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

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:

143423832234311313.

  1. Does this process always terminate?

  2. 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

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