How irrational quantities physically exist in nature?

We know that an irrational no has well defined decimal values upto infinite decimal places. These irrational quantities exist in nature in some kind of measurements. For an example, circumference of a circle is '2πr' , so if radius is rational then circumference will be irrational ,and this case is quite natural. But I am unable to understand , how can a physical measurement exist with infinite precision (due to precision of infinite decimal values in an irrational no) ?? Isn't it in contrary with the laws of Physics??

Solution 1:

I've got bad news for you: Even rational quantities don't exist in nature. Numbers, of whatever type, are just a way to describe our observations. But when we measure, we always have a measurement error, so we cannot really say "this measured quantity has the value $x$", all we can say is "this measured quantity is close to the value $x$". Well, we can do a bit more: Namely quantify how close it is. This is why measurement results are normally written in the form $x\pm y$. For example, a length might be measured to be $1.3\pm 0.5~\rm mm$.

Now we pretend that there is a "true value" of the measured quantity. For example, in the above example, we assume that there is a true value for the length, which is likely(!) less than $0.5~\rm mm$ away from the value $1.3~\rm mm$.

Exact numbers are therefore nothing we can measure, and nothing which we can even prove really exist in nature. They are abstract concepts we made up in our mind in order to describe what we observe. And they work very well for this purpose. But whether they really correspond to something "out there", nobody can really know.

But wait, you say, there are circles, and for those we can prove the circumference over diameter is $\pi$. So $\pi$ is part of our world, right? Wrong: You never have seen a circle. You've certainly seen lots of shapes that closely fit our concept of a circle. But likely if you look close enough, you'll find that it is not really a circle. And even if you cannot detect the difference, how do you know that if you had looked just a little bit closer, you'd have found a deviation?

And ultimately, those circle-lookalikes you've seen probably were all made of atoms anyway, which means they could not have been a circle, as they have volume, while the circle is an infinitely thin line.

Solution 2:

One odd thing about the question is the apparent belief that the decimal notation of a number is somehow fundamental to its meaning, and so an unending decimal expansion is a sign that the number itself is somehow unending or imprecise.

First, a simple factual point: the question claims:

We know that an irrational no has well defined decimal values upto infinite decimal places

I'm not exactly sure what this means, but I suspect it's not true. For example, we do have formulae that will tell us what the $n^\mathrm{th}$ decimal digit of $\pi$ is – in other words, we have a perfectly precise specification of any digit of $\pi$ you care to ask for. So there's no fundamental sense in which $\pi$ is less well-specified than, say $4.6$, which after all has an infinite decimal expansion as well, namely $4.600000\dots$

Anyway, this fixation on decimal expansions is unnecessary. Here's how I choose to imagine the "true nature" of a real number: the essence of a real number is that it measures some continuous quantity – for example, each real number is the length of some idealized line segment.

Then decimal expansions are a way to understand these lengths by comparing them to fixed rational lengths that we know about, namely $1$, $0.1$, $0.01$, and etc. When we say $\pi = 3.141\dots$, we can view this as a series of comparisons, comparing $\pi$ to $3$ and saying it is larger, comparing it to $3.15$ and saying it is smaller, etc.

In this way, the decimal expansion isn't the "true nature" of a number, but simply as the view of a number you get when you look at it through the lens of powers of ten. This becomes more obvious when you realise that you can write the same number in different bases, view it through many different lenses, and indeed in some cases you get very different looking results, e.g. $1/3$ has an unending decimal expansion but a very simple ternary or nonary one.

Now ask again what it means for some real number, the length of an idealized line segment, to be a rational number. Well, to be rational is to be $p/q$ for some whole numbers $p$ and $q$, so a number is rational precisely if some multiple of it is a whole number (namely, $q$ times it is $p$). What that means is you can take your line segment and duplicate it $q$ times, you get a line segment that is exactly $p$ times a segment of length $1$.

Now the question becomes: why should it be rational? Why should that relationship ever hold exactly?

Solution 3:

Philosophically, you could consider an "irrational quantity" as the limit of a sequences of "rational quantities", so e.g. 2pi is what the circumference of a unit circle approaches as you make it more and more perfect.