# Triple fractions (and more complex fractions)

Usually $$\frac{a}{\frac{b}{c}} = \frac{ac}{b}$$ i.e. $b/c$ is seen as the denominator, and $a$ is the numerator.

If you have $a/b/c/d$, what do you choose to take as the denominator? $$\frac{b}{\frac{c}{d}} , \frac{c}{d},\text{ or } d\text{?}$$ and why?

And what about for $a/b/c/d/e$?

• A common convention is that you carry out divisions left to right. So $a/b/c$ would mean $(a/b)/c$. Similarly, $a/b/c/d$ would be $((a/b)/c)/d$. But it is much better to use parentheses to avoid ambiguity. – almagest Apr 26 '16 at 10:42
• Usually $a/b/c/d= \frac{a}{bcd}$ etc – gammatester Apr 26 '16 at 10:44
• I wish math teachers would teach operator precedence and associativity. – Kamil Jarosz Apr 26 '16 at 11:35
• It does not follow proper conventions of notation, and you should avoid such things. Most calculators and computers will work left to right, apply the first operation, find and answer and then divide by the next number in the line. That doesn't mean that that is necessarily correct. – Doug M Apr 26 '16 at 20:21
• There is no correct doug, convention is simply what people do. If everyone is going to assume that a/b/c/d is equivalent to ((a/b)/c)/d) it seems reasonable that's what you should use if that's what you intend to communicate. This would be first time I've heard where if they're all multiplication or division that they don't happen from left to right, but I'm aware that different countries and communities have different conventions. In what context is that that proper convention of notation and do you have any resources that affirm that? – VoronoiPotato Apr 26 '16 at 20:53

The trouble is $$a/(b/c) \ne (a/b)/c$$since $a/(b/c) = \frac{{ac}}{b}$ and $(a/b)/c = \frac{a}{{bc}}$. So, without brackets, the notation is ambiguous. Introducing brackets, for example, we could choose between $$a/b/c/d=a{(b/c/d)^{ - 1}}$$ and $$a/b/c/d=(a{/b/c)d^{ - 1}}$$The first leads to $$a{(b{(c/d)^{ - 1}})^{ - 1}} = a{(b{c^{ - 1}}d)^{ - 1}} = \frac{{ac}}{{bd}}$$The second leads to $$(a/b){c^{ - 1}}{d^{ - 1}} = a{b^{ - 1}}{c^{ - 1}}{d^{ - 1}} = \frac{a}{{bcd}}$$ which is less interesting, it seems to me. The first expression leads to $$a/b/c/d/e = a{(b(c/d/e))^{ - 1}} = a{\left( {\frac{{bd}}{{ce}}} \right)^{ - 1}} = \frac{{ace}}{{bd}}$$ Collecting results, the following interesting pattern emerges:$$a/b = a{b^{ - 1}} = \frac{a}{b}$$ $$a/b/c = a{b^{ - 1}}c = \frac{{ac}}{b}$$ $$a/b/c/d = a{b^{ - 1}}c{d^{ - 1}} = \frac{{ac}}{{bd}}$$ $$a/b/c/d/e = a{b^{ - 1}}c{d^{ - 1}}e = \frac{{ace}}{{bd}}$$ $$a/b/c/d/e/f = a{b^{ - 1}}c{d^{ - 1}}e{f^{ - 1}} = \frac{{ace}}{{bdf}}...$$ and so on.
As @almagest points out, the convention is that operations at the same level of precedence (e.g. multiplication/division) are carried out from left to right. So $a/b/c/d$ is interpreted as $((a/b)/c)/d$, and similarly $1-2-3-4$ is interpreted as $((1-2)-3)-4$. It is best to use parenthesis in any case, or even better, to use prefix or postfix notation.
$a/(b/c)$ or $\frac{a}{\frac{b}{c}}$ is indeed $ac/b$. We also have $$\frac{a}{\frac{b}{\frac{c}{d}}} = a/(b/(c/d)) = a/(bd/c) = ac/bd$$ and so on.