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Linelites

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Registered: 2016-10-25

Posts: 2

Irrational Philosophical conundrum in Pythagorean Triangles

In trying to fully grok the concept of irrational vs. rational, I try to visualize the difference between them represented by two Pythagorean Triangles.  One triangle, like the 3,4,5, will have commensurate, rational values between the legs and hypotenuse.  Change the aspect slightly, so as to make it 3,1, and the sqrt of 10, and the legs and hypotenuse are now incommensurate and irrationality introduced.  My question is: by what mechanism does this occur?  Why should a slight shift in aspect of the legs take us from one realm of numbers to another? 
I am a hobbyist, this is my first post.  Perhaps the answer is obvious, but it eludes me and I hope someone can help me.  Perhaps it is purely a numerical property that is detached from the physical representations of geometry and therefore lacks any mechanism.

thickhead

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Registered: 2016-04-16

Posts: 1,086

Re: Irrational Philosophical conundrum in Pythagorean Triangles

Beware! You may be drowned.
https://en.wikipedia.org/wiki/Hippasus

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Linelites

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Registered: 2016-10-25

Posts: 2

Re: Irrational Philosophical conundrum in Pythagorean Triangles

Thrown back into the subconscious....

IF
Vasudhaiva Kutumakam.{The whole Universe is a family.}  Feels like lately  we're in the middle of a cosmic Thanksgiving 

Agnishom

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Registered: 2011-01-29

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Re: Irrational Philosophical conundrum in Pythagorean Triangles

In trying to fully grok the concept of irrational vs. rational, I try to visualize the difference between them represented by two Pythagorean Triangles.  One triangle, like the 3,4,5, will have commensurate, rational values between the legs and hypotenuse.  Change the aspect slightly, so as to make it 3,1, and the sqrt of 10, and the legs and hypotenuse are now incommensurate and irrationality introduced.  My question is: by what mechanism does this occur?  Why should a slight shift in aspect of the legs take us from one realm of numbers to another? 
I am a hobbyist, this is my first post.  Perhaps the answer is obvious, but it eludes me and I hope someone can help me.  Perhaps it is purely a numerical property that is detached from the physical representations of geometry and therefore lacks any mechanism.

This should not be posted in Help Me!

Because you're being philosophic here, I'd want to point out to you that the rational numbers are dense in the reals. Or put in a more fancy way, there is a rational between any two irrationals and an irrational between any rationals. So, there is not much work to be done to switch over from rationals to irrationals.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Irrational Philosophical conundrum in Pythagorean Triangles

Thrown back into the subconscious....

IF
Vasudhaiva Kutumakam.{The whole Universe is a family.}  Feels like lately  we're in the middle of a cosmic Thanksgiving 

What I meant was that Hippasus who invented irrational number was drowned as punishment.

Last edited by thickhead (2016-10-28 03:09:38)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}