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I'm happy to have the nits picked off it, but it might just convince a schoolkid.
However the locus of the arc is closer to the chord than the "two tangents" construction, so it must be shorter than the locus of the escribed figure.
Therefore however closely we approximate the circumference with rational fractions of r, the arc will lie between them and cannot be a rational fraction of r.
What I was trying to get at is that however small we make θ, if the uprights are rational fractions of r, s lies between them and must therefore be something other than a rational fraction of r.
Vague approximations to straight lines by reference to locus:Quote from: alancalverd on Today at 00:39:51However the locus of the arc is closer to the chord than the "two tangents" construction, so it must be shorter than the locus of the escribed figure. I can draw a line which is a dense series of zig-zags (or tight curves if I want a smooth line). If I put enough zig-zags or tight turns in then I can get it to any length I want. I can do this while keeping the locus of the line "close" to the chord. It can all be kept inside of the escribed figure on your diagram but still be longer than the escribed tangents.
And now another interesting question rears its head! There being more irrationals than rationals, why are so few of them important? Apart from e, π, √2 and an occasional visit from √3, which dominate my working hours, I can't think of another that has swum into the ken of this ancient physicist in the last 60 years!
tape measures aren't callibrated finely enough to measure irrational lengths.
This is probably because we don't tend to use them as an input. For example, if you were making an aeroplane wing you probably wouldn't make it or measure it to be √2 metres long,
In reality, an aeroplane wing is overwhelming more likely to be an irrational number of metres than a rational number of them - but tape measures aren't callibrated finely enough to measure irrational lengths.
In reality, no object is any particular length at all.
One of the oldest yard-rods in existence is the clothyard of the Worshipful Company of Merchant Taylors. It consists of a hexagonal iron rod 5⁄8 inch in diameter and 1⁄100 inch short of a yard, encased within a silver rod bearing the hallmark 1445.....The US survey yard is very slightly longer....
Hopefully, everyone knows the tape measure with markings to show all irrational fractions of a metre is not something I seriously intended to produce.
Using the old standard for length was quite clever.
There being more irrationals than rationals, why are so few of them important? Apart from e, π, √2 and an occasional visit from √3
Quote from: Eternal Student on 07/09/2021 00:42:31Using the old standard for length was quite clever.I was reminded of a quote from a guy who worked at the BIPM something like:"If I scratch the prototype kilogram and remove some of the metal, it is still a kilogram and everything in the universe suddenly gains mass.".