# What happens when we (incorrectly) make improper fractions proper again?

Many folks avoid the “mixed number” notation such as $$4234\frac{2}{3}$$ due to its ambiguity. The example could mean “$$44$$ and two thirds”, i.e. $$4+234+\frac{2}{3}$$, but one may also be tempted to multiply, resulting in $$83\frac{8}{3}$$.

My questions pertain to what happens when we iterate this process — alternating between changing a fraction
to a mixed number, then “incorrectly” multiplying the mixed
fraction. The iteration terminates when you arrive at a proper
fraction (numerator $$≤\leq$$ denominator) or an integer. I’ll “define” this process via sufficiently-complicated example:

$$143→423→83→223→43→113→13.\frac{14}{3} \rightarrow 4 \frac{2}{3} \rightarrow \frac{8}{3} \rightarrow 2 \frac{2}{3} \rightarrow \frac{4}{3} \rightarrow 1\frac{1}{3}\rightarrow \frac{1}{3}.$$

1. Does this process always terminate?

2. For which $$(p,q)∈N×(N∖{0})(p,q)\in\mathbb{N}\times(\mathbb{N}\setminus\{0\})$$ does this process, with initial iterate $$pq\frac{p}{q}$$, terminate at $$\frac{p \mod q}{q}\frac{p \mod q}{q}$$?

Consider the mixed number $$a\frac{b}{c}a\frac{b}{c}$$, where $$0 \le b < c0 \le b < c$$ and $$a > 0a > 0$$. Then, it is clear that $$ab < ac+bab < ac+b$$, and so the process always continues to lead to smaller and smaller fractions with the same denominator $$cc$$ until the numerator finally becomes smaller than $$cc$$.
In case of a negative mixed number $$-a\frac{b}{c}-a\frac{b}{c}$$, remember that this means "$$-(a+\frac{b}{c})-(a+\frac{b}{c})$$", not "$$(-a)+\frac{b}{c}(-a)+\frac{b}{c}$$". But one can easily ignore the negative sign, so without loss of generality, one can consider positive mixed numbers only.