3}}$. – DeltaXY Jan 29 '21 at 14:53