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What is the pythagorean theorem?

'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.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 32,080

In mathematics, the **Pythagorean theorem**, also known as **Pythagoras' theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":

where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras.

The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps and cartoons abound.

For details, see the link Pythagoras' Theorem.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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Hi Ganesh;

How are you?

My concern is that I am not sure what the pythagorean theorem is saying in terms of modern mathematics? For example, is it a statement about metric spaces? Well, if we are talking about R^2 with the l_2 norm, then the statement is trivial, if we are not talking about this particular metric on R^2, then the "theorem" seems to be false.

Is it then telling us that l_2 norm is the God-given norm for R^2? What does that mean?

I think the correct answer is that the pythagorean theorem somehow tells something about inner-product spaces. But I still don't get what it is.

*Last edited by Agnishom (2019-12-14 15:43:06)*

'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.

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**ganesh****Administrator**- Registered: 2005-06-28
- Posts: 32,080

Agnishom wrote:

Hi Ganesh;

How are you?

Hi Agnishom,

Fine, Thanks.

Regarding your query, you may PM bob bundy, and zetafunc. They may help you.

Good wishes.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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