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What do you call a fraction that has one fraction in the numerator and also one in the denominator? I mean (a/b)/(c/d). The word by word translation from my native language would be: composite fraction. But there is no entry on Google search page for that.

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    $\begingroup$ I believe there is no widely used "named" term for a ratio of ratios in English. I have never seen a special adjective for it, so even if such a term exists I think it probably has to be considered obscure: you could not expect that people who know mathematics will be familiar with the term in the sense you have in mind. $\endgroup$
    – KCd
    Aug 28 at 12:12
  • $\begingroup$ You might want to name the term in your native language. $\endgroup$
    – Tommi
    Aug 28 at 12:42
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    $\begingroup$ My native language calls it a "double fraction", though looking closely, I find three ... $\endgroup$
    – TAR86
    Aug 28 at 20:32
  • $\begingroup$ "Compound fraction" is my recollection from decades ago. "Rational expression" would be what I'd call it now, for sort-of-grown-up accuracy, as opposed to any doctrinaire requirements "in school"... $\endgroup$ Sep 1 at 19:46
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I like your term. The wikipedia article on fractions also mentions they are called complex fractions or compound fractions. Personally, I dislike the term complex fraction as it is obviously going to be interpreted as things like $\frac{3+i}{2-i}$.

I think I call them "fractions of fractions" which is really in tune with your term composite or the synonymous term compound. Furthermore, I tend to use the following language for $$ \frac{ \ \frac{a}{b} \ }{\frac{c}{d}} $$ I say

  • $a$ is the numerator of the numerator
  • $b$ is the denominator of the numerator
  • $c$ is the numerator of the denominator
  • $d$ is the denominator of the denominator

For some variety I also speak of the numerator as "upstairs" and the denominator as "downstairs" when I'm talking through the math I'm writing.

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  • $\begingroup$ Since 1/(1/d) = d, the downstairs of downstairs is upstairs? $\endgroup$
    – Stef
    Aug 30 at 12:51
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    $\begingroup$ @Stef yep. That is how it works. However, in terms of practical architecture it is not an easy design to implement in our spacetime. You'd need to hire Escher. $\endgroup$ Sep 1 at 10:43
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The term complex rational expression (or complex fraction) is commonly used, in U.S. algebra/college algebra texts, to refer to rational expressions where the numerator and/or denominator contain sums or differences of other rational expressions.

Note that technically this would not apply to an expression $\frac{a/b}{c/d}$, which has no sums or differences in it; but perhaps by extension you could use the term, and others would know what you're talking about. Here's the presentation from Sullivan, Algebra & Trigonometry:

Sullivan definition of complex rational expressions

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    $\begingroup$ +1 I think your answer is correct according to current algebra books, at least this agrees with Wikipedia. But, I also must say, the term compound would be a far better choice. $\endgroup$ Aug 28 at 15:10
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    $\begingroup$ "Complex rational expressions" is how I have seen it phrased in textbooks. Though I'll agree that I like "compound" more. $\endgroup$
    – Carser
    Aug 28 at 15:46
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    $\begingroup$ This usage of "complex fraction" appears in older algebra texts, although I haven't looked just now to see how far back it goes. In my mother's college algebra text -- College Algebra by Moses Richardson (1947), which is one of several similar algebra texts I struggled with when trying to learn algebra -- Article 54 (pp. 110-113) is titled Complex fractions. Incidentally, the footnote about "mixed expression" on p. 111 is worth looking at! $\endgroup$ Aug 28 at 18:50
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    $\begingroup$ Every textbook I've ever used called these complex fractions, and I never noticed the thing about a sum or difference needing to be part of it. $\endgroup$
    – Sue VanHattum
    Aug 28 at 22:40
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    $\begingroup$ I'd say OP's expression is a rational expression, and avoid the word complex, to avoid confusion with complex numbers. Off-topic rant: why focus on the effort to work with it? It's just an expression, a member of a much larger family. $\endgroup$
    – Pablo H
    Aug 30 at 13:39
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In addition to "complex fraction" and "compound fraction", they are also sometimes called "nested fractions", as can be shown (for example) here.

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  • $\begingroup$ "Nested fraction" seems a better phrase choice for this than anything else I've seen thus far here. However, I don't know how well the natural language usage for what corresponds to "nested" in other languages conveys the idea we want, and if I were making up a phrase for this (assuming that none existed yet), I'd probably want to investigate that aspect. My recollection from teaching (has been 16 years now) is that I often said things like "... and for the test make sure you can do problems like those having fractions within a fraction" (i.e. I did NOT stress vocabulary all that much). $\endgroup$ Aug 31 at 17:26
  • $\begingroup$ In fact, I'd never heard of "complex fraction" (and probably not "compound fraction" either) prior to this question post. "Nested fraction", I've encountered, and I agree communicates the concept most clearly (directly and unambiguously). $\endgroup$
    – ryang
    Aug 31 at 18:35
  • $\begingroup$ I have seen "complex fraction" in texts, but I try very hard to avoid that term, as it sometimes causes confusion with "complex numbers". I typically use the phrase "compound fraction". $\endgroup$ Aug 31 at 22:13
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    $\begingroup$ I agree. This is the best choice. I forgot about this term when I wrote my answer. Of course, one might think continued fractions are also nested fractions. So, there is that danger, but is it a real danger with college algebra ? As I think upon it, the times I have had a college algebra student bring up continued fractions equals zero. $\endgroup$ Sep 1 at 10:47
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    $\begingroup$ @James S. Cook: the times I have had a college algebra student bring up continued fractions equals zero. This was a standard topic in coll. alg. (U.S. and most other countries) before WW I, but in the U.S. at least I think it was not often covered beginning in the 1920s, although most of the standard coll. alg. texts had a chapter near the end of the book on it (which very few instructors covered, I believe), even up to the 1950s (and more infrequently, in the next few decades). However, I've had 0 students mention the term in coll. algeb. courses I taught from the mid 1980s to mid 2000s. $\endgroup$ Sep 1 at 19:22
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I googled fraction over fraction and was led to the term "complex fraction". Note that this includes more different cases than the one you identified, but does include fraction over fraction. Lots of hits on that term and it is commonly used, for example at Khan Academy.

See: https://www.merriam-webster.com/dictionary/complex%20fraction

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