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The Mighty Fool
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Thed famous pythagorean theorm (a2+b2=c2)Was actually already in use in china, egypt and a number of other countries, and it was 'discovered' later by other cultures who'd never heard of it. What really makes pythagoras the 'father of modern mathematics' is not that he came up withy the formula, but that he found a FLAW in it. Under what circumstances will the pythagorean formula NOT work?
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Scott Cram
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There are plenty of circumstances where the pythagorean theorem won't work. For example, when finding the circumference of a circle in relation to its diameter, the pythagorean theorem doesn't work under these conditions.

The most frequently quoted flaw in the pythagorean theorem is that it doesn't work for triangles that aren't right triangles.
Philemon Vanderbeck
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I imagine it doesn't work on a curved surface either.
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Nir Dahan
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Quote:
On 2008-10-14 02:33, Scott Cram wrote:
There are plenty of circumstances where the pythagorean theorem won't work. For example, when finding the circumference of a circle in relation to its diameter, the pythagorean theorem doesn't work under these conditions.

The most frequently quoted flaw in the pythagorean theorem is that it doesn't work for triangles that aren't right triangles.


I don't think there is a "flaw" - it is **defined** for right triangles and euclidian geometry. If you want non right triangle you might wanna consult the cosine theorem (of which the pythagorian theorem is a special case http://en.wikipedia.org/wiki/Cosine_theorem)
I am sure there are also generalizations for non euclidian geometries.

but I wouldn't define it as a "flaw"

for mathematical correctness don't trust me though - let's see what stan has to say about it.
The Mighty Fool
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*ahem* Let me be a little more specific....

When does the pythagorean formula for finding the hypotonuse of a right triangle not work?

(there are different geometric formulae for circles, non-right triangles & so-forth...the pythagorean one is...as Nir Dahan said...defined for a right triangle)
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Nir Dahan
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Maybe in a triangle defined in taxicab or Manhattan geometry...
if you can call it a triangle...
stanalger
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Nir already answered this. I agree with him.
The Pythagorean Theorem works for all right triangles in Euclidean geometry.
Is The Mighty Fool thinking that the fact that rational legs do not necessarily
imply a rational hypotenuse is a "flaw" in the theorem?
The Mighty Fool
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Hmm...I went to the wikipedia link, but it said there was no article of that name?

The flaw is this: (taken from one of the greek histories, not Herodotus) what if you have a right triangle who's legs are both 1? If we plug it into the formula, we get the answer that C=the square root of 2. What do you get if you take the square root of 2? You get an infinite irrational number. Are there any two numbers which when added will give you an irrational result? Is the square root of 2 the RATIO of two whole numbers?!? Of course not & Pythagoreas knew it. Pythagoreas did come up with a solution in which he established the existense of irrational & non-existant numbers, from which illustrious characters such as PI, e, and imaginary numbers would spring. Pythag beleived (correctly) that this was such a monumental discovery, he sacrificed 12 oxen that very day.
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stanalger
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Quote:
On 2008-10-14 23:25, The Mighty Fool wrote:
Hmm...I went to the wikipedia link, but it said there was no article of that name?


Remove the parenthesis from the end of the url.

Quote:
The flaw is this: (taken from one of the greek histories, not Herodotus) what if you have a right triangle who's legs are both 1? If we plug it into the formula, we get the answer that C=the square root of 2. What do you get if you take the square root of 2? You get an infinite irrational number. Are there any two numbers which when added will give you an irrational result? Is the square root of 2 the RATIO of two whole numbers?!? Of course not & Pythagoreas knew it.


Yes, a right triangle with legs each of length 1 has a hypotenuse of length sqrt(2).
How is that a flaw? (The square root of two IS irrational, but it's not
infinite.)

Quote:
Pythagoreas did come up with a solution in which he established the existense of irrational & non-existant numbers, from which illustrious characters such as PI, e, and imaginary numbers would spring. Pythag beleived (correctly) that this was such a monumental discovery, he sacrificed 12 oxen that very day.


Pythagoras established the existence of nonexistent numbers? What does that mean?

You asked:
"When does the pythagorean formula for finding the hypotonuse of a right triangle not work?"
What's the answer?
Nir Dahan
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As far as I know, the CONCEPT of a number was a bit different back then. And especially to Pythagoras. They (Pythagoras and his cult) refused to recognize irrational numbers at all! They twisted and turned just to avoid them in any possible way.

I believe this was settled later on in history.

I still don't see the "flaw" - but maybe it is just me...

one more thing, if you consider this a flaw with a square root of two - what would you say about transcendental numbers like "e" - after all it is equal to 1/0! + 1/1! + 1/2! + 1/3! + 1/4! ...
no sane person would consider a creature like "e" would be the result of a mere addition of simple integer fractions - or would he...

N.
The Mighty Fool
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It's a flaw because it's incorrect. What you get on paper (1.4142135623730950488016887242097...etc.) is not what you'll get if you actually measure the hypotenuse. It's the sort of paradox like the one found in'achillees & the tortise' in which a set of correct and logical steps somehow lead to an incorrect result. If we were to take the formula as correct, then the hypotenuse would never actually REACH the other end of the triangle. It would get infinetly CLOSER, but never touch.
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Nir Dahan
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...and for this infinite sequence of numbers we have a special notation - sqrt(2)
the SEQUENCE is infinite - I see nothing ***wrong*** with it. I am not convinced it is a flaw.

as for closer but never touch. I agree this is confusing but the foundations of calculus show us that an infinite summing of numbers CAN have a finite result. But, you really have to be careful when adding up an infinite number of times. For this mathematicians invented the concept of convergence (which in turn has many finer sub-categories).
some useful links (don't click if you are scared of nasty notations):
http://en.wikipedia.org/wiki/Convergent_series
http://en.wikipedia.org/wiki/Convergence_tests

N.
stanalger
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Quote:
On 2008-10-15 01:50, The Mighty Fool wrote:It's a flaw because it's incorrect.  What you get on paper (1.4142135623730950488016887242097...etc.) is not what you'll get if you actually measure the hypotenuse.  It's the sort of paradox like the one found in'achillees & the tortise' in which a set of correct and logical steps somehow lead to an incorrect result.  If we were to take the formula as correct, then the hypotenuse would never actually REACH the other end of the triangle.  It would get infinetly CLOSER, but never touch.

In Euclidean geometry:
(i) If a right triangle has legs of length 3 and 4, then it has a hypotenuse of length 5.

(ii) If a right triangle has legs of length 1 and 1, then it has a hypotenuse of length sqrt(2).

Does The Mighty Fool think that statement (i) is "correct", but statement (ii) is
"flawed"? That's ridiculous.
stanalger
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I think The Mighty Fool's "problem" with sqrt(2) isn't even related to it being irrational. 1/3 is rational, but its decimal expansion is .333333333333333333333....Does The Mighty Fool think that you can't have a line segment of length .33333333333333.... because you would "never actually REACH the other end"?

Can you not have a hypotenuse of length 5 since 5 = 4.99999999999999...?

"Zeno's Paradox" should be called "Zeno's Fallacy."
The Mighty Fool
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Been awhile I know, but I just found an interesting solution to Zeno's paradox. The problem with 'Achilles & the tortise' is that the race is only measured in 3 dimensions. If the 4th dimension (time) is added, everything makes sense.
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Philemon Vanderbeck
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The 4th dimension is not necessarily "time."
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Nir Dahan
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Quote:
On 2008-12-31 12:53, The Mighty Fool wrote:
Been awhile I know, but I just found an interesting solution to Zeno's paradox. The problem with 'Achilles & the tortise' is that the race is only measured in 3 dimensions. If the 4th dimension (time) is added, everything makes sense.


no connection to the "4th dimension"
it all boils down to accepting that adding an infinite number infinitesimal values can be a finite number - or in other words we have to look up at the definition of convergence of a series
Philemon Vanderbeck
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All numbers are real... even the imaginary and irrational ones.
Professor Philemon Vanderbeck
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