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We know that the use of rational numbers is easy to imagine when we use them in real life. Like dividing a particular area of land by $2.25$ will give us an exact quantity which can actually be used to physically divide the land (for example by fencing) into tangible equal parts which when combined will return the whole physical and tangible area of the land.

But what if we divide it by a repeating irrational such as $\pi$? The number we will get will itself be repeating ad nauseam and hence can't be used to physically divide the land. Or can it be? So my question was how do we reconcile irrational numbers with how we normally use or imagine numbers in reality?

MathKnight
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  • Grow (or lose) some fingers and then you can choose a "realistic" radix where the fraction is finite (non repeating). That a fraction repeats in some radix is an artefact of the choice of radix, e.g. $1/3$ doesn't repeat in radix $3.\ $ As such there in not much to imagine (except twiddling fingers). – Bill Dubuque Feb 08 '22 at 19:17
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    Take a piece of string of length $c$ and place it in a circle. Measure the diameter of the circle and it will be $\frac c\pi$. So that suggests you can divide by $\pi$ – Henry Feb 08 '22 at 19:18
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    Have you ever tried measuring division of land by fencing? I don't think there is any difference in practice (and "in practice" is what your reasoning is based on) between dividing by $2.25$ and dividing by $2.25 + \frac{\pi}{1000000000}.$ Note that $2.25 + \frac{\pi}{1000000000}$ is irrational. And by the way, in what way is $\pi$ a repeating irrational? I think with "repeating" you're using the wrong word. – Dave L. Renfro Feb 08 '22 at 19:25
  • @Henry - So that means it is possible for a physical object to exist with length equal to pi? – MathKnight Feb 08 '22 at 19:30
  • @BillDubuque But isn't pi irrational in every base? What about then? – MathKnight Feb 08 '22 at 19:30
  • @MathKnight Repeating decimals are rational – Bill Dubuque Feb 08 '22 at 19:31
  • @DaveL.Renfro But it is not possible to physically demarcate land in exact units of pi? Or does some method exist for that? – MathKnight Feb 08 '22 at 19:32
  • @BillDubuque - I am sorry... I am meant to say about non terminating non recurring irrational numbers... I will edit post to reflect that. – MathKnight Feb 08 '22 at 19:35
  • not possible to physically demarcate land in exact units of pi? --- My view is that it is not possible to physically demarcate land in exact units of $0.5.$ Incidentally, the relevant issue with what I think you're getting at (division of something into exactly $n$ equal parts for some positive integer $n)$ is rational vs. irrational, and getting obsessed about types of decimal expansions (or expansions in other bases) seems to serve no real purpose here. – Dave L. Renfro Feb 08 '22 at 19:46
  • @MathKnight "it is possible for a physical object to exist with length equal to pi?" $;-;$ Yes, assuming you can somehow measure a "physical object" with infinite precision (which you cannot in practice, not even when halving it). But if you could then, yes, the length of a circle of diameter $1$ would be exactly $\pi$. And the diagonal of a square of side $1$ would be exactly $\sqrt{2}$ which is also irrational. – dxiv Feb 08 '22 at 20:00
  • We cannot distinguish physically an irrational number from a rational number very close to it. Therefore, irrational numbers are not relevant in physics. Physically , you cannot claim that a square has side length exactly $1m$ , not even that it is actually a square. Therefore , physically it makes no sense to claim that the diagonal has length $\sqrt{2}$. – Peter Feb 09 '22 at 13:20

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You have a fundamental, but very common, misconception about physical measurements. There is no such thing as a physical object of exactly length $\pi$, no matter what the unit-of-measure for length. Not $\pi$ meters, not $\pi$ feet, not $\pi$ femtofurlongs.

But this is not because $\pi$ is irrational. There is also no such thing as a physical object of length exactly $1$. The actual cause is that physical measurements are not defined to perfect accuracy. Once you start looking close enough, the very definition of what you are measuring no longer makes sense. At a macroscopic scale, the boundaries of the object seem quite precise. But by the time you get to the atomic scale, they are not clear at all. Is this atom a part of the object or not? The distinction becomes fuzzy. Physical objects exchange molecules with their environment constantly.

Look a little closer, and the picture becomes even less clear. So you've decided this atom is part of the object. But what counts as "inside" the atom and thus part of the object, and what counts as out? You cannot decide by particle position, because the particles do not have well-defined positions. Instead they exist in a quantum-mechanical smear.

All physical measurements suffer similar drawbacks. There is some limit to their precision. This limit is not just because of our inability to measure more precisely, but also because the quantities we are measuring are not even well-defined below some level. The only measurements we know precisely are either because they are fundamentally discrete and thus can only take on certain values ("I have 2 eyes") or else because we have defined the units they are being measured in to enforce a specific value. Thus we know exactly what the speed of light in a vacuum is in meters per second. Instead it is what constitutes a "meter" and a "second" that are a bit fuzzy.

So no, you cannot divide the area of a piece of land by exactly $2.25$ or even by exactly $2$, any more than you can divide it exactly by $\pi$. The area of the piece of land is not exact in and of itself, so neither can any division of it be.

The physical universe is not mathematical in nature. Rather, mathematics is a tool we can use to model it. But those models are at best approximate representations. They cannot reflect it exactly.

Paul Sinclair
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