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In real life, we can have a pencil of length 2 cm. Can we have pencil of length $\sqrt{2}$ cm?

My answer to that was no , we cannot even make 2 cm pencil.

My argument was that when are working theoretically in mathematics we have axioms and certain assumptions. In real life we approximate things, therefore every lengths and measurements are just an approximation.

Qmechanic
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    "we can have a pencil of length $2\ \operatorname{cm}$." Are we sure about that? – pancini Feb 16 '24 at 07:34
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    In my humble opinion, you're almost right. In the most extreme case, one can define one centimeter as half the length of your pencil. That would make it exactly two centimeter. But as you state, any other measurement (like $\sqrt{2}$ cm) would be an approximation. – Dominique Feb 16 '24 at 07:35
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    Yes, of course: put your 1 cm pencil on the origin of the cartesian axis and draw a square of 1 cm side; then use a 2 cm pencil and put it on the diagonal of the square and cut it at the intersection point of the two sides. – Mauro ALLEGRANZA Feb 16 '24 at 07:38
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    I think this is a question about physics and not mathematics :) For the records, I'd like my pencil to be $\pi$ cm. –  Feb 16 '24 at 07:40
  • The question about whether we can have a pencil of length 2cm brings in all kinds of non-mathematical issues and depends on definition. For a long time we could have a piece of metal exactly 1 metre long because there was a particular piece of metal used to define a metre length. The mathematical question would be about whether given one length another different length can be constructed. And given a length of $2$ the length $\sqrt 2$ can be contracted via the diagonal of a square. –  Feb 16 '24 at 07:43
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    I agree with the original proposer. We can't have a pencil of any exact length, because it isn't possible to define to arbitrary precision (e.g. less than the size of a "pencil molecule") where the pencil starts and ends. Somehow saying something is "exactly $\sqrt 2$ cm long" sounds ridiculous because of the implied infinite decimal precision, whereas we use "exactly 2 cm" in everyday English to mean "2 cm, to as high a degree of accuracy as I can be bothered to measure". –  Feb 16 '24 at 07:45
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    @SeverinSchraven I am less fussy but I insist that my pencils have a transcendental length. Ideally, a non-computable length. – badjohn Feb 16 '24 at 11:31
  • Possible duplicates: https://physics.stackexchange.com/q/52273/2451 and link therein. – Qmechanic Feb 16 '24 at 15:54

2 Answers2

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It is philosophical in nature. Here is my intuitive opinionated answer.

(1) We can easily define a certain Pencil to have certain length. When we say it is $4$ cm then we will get $2$ cm when we break it into 2 . . . .

We can also define it to be $\sqrt{2}$ cm or $\pi$ cm : In that case , it is "harder" to generate $2$ cm.

(2) A Metal bar of size $1$ cm can be heated or cooled to get $1.1$ cm or $0.9$ cm. Without quantization , we can assume that the Bar goes through all lengths , rational & irrational between original length & final length.
With that view , $\sqrt{2}$ cm or $\pi$ cm will be nothing special . . . .

With quantization , we can take the least change to be $\delta$ : that value will dictate what lengths we can generate , whether irrational or not.

(3) When heating & cooling curves & angles & curved angles , we might generate more irrational values.

Using 3D Volumes like blowing up balloons , we might generate $\sqrt[3]{2}$ & such . . . .

(4) Eventually , it all rests on what values we Define , what values we have , what "operations" we can use & what values Physics allows us to generate.

Electro-Magnetism / gravity / space-time curvature at cosmic levels / macro levels / quantum levels will allow various values while preventing other values.

Prem
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Yes. We can construct a square with a unit edge length, then the diagonal of that square (the hypotenuse of each isosceles right triangle the diagonal splits the square into) has a length of $\sqrt{2}$. So it's a constructible length which could be manufactured if someone really wanted to. Even taking into account imprecision in measurements, if enough machines were manufacturing enough pencils, one of them could have a length of $\sqrt{2}$ by chance, so it is possible, just not probable.