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When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of times)? In other words, why does 1 keep appearing? I'm looking for the visual intuition.

enter image description here

Here's my attempt to answer this:

  1. Start with this definition: Division is "equal distribution."

  2. Consider $\frac{2}{3}$ as equally distributable, as part of $\frac{1}{3}\times{3}$.

  3. Realize that 1 frequently appears because (probably) it demonstrates #1. If you divide something into 2 equal parts, each part is $\frac{1}{2}$ -- that is, its multiplicative inverse. So $\frac{1}{2}$ $\times$ $\frac{2}{1}$ = 1 which indicates that you've equally distributed.

Here is a sketch: enter image description here

Update: This works because $\frac a b = c$ (where a,b, and c are rational numbers and b is not zero) can be rewritten as $a = b \times c$

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  • $\begingroup$ Duplicate of math.stackexchange.com/questions/1339927/… ? $\endgroup$ – Andrea Mori Jun 26 '15 at 14:14
  • $\begingroup$ I like this question. A comment, when you mention "groups of the numerator", I have seen this referred to as the partitive view of division (in contrast with the "repeated subtraction/measurement" view). My instinct would be that 1 shows up in reference to a numerator, because it's comparatively easier to see that $\dfrac{1}{1/4} = 4$, so that we view $\dfrac{2}{1/4}$ as $2\left(\dfrac{1}{1/4}\right) = 2 \cdot 4 = 8$. $\endgroup$ – pjs36 Jun 26 '15 at 15:41
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Well, your pictures are not correct. I recommend you to watch these Khan Academy videos:

Edit: Or I am not able to see what you have meant by those pictures.

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