How can an irrational number, represent a finite length between two points?

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Stop replying to blatant baits.

[math] \exists x, \space \{x^{2} = 2 \space| \space x \in \mathbb Q\}[/math]
[math]x \in \mathbb Q \Rightarrow (x = \frac{a}{b}\wedge gcd(a,b) = 1 ) \Rightarrow (\frac{a}{b})^2 = 2 \Rightarrow a^2 = 2b^2 \Rightarrow 2|a^2 \Rightarrow 2|a[/math]
[math]a=2c,\;4c^2=2b^2 \Rightarrow2c^2=b^2 \Rightarrow 2|b\Rightarrow gcd(a,b)\neq1\Rightarrow x\notin \mathbb Q[/math]
[math]\square[/math]

>draw a line that is 1 unit long
>draw another line that is 1 unit long, perpendicular to the first and touching one vertex
>connect the lines

congrats I drew a line of length sqrt(2), also you could "arguably enter" Zeno's paradox when drawing a line of length 1, since you have to draw a half, then a quarter, etc.

brainlet here
I never understood how can a finite number be indefinite

Irrational doesn't mean incomprehensible or indeterminable. It means not the ratio of two integers.

>How can an irrational number, represent a finite length between two points?
There's a maths guy in Australia who would be right up your alley: Dr. Norman Wildberger.
He has a youtube channel and has attempted to rebuild math from the ground up without invoking the assumption that there are any numbers besides the rationals.

Well yes, you enter the problem again when drawing a line 1 unit long, drawing a line that has length of sqrt(2) like in your example is moreso drawing a triangle and then defining the two legs to be ideally equal to 1 and the hypotenuse is consequently 1. You haven't addressed the problem, just named a way around it using idealized models.

the sqaure root of 2 is a measure of a Euclidean Geodesic between 2 points. If you are on a quantized surface then obviously your Geodesics are quantized and the theory of length breaks down into a degenerate approximate form.

The mistake is thinking of geometry with geodesics as the primary notions of operation between points.

The proper notion is quadrance. Given a vector V = (a,b), and a symmetric Matrix a (the metric, the identity matrix is the euclidean metric) then define the quadrance of the vector to be VA(Vt). Once you have the quadrance, define a Vector U = (c,d). to find the metric separation between the two vectors, you can use ((VAUt)^2)/((UAUt)(VAVt) to be equal to the (cosine of the angle between the vectors) squared. From there the laws of rational trig hold any symmetric matrix A, are valid over a general field (finite, complex), suggest non euclidean geometries, and make projective geometry easier.

think of this. you can never shoot a bullseye on its absolute center. you know it is there, you can get indefinitely close, but you aint never getting it exactly right.
this is an irrational number

>Brainlets itt
Including you.

What would it take to "address the problem," and why doesn't an actual line on paper with a length of √2 centimeters or feet or whatever solve it?

zeno's ""paradox"" isn't worth thinking about anyway. all it says is that some infinite sequences have a finite sum. shocker. you should have seen a proof of that in calculus 2

You can argue a line gets asymptotically close to a length of sqrt(2), in real life a line would be measurably the same as sqrt(2), but in theory the line would have to terminate at some point while sqrt(2) wouldn't. Essentially the line would deviate from sqrt(2) at some point even if that point is at 10^-9999 place.
Zeno's paradox relates to this as yes a line of 1+.4+.01... would approach sqrt(2) as much as 1+1/2+1/4... approaches 2 but it still never reaches it, just approaches a point that is immeasurably different.

>break a stick
>it's exactly sqrt(2) inches long
>never know because irrational
tfw

So what's the point of bringing in sqrt(2)? All you're saying is that no measurement can be 100% accurate.

Empty space makes it theoretically possible to have a line of exactly sqrt(2) centimeters in length. Though it' s probably impossible to have a measuring instrument that precise.

Because even if we define some line to be ideal 1 you could've construct sqrt(2) from segments of that line. sqrt(2) as a value exists as a consequence of mathematics rather than a construction.

I think what OP meant is that we cannot represent irrational lengths in finite space (like our own world)

perfect explanation