Determining the square root of 2 via geometry

[HAGD]

I know we can hone in on pi using rounder and rounder regular polygons and blah blah but is there a similar method to perhaps hone in on the square root of 2? I ask because I'm aware that the square root of 2 is what you get when you try to find the diagonal of a square with a width of 1 using Pythagoras theorem.

also since there arent many math topics here... WHAT OTHER EUCLID GEOMETRY THINGS U GUYS GOT?

[Vannak]

when it comes to pi, the much more interesting thing is the algorithms that produce it, infinite sums and the such.

for the square root of two, it is form 1+1/n, where n is defined recursively as 2+1/n. Basically
1+[2+1/(2+1/...)]

Others exist, definitely.

[Layra-chan]

I'm confused. We can construct the square root of two precisely using geometry. As you said, it's the diagonal of a unit square. Do we need a more complicated geometric construction? It's not like the approximation of pi via polygons doesn't use the Pythagorean theorem all over the place in some form or other.

[HAGD]

Layra-chan

I'm confused. We can construct the square root of two precisely using geometry. As you said, it's the diagonal of a unit square. Do we need a more complicated geometric construction? It's not like the approximation of pi via polygons doesn't use the Pythagorean theorem all over the place in some form or other.

from what i understand, your contention is that it would be more complicated to represent the square root of 2 numerically?

[HAGD]

Vannak

when it comes to pi, the much more interesting thing is the algorithms that produce it, infinite sums and the such.

for the square root of two, it is form 1+1/n, where n is defined recursively as 2+1/n. Basically
1+[2+1/(2+1/...)]

Others exist, definitely.

so because n is defined as a formula which contains itself, the calculation is infinite and implies the nature of pi, huh? am i on the right track or have i completely misunderstood?

thanks for the post ill have to look into pi algorithms

[Layra-chan]

HAGD

Layra-chan

I'm confused. We can construct the square root of two precisely using geometry. As you said, it's the diagonal of a unit square. Do we need a more complicated geometric construction? It's not like the approximation of pi via polygons doesn't use the Pythagorean theorem all over the place in some form or other.

from what i understand, your contention is that it would be more complicated to represent the square root of 2 numerically?

Yes? It's not rational, so neither its decimal expansion nor its continued fraction expansion terminate, although its continued fraction expansion does have a simple pattern. Geometrically, you draw three lines. Not exactly sure how this is more complicated.

[HAGD]

Layra-chan

HAGD

Layra-chan

I'm confused. We can construct the square root of two precisely using geometry. As you said, it's the diagonal of a unit square. Do we need a more complicated geometric construction? It's not like the approximation of pi via polygons doesn't use the Pythagorean theorem all over the place in some form or other.

from what i understand, your contention is that it would be more complicated to represent the square root of 2 numerically?

Yes? It's not rational, so neither its decimal expansion nor its continued fraction expansion terminate, although its continued fraction expansion does have a simple pattern. Geometrically, you draw three lines. Not exactly sure how this is more complicated.

its simpler, and i think i understand that you've embraced these irrational numbers with open arms. however, it is much more useful in my opinion to understand the square root of two as close to 1.4 because we can only intuit rational numbers. i see your point, but surely you sympathize with mine

[Vannak]

Layra-chan

HAGD

Layra-chan

I'm confused. We can construct the square root of two precisely using geometry. As you said, it's the diagonal of a unit square. Do we need a more complicated geometric construction? It's not like the approximation of pi via polygons doesn't use the Pythagorean theorem all over the place in some form or other.

from what i understand, your contention is that it would be more complicated to represent the square root of 2 numerically?

Yes? It's not rational, so neither its decimal expansion nor its continued fraction expansion terminate, although its continued fraction expansion does have a simple pattern. Geometrically, you draw three lines. Not exactly sure how this is more complicated.

I think he's coming at this from a formalism point, rather than an experimentally determined value point. He wants more methods like the circular approximation by increasing the n-gon count.

The fraction form I gave you is called the continued expansion form, and there are lots of aesthetically neat ones. Though I'm pretty sure not every thing with a continued fraction expansion is irrational, I know many irrational numbers have very regular continued fraction expansions.

[VorpalNeko]

It still doesn't really make sense, though. Constructable polygons are just about the geometrically simplest objects next to circles. So from the geometrical POV, an isosceles right triangle is already 'mission accomplished' for √2.

Anyway, for a sequence of rational approximants, your suggestion of a continued fraction gives the best ones--provably 'best' in the sense that a convergent of the continued fraction (i.e. truncated), when simplified, comes closer to √2 than any other fraction of lesser or equal denominator. So 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ... . (Actually there's also a slightly stronger sense of 'best' that also applies for them.)

It's notable that the better rational approximants to π produced by the n-gon method, 22/7 and 355/113 (the latter of which used a 12288-gon), are both found as convergents of a continued fraction for π.

[Layra-chan]

Are we doing geometry or algebra? Algebraically, rationals are simpler. Geometrically, (in the compass-and-straightedge sense) it basically goes integers -> constructible numbers. It takes about as much effort to divide geometrically as it does to take a square root, so from a geometric standpoint, it doesn't make any sense to stop at the rational numbers.
Given that you bring up the polygon approximation to pi, which for most polygons is not a rational number at all, I was under the impression that we were doing geometry, in which even from a formalist standpoint sqrt(2) is no more complicated than, say, 7/5.

[HAGD]

Layra-chan

Are we doing geometry or algebra? Algebraically, rationals are simpler. Geometrically, (in the compass-and-straightedge sense) it basically goes integers -> constructible numbers. It takes about as much effort to divide geometrically as it does to take a square root, so from a geometric standpoint, it doesn't make any sense to stop at the rational numbers.
Given that you bring up the polygon approximation to pi, which for most polygons is not a rational number at all, I was under the impression that we were doing geometry, in which even from a formalist standpoint sqrt(2) is no more complicated than, say, 7/5.

well, im only satisfied with 7/2 because i can convert it to 3 1/2. similarly, ill only be satisfied with the square root of 2 once i can convert it to a rational number. not that i dont see what youre saying

[Layra-chan]

HAGD

Layra-chan

Are we doing geometry or algebra? Algebraically, rationals are simpler. Geometrically, (in the compass-and-straightedge sense) it basically goes integers -> constructible numbers. It takes about as much effort to divide geometrically as it does to take a square root, so from a geometric standpoint, it doesn't make any sense to stop at the rational numbers.
Given that you bring up the polygon approximation to pi, which for most polygons is not a rational number at all, I was under the impression that we were doing geometry, in which even from a formalist standpoint sqrt(2) is no more complicated than, say, 7/5.

well, im only satisfied with 7/2 because i can convert it to 3 1/2. similarly, ill only be satisfied with the square root of 2 once i can convert it to a rational number. not that i dont see what youre saying

Okay, so you just don't like irrational numbers. You're not even really happy with rational numbers except in the sense that they can be converted from "improper" fractions to "proper" fractions.
Just to check, are you actually happy at all with the polygon approximation to pi? Because that doesn't spit out rational number approximations most of the time.

As an aside, can I just say that the distinction between proper and improper fractions is about the most mathematically inane thing I've ever seen in a maths class? With the caveat that I never actually took algebra, geometry or calculus as classes in grade school, so there might be more meaningless things there.

[Morberticus]

HAGD

ill only be satisfied with the square root of 2 once i can convert it to a rational number

You might be waiting for some time.

[Slutty_Eddie]

HAGD

]well, im only satisfied with 7/2 because i can convert it to 3 1/2. similarly, ill only be satisfied with the square root of 2 once i can convert it to a rational number. not that i dont see what youre saying

Math is about solving the problems, not just throwing your hands up and choosing "good enough" answers. For example, by accepting the square root of -2 as imaginary, it opens up the world to complex numbers, and manifold space and all sorts of interesting mathematical constructs.

By limiting yourself to rational numbers, you get an answer, but not the whole answer.

[Layra-chan]

In fact, by limiting yourself to rational numbers, you get a very difficult answer. The integers, and by extension the rationals, are very, very complicated, much more so than the so-called complex numbers. There are entire branches of mathematics, very big, very active, devoted to dealing with the fact that the rationals are terrible from basically every perspective, algebraic, geometric, analytic, topological. The only "nice" thing I can really think of about the rational numbers is that they're the initial object in the category of characteristic 0 fields, but that doesn't really give a good characterization of the internal structure at all.