If I cut a string, will its length be rational or irrational? [duplicate]

If I cut a string (lets say approximately halfway), will the length of each side (in inches) be rational or irrational?

marked as duplicate by Qmechanic♦Aug 23 '17 at 17:56

• Your question seems a bit unclear. Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. What do you want to know by saying whether the length is rational or irrational? If you cut it, and measure the two parts, the length measured will depend on accuracy and least count of instrument. – Wrichik Basu Aug 23 '17 at 17:11
• I think that Jerry Schemer's answer to the question "Is it possible for a physical object to have a irrational length" linked to above is a pretty good one. Both the set of rational numbers and the set of irrational numbers densely fill the real number line. However, the set of irrational numbers is a "higher order" of infinity that the infinite number of rational numbers. (See works by Georg Cantor on infinite sets). So if one were to randomly choose a number on the real number line, the odds of it being irrational are virtually 1. – Samuel Weir Aug 23 '17 at 18:28

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.

• This answer seems to assume that $1\mathrm s/N \approx 0.11\rm ns$ is the briefest reliably measurable time, which is not really correct. – rob Aug 23 '17 at 18:01
• I had in mind a fraction for n but you're right, it comes across as an integer the way I wrote it. – user154997 Aug 23 '17 at 18:23
• This also seems to assume that we are concerned with measurable lengths, while the question itself may not necessarily be asking about measured values; but actual ones. See for example this answer physics.stackexchange.com/a/52281/127931 – JMac Aug 23 '17 at 18:24
• Yes, sure the a Mathematician answer would be it is nearly certain the original length is irrational, so both half too. – user154997 Aug 23 '17 at 18:26
• Between any pair of rational numbers, there is an irrational. Possibly more surprising is that the reverse is also true: between any pair of irrational numbers, there is a rational. So, unless you have absolutely zero tolerance, you can't know. Algebraic numbers, in general, are irrational but not so bad: e.g. $\sqrt 2$, they are roots of polynomials with rational coefficients. You might want the term "transcendental", these are not the root of any polynomial with rational coefficients e.g. $\pi$ and $e$. However, irrelevant in this case: again you need zero tolerance to know which. – badjohn Aug 24 '17 at 6:33