Can Pythagoras theorem be proven wrong?

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re: Can Pythagoras theorem be proven wrong?

Post by ME »

johannesbender wrote:pythogorean theorem applies to pythogorean triangles , that is all i know.
That 7x7 square has one point in [0,0] and the opposite in [7,7]
The triangle is implied by the corner of the square and exactly as raj already drew [0,0]-[7,0]-[7,7]
The circle with radius 10 has its origin in [0,0]

Would the vector from [0,0] to [7,7] be of length 10 then the circle would intersect perfectlyat point[7,7].

It does not.

(yet it somehow does in raj's drawing)

Because of Pythagoras, we know that length [0,0]-[7,7] is almost 0.1 unit short of 10, (√98 ≈ 9.8995 ≈ 9.89949494).
We can draw a small circle around point [7,7] with a radius 0.1.
This small circle does intersect this radius-10 circle. It graphically confirms that the circle with radius 10 outdistances that hypotenuse of the square-box by 0.1 units.

Raj,
Please check coordinates [8, 6] and [9.6, 2.8].
What do you think of coordinates [7.6, 6.5] ?

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johannesbender wrote:May i just ask how are you guys inserting your images mid way in the replies ?

i could never get that done with bbcode image urls here.
Inline images
While secured "https" was added later, site owner Scott decided to automatically convert "http" to "https" (or something like that) to keep former inline images, and thus posts, intact. As a result you have to reference the image with "http". The converter does the rest.
Unfortunately, large images like I added can break the site, or add annoying horizontal scrollbars for the entire page. So rather not. But raj is ignoring me. So I still did.
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re: Can Pythagoras theorem be proven wrong?

Post by raj »

THIS THREAD started in the form of question from a non-scientific person, from an odd observation made during my working on my pendulums wheel concept.

Un point a la ligne. C'EST TOUT.

Thank you for joining me on this futile discussion.

Raj
Keep learning till the end.
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Re: re: Can Pythagoras theorem be proven wrong?

Post by johannesbender »

ME wrote:
johannesbender wrote:pythogorean theorem applies to pythogorean triangles , that is all i know.
That 7x7 square has one point in [0,0] and the opposite in [7,7]
The triangle is implied by the corner of the square and exactly as raj already drew [0,0]-[7,0]-[7,7]
The circle with radius 10 has its origin in [0,0]

Would the vector from [0,0] to [7,7] be of length 10 then the circle would intersect perfectlyat point[7,7].

It does not.

(yet it somehow does in raj's drawing)

Because of Pythagoras, we know that length [0,0]-[7,7] is almost 0.1 unit short of 10, (√98 ≈ 9.8995 ≈ 9.89949494).
We can draw a small circle around point [7,7] with a radius 0.1.
This small circle does intersect this radius-10 circle. It graphically confirms that the circle with radius 10 outdistances that hypotenuse of the square-box by 0.1 units.

Raj,
Please check coordinates [8, 6] and [9.6, 2.8].
What do you think of coordinates [7.6, 6.5] ?

-
johannesbender wrote:May i just ask how are you guys inserting your images mid way in the replies ?

i could never get that done with bbcode image urls here.
Inline images
While secured "https" was added later, site owner Scott decided to automatically convert "http" to "https" (or something like that) to keep former inline images, and thus posts, intact. As a result you have to reference the image with "http". The converter does the rest.
Unfortunately, large images like I added can break the site, or add annoying horizontal scrollbars for the entire page. So rather not. But raj is ignoring me. So I still did.
Thank you ME , always wondered but never asked how to so with the images like that , i will keep them "internet/bandwidth friendly" if i ever want to use it.
Its all relative.
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Re: re: Can Pythagoras theorem be proven wrong?

Post by ME »

raj wrote:THIS THREAD started in the form of question from a non-scientific person, from an odd observation made during my working on my pendulums wheel concept.

Un point a la ligne. C'EST TOUT.

Thank you for joining me on this futile discussion.

Raj
There's a difference between a weird observation (and ask a question about it) and disproving a complete theory because of it.
Earlier raj wrote:I have found a way to prove Pythagoras theorem wrong.

I can prove this in minutes.

My problem now is how do I claim this discovery proof as MINE, before disclosing the proof to the world.
With a "degree in Maths and Computer Science" you, of all people, should understand how these kinds of abstractions work.
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re: Can Pythagoras theorem be proven wrong?

Post by Fletcher »

Mornin Raj ..

IMO your alternative to Pythagoras theorem has to be robust. That is, work at any scale including the very large, where things are easier to see.

Part of the problem with these types of exercises is making sure you have an exact (perfect) Right Angle to work from. And Square etc.

Many builders when laying out a building footprint or lining up an internal wall etc use a Pythagorean 3,4,5 Triangle where the hypotenuse is 5 (or multiples of) etc.

In the attachment below is an alternative to using Pythagoras for finding a perfect Right Angle at large scales that I actually once used to mark out a new implement shed in Fiji (13 x 8 meters).

The method was used by the Ancient Egyptians for finding the corners of their pyramids at very large scales that I stumbled upon many years ago. It just uses pegs, ropes, circles, and measuring tape. No 3,4,5 Triangles.

Note that I drew in the Large Grey circle of radius 10.000 units and inside it a Large Orange circle of radius 9.899 units (the hypotenuse). This inner touches the corner of the 7.000 x 7.000 square formed once I established the perfect Right Angles.

If still in doubt about accuracy of Pythagorean Geometry and Theorem I suggest you head down to the beach with your grand kids and start making lines and circles and see if you can still robustly disprove Pythagoras at the larger scale with accurate Right Angles to work from.
Attachments
Raj - Pythagoras Theorem
<br />
<br />Forming Perfect Right Angles Using Overlapping Circles on a Line.
Raj - Pythagoras Theorem

Forming Perfect Right Angles Using Overlapping Circles on a Line.
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re: Can Pythagoras theorem be proven wrong?

Post by Fletcher »

OK .. here's a simple visual proof raj. To 3 decimal places.

Just follow the doubling meno method of the original top left black squares thru 3 steps until the square has been doubled in length of sides and the area quadrupled.

You can't get from 7.000 to 14.000 side lengths with a 10.000 side for the hypotenuse.

Look closely at the background grid reference to see the divergence which gets bigger and bigger with each area doubling thereafter.
Attachments
Raj - Pythagoras Proof 4
<br />
<br />Meno Square Doubling
Raj - Pythagoras Proof 4

Meno Square Doubling
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re: Can Pythagoras theorem be proven wrong?

Post by Fletcher »

Here's a visual proof using just Squares and their relationships between side lengths and areas.
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Raj - Pythagoras
<br />
<br />Using Squares
Raj - Pythagoras

Using Squares
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re: Can Pythagoras theorem be proven wrong?

Post by WaltzCee »

This reminds me of Homer Simpson.

https://m.youtube.com/watch?v=pSh-lY6XoKY

Don't laugh too hard.
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re: Can Pythagoras theorem be proven wrong?

Post by raj »

I have just put this video on youtube.

Raj
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RAJ - pythagoras work - 260720.jpg
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re: Can Pythagoras theorem be proven wrong?

Post by Fletcher »

YouTube

https://www.youtube.com/results?search_query=Raj+Balkee

There appears to be something wrong with your presentation Raj (as per your screen shot in the previous post)

Explanation :

You have used an on line Right Angle Triangle Calculator (one of many online).

The presentation in your screen shot is upside down compared to most calculators.

a height = 1 , b base = 10, c = hypotenuse length.

Note that a & b form a right angle of 90 degrees.

Length of c (hypotenuse) = 10. 04988 (5 decimal places accuracy/precision limit available from you)

The math that Pythagoras uses is ...

=> a^2 + b^2 = c^2
=> 1^2 + 10^2 = c^2
=> 1 + 100 = c^2
=> c^2 = 101
=> c = sqrt(101)
=> c = 10.0498756 (stopped at 7 decimal places by me). N.B. 10.0498756^2 = 100.99999957547536 (calculator runs out)

You math is as reproduced ..

=> a^2 + b^2 = c^2
=> (1)^2 + (10)^2 = (101.04988)^2 N.B. 5 decimal places precision provided by you
=> 1 + 100 = 101.0000880144 N.B. answer is to 10 decimal places precision

Conclusion : => 101 =/= 101.0000880144 i.e. False


As you well know raj math is only as accurate to the least number of decimal places used ! It's called Decimal Precision and is a basic premise of Math learned at school !
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re: Can Pythagoras theorem be proven wrong?

Post by johannesbender »

If i remember correct your suppose to use integers for pythogorean right triangles ? meaning whole numbers only .
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Re: re: Can Pythagoras theorem be proven wrong?

Post by Fletcher »

Oystein wrote:
When all sides in a right triangle is whole numbers, it's called a Pythagorean triplet,

The basic triplets are:

3,4,5
5,12,13
8,15,17
7,24,25

Nowhere can 7,7,10 be found. Because it can't be done.

Best
ØR
When Triplets (whole numbers only) are used Decimal Precision is redundant. Any other time it applies.
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Re: re: Can Pythagoras theorem be proven wrong?

Post by ME »

johannesbender wrote:If i remember correct your suppose to use integers for pythogorean right triangles ? meaning whole numbers only .
For the Pythagorean triplets you need natural (whole) numbers, to get them aligned to a grid.


For the Pythagorean formula (a² + b² = c²) you can use irrational numbers too .... (numbers with many decimals)

To start we can multiply or divide any of those Pythagorean triplet by a number.

For example:
3² + 4² = 5²
Multiply by 2:
6² + 8² = 10²
Divide by 10:
0.6² + 0.8² = 1.0²

You can measure that the angle of the triangle between side 0.8 and the hypotenuse 1.0 is about 36.87°. This angle remains the same despite those multiplication factors, even when using side 4 and hypotenuse 5 the angle is about 36.87°

The Sine of 36.87…° is (close to) 0.6, The Cosine of 36.87…° is 0.8.

We can actually replace this into the Pythagorean formula:
Sin²&#952; + Cos²&#952; = 1²

It is a unit circle.
When we factor in the radius r then:
(r·Sin &#952;)² + (r·Cos &#952;)² = r²

Et voila, trigonometry!
We can use any number for radius r, or angle &#952;:

To write our initial example a bit more complex yet more universally:
(5·Sin 36.87…°)² + (5·Cos 36.87…°)² = 5²

Hence, when 2·7² would equate to 100 then not only would the Pythagorean stuff be broken but also trigonometry.... and more.

Add:
I had to explicitly add those triple dots to indicate a rounding error. Seems important somehow.
For example:
10.050² = 101.0025
10.0499² = 101.0004900…
10.04988² = 101.0000880…
10.049876² = 101.0000076…
10.0498756² = 100.9999996…
10.049875621121² = 101.0000000…
WaltzCee wrote:This reminds me of Homer Simpson.
https://m.youtube.com/watch?v=pSh-lY6XoKY
Don't laugh too hard.
While our topic-starter tries to fit irrational lengths onto natural grids, this recent one seems to apply too:
https://www.youtube.com/watch?v=sDfzCIWpS7Q
Marchello E.
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re: Can Pythagoras theorem be proven wrong?

Post by johannesbender »

Theodorus of Cyrene looked in to the irrational numbers apparently.

https://www.intmath.com/blog/mathematic ... mbers-4948

i dont know squat about these things though lol.
Its all relative.
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