Many folks avoid the “mixed number” notation such as 423 due to its ambiguity. The example could mean “4 and two thirds”, i.e. 4+23, but one may also be tempted to multiply, resulting in 83.

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 ≤ denominator) or an integer. I’ll “define” this process via sufficiently-complicated example:143→423→83→223→43→113→13.

Does this process always terminate?

For which (p,q)∈N×(N∖{0}) does this process, with initial iterate pq, terminate at \frac{p \mod q}{q}?

**Answer**

Yes, the process does always terminate.

Here’s why:

Consider the mixed number a\frac{b}{c}, where 0 \le b < c and a > 0. Then, it is clear that ab < ac+b, and so the process always continues to lead to smaller and smaller fractions with the same denominator c until the numerator finally becomes smaller than c.

In case of a negative mixed number -a\frac{b}{c}, remember that this means "-(a+\frac{b}{c})", not "(-a)+\frac{b}{c}". But one can easily ignore the negative sign, so without loss of generality, one can consider positive mixed numbers only.

**Attribution***Source : Link , Question Author : Zim , Answer Author : Geoffrey Trang*