Non-geometric Proof of Pythagorean Theorem Is there a purely algebraic proof

Janessa Olson 2022-07-07 Answered
Non-geometric Proof of Pythagorean Theorem
Is there a purely algebraic proof for the Pythagorean theorem that doesn't rely on a geometric representation? Just algebra/calculus. I want to TRULY understand the WHY of how it is true. I know it works and I know the geometric proofs.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Jenna Farmer
Answered 2022-07-08 Author has 17 answers
The "modern" approach is this : first we define the field R (for instance, it's the only totally ordered field with the supremum property).
Then we define what a R -vector space is : it's an abelian group with an external action of R satifying some axioms.
Then there is a notion of dimension : we can define a vector space of dimension 2.
The notion of Euclidean distance is obtained by defining what an inner product is : it's a symmetric bilinear form such that x , x > 0 if x 0. The distance is then | | x | | = x , x . We also have the notion of orthogonality from this inner product.
Well, once you did that, then the Pythagorean theorem is a triviality : | | x y | | 2 = x y , x y = x , x 2 x , y + y , y = | | x | | 2 + | | y | | 2 (assuming of course that x , y = 0, the orthogonality hypothesis).
Of course all the work went into the definitions, which is contrary to the basic approach of geometry which deduces properties of distance from a set of axioms (generally ill-defined, but it can be made precise with a little work).
The interesting thing about this modern approach is that algebraic structures come before geometric content. This is powerful because algebraic structures have enough rigidity. For instance, if you start with a set of points and lines satisfying some incidence axioms, it's very hard to define what it means that it has a certain dimension. But if you have a vector space structure, then it's easy.
Of course it can be a little disappointing beacause it feels like we "cheated" : we made the theorem obvious by somewhat putting it in the definitions. But on the other hand, it's very clear and precise : can you properly define what distance or an angle is using "high school geometry" ? Not so easy. Even in Euclide's Elements, this is kind of put under the rug as "primitive notions". This approach makes everyting perfectly well-defined and easy to work with.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-09-01
For each of the following, can the measures represent sides ofa right triangle? Explain your answers.a. 3 m, 4 m, 5 mb. 2 c m , 3 c m , 5 c m
asked 2022-07-11
pythagorean theorem extensions
are there for a given integer N solutions to the equations
n = 1 N x i 2 = z 2
for integers x i and zan easier equation given an integer number 'a' can be there solutions to the equation
n = 1 N x i 2 = a 2
for N=2 this is pythagorean theorem
asked 2022-06-04
Deduce the Pythagorean Theorem [duplicate]
Let V be an inner product space,and suppose that x and y are orthogonal vectors in V.We also know x + y 2 = x 2 + y 2 .My question is how can we deduce the pythagorean theorem in R 2 from it.If possible give geometrical views.
Any help will be greatly appreciated.
thanks!! in advance.
asked 2022-05-28
How does the attached figure prove the Pythagorean theorem?
asked 2022-06-13
Can the Pythagorean Theorem be extended like this?
What is the significance, if any, of the fact that
3 3 + 4 3 + 5 3 = 6 3
?
How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.
asked 2021-01-15

An airplane starting from airport A flies 300 km east, then 350 km at 30 degrees west of north and then 150 km north to arrive finally at airport B. The next day, another plane flies directly from A to B in a straight line. In what direction should the pilot travel in this direct flight? How far willthe pilot travel in the flight? Assume there is no wind during either flight.

asked 2022-05-03
Pythagorean theorem does require the Cauchy-Schwarz inequality?
In Wikipedia I found a proof of Schwarz inequality making use of the Pythagorean Theorem. Anyway I'm wondering... isn't that a petitio principii? Isn't the Pythagorean Theorem in Hilbert spaces proved by the Cauchy-Schwarz inequality? Or effectively is the Pythagorean Theorem a stronger if not equivalent notion?

New questions