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I'm currently studying physics and contemplating the nature of measurements. It seems to me that in practical scenarios, measurements are often represented using rational numbers. However, I'm curious whether this limitation is inherent to the nature of physical measurements or simply a consequence of our methods of measurement and representation.

I have two reasons for considering this:

  • The readings on measurement devices can physically never be a real number, e.g. $1.030405102....$ etc.

  • All measurements are relative to a standard unit that we have agreed upon, such as the meter. Consequently, this tends to render every measurement inherently rational.

Are there any fundamental reasons why measurements in physics can only yield rational numbers? Or are there instances where measurements can result in irrational or even complex numbers?

I would appreciate insights and examples from various branches of physics to shed light on this matter.

Qmechanic
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bananenheld
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  • Possible duplicates: https://physics.stackexchange.com/q/52273/2451 , https://physics.stackexchange.com/q/64101/2451 , https://physics.stackexchange.com/q/2010/2451 and links therein. – Qmechanic Feb 11 '24 at 11:13
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    @Qmechanic I disagree with this being a duplicate of any of the three linked questions. Those are about the inherent (ir)rationality of physical quantities, while this one is about the (ir)rationality of measurements. These are not the same. – Vercassivelaunos Feb 11 '24 at 17:42

2 Answers2

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Any measured value has a finite precision, so, by definition, it must be a rational number. If the "true" value of an observable were an irrational number - $\sqrt 2$ for example - then we could never measure it exactly. We might measure a value of $1.414$ or $1.414213$ or $1.414213562373$ etc., but our measured value would always be a rational approximation to $\sqrt 2 $ (assuming our measuring apparatus was accurate).

In fact, the whole notion of a measurement yielding a single value is misleading. Any measurement involves some amount of experimental error. So to be totally transparent every measured value should include a tolerance e.g. $1.414 \pm 0.001$ or $1.414213 \pm 0.000001$ etc.

gandalf61
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  • So the only thing we can measure about the world are rational intervals? – bananenheld Feb 11 '24 at 10:28
  • That is a misleading statement, @bananenheld. The number by which you quantify "a thing" through a measurement might have limited precision, but this shouldn't mean the world has finite granularity. You don't measure a number, the number is the outcome of a measurement. – Albert Feb 11 '24 at 10:57
  • Yes but all we have are outcomes of measurements. This seem we can only look at the world with granularity – bananenheld Feb 11 '24 at 16:08
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There are several ways to answer your question:

First, as you yourself noted, any practical measurement will always have a finite precision, i.e., finite number of decimal digits, so will always be a rational number. Even if you continue to improve your measuring technique and measure 100 digits of precision, it is still possible that the number you're measuring only has 1000 digits of precision and not more, and is rational.

Second, in most (although not all) cases, measurements have units, and it is arbitrary choice what this unit will be. Imagine that for some unknown reasons, if you measure in kilograms, every measurement must be rational. Then, if you take as your unit of mass kilogram/PI, let's call it pilogram, then in those units the mass a one kilogram will be PI pilograms - an irrational number.

Third, quantum mechanics usually causes things to come in discrete, integer, quantities. So if the fundemental particle of charge has some charge, you can call it 1 (or 1/3, if you include quarks), you can have just integer multiples of this charge. So charge will always be a rational number and have limited precision - you won't be able to measure 1/10th of an electron charge even if you improve your measuring technique.

Nadav Har'El
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