If I cut a string (lets say approximately halfway), will the length of each side (in inches) be rational or irrational?
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A physicist answer: the most precise way to measure each length would be to time the round trip of a light signal in vacuum. After all, there is a reason for the definition of the meter being based on exactly such a procedure. So ultimately the length $l$ will be measured as
$$l = c \frac{1}{2}\Delta t$$
where $\Delta t$ is the round-trip time and $c= 299 792 458\, \mathrm{m}/\mathrm{s}$. The best we can do is to measure $\Delta t$ as a multiple of some stable period. Since the second is defined as $N=9 192 631 770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom, we shall go with that and get
$$l = c\frac{1}{2}\frac{n}{N} \tag{*}$$
where $n$ is the number of periods measured for the round-trip. In fact, one will get an sandwiching:
$$c\frac{1}{2}\frac{n}{N} \le l \le c\frac{1}{2}\frac{n'}{N}$$
with $n<n'$. In this case, the quoted answer would be the average of the two bounds, which has the same form as eqn (*). Thus we can say confidently that this most precise measure will yield a rational number. I used SI units but since the conversion factor between an inch and a meter is a rational number, the answer is the same.