Non-terminating values of physical quantities in reality

Question

My question is: Can the true value of a physical quantity be non-terminating?

It is not possible to measure the true value of a physical quantity due to error (also called uncertainty) in the measurement. As a result, the measured value is never a non-terminating number, with accuracy of the instrument deciding the number of significant figures.

True value ($T$) = Measured value ($M$) $\pm$ Error ($E$)

Suppose, for the length of a particular object measured with a metre scale, $M = 50.2$ cm and $E = 0.1$ cm $=0.1000 \ldots$ cm (Right?). Then we have, \begin{equation} 50.1\ \text{cm} \leq T \leq 50.3\ \text{cm} \end{equation} Or more precisely, \begin{equation} 50.1000\ldots\ \text{cm} \leq T \leq 50.3000\ldots\ \text{cm} \end{equation} But does the above inequality mean that $T$ can be non-terminating also?

Suppose, \begin{equation} T= 50.20789 \ldots =50+0.2+0.007+0.0008+\ldots \end{equation}

The infinitely many terms added to make the true value do not seem to make any sense to the finite length of the object.

PS: If non-terminating true value possible, any comment on its type (rational but non-terminating, surd, transcendental, etc.)?

Thank you in advance :)

Answer 1

But does the above inequality mean that $T$ can be non-terminating also?

In typical models parameters are thought of as real numbers with continuous distributions; if this is the case, then in a measure-theoretic sense the probability of finding a rational number (which encompasses all terminating decimals) is exactly zero.

However, modeling parameters with real numbers is just a convenient choice, and it is experimentally indistinguishable from them only taking, say, rational values.

So, in a deep sense, I think your question is unanswerable: both possibilities are consistent with observations.

A caveat is about units of measurement: by redefining them we can make rationals into non-rationals and vice versa. Restricting ourselves to dimensionless constants, however, solves this.

The infinitely many terms added to make the true value do not seem to make any sense to the finite length of the object.

An object with finite length can be described by a non-terminating decimal expansion: the series you wrote is convergent, so there is no issue there.

As a final point, it's not what you asked, but I think you have a misconception about the meaning of "error": if we state that $x = a \pm b$, we typically do not mean that $a - b \leq x \leq a + b$, but instead that our belief about $x$ is distributed with mean $a$ and standard deviation $b$. This is a compactified notation, and it is a complete description of the distribution only for certain ones such as Gaussians. Still, for many purposes, it is enough.

The point is that there are no hard constraints: if I state that $x = a \pm b$ and another experiment measures $x = a + 1.5 b$ that's a rather weak disagreement. If they find $x = a + 10b$ it's a very strong disagreement, but we can't call it impossible strictly speaking.