Why is pythagoras theorem what it is?

[blah]

Why is the absolute distance sqrt(x^2 + y^2)?

I've found several proofs on the internet but none really tell me much.
I believe I have noticed something about pythagoras theorem and that is it allows all inertial reference frames in physics to be equivalent. In physics you can describe objects in any inertial frame and they will have the same relative speeds and distances (in Newtonian physics) and there is no reason to say one inertial frame is more correct than any other. But say the absolute distance was x + y instead of sqrt(x^2 + y^2), then relative speeds and distances would change has you rotated the frame of reference and the laws of physics wouldn't be equivalent in all reference frames. Also, I believe you can say special relativity extends pythagoras therem to include time, and SR finds a way to make time and distances change with velocity and still keep all inertial reference frames equiviilent.

Is this right? and if so why?

 

[dextercioby]

It's correct.Euclidean space is rotationally symmetric and for a vector its modulus (given the Pythagorean theorem in 3 dim-s) is constant both under space translations & under space rotations...

As for flat Minkowski space time,it's the (relativistic) interval that is constant under the rotation group SO(1,3).

Daniel.

 

[tacman]

Pythagoras theorem is a fundamental mathematical concept that is based on the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be represented algebraically as a^2 + b^2 = c^2, where a and b are the lengths of the two shorter sides and c is the length of the hypotenuse.

This theorem is important because it allows us to calculate the length of one side of a triangle if we know the lengths of the other two sides. It also has many real-world applications, such as in construction and navigation.

As for why the absolute distance is represented as sqrt(x^2 + y^2), this is because of the Pythagorean theorem. In a Cartesian coordinate system, the coordinates of a point are given by the distances from the point to the x and y axes. These distances can be represented by the sides of a right triangle, with the hypotenuse being the distance from the point to the origin. Therefore, the absolute distance can be calculated using the Pythagorean theorem.

In regards to your observation about inertial reference frames in physics, you are correct. The Pythagorean theorem is essential in maintaining the equivalence of all inertial reference frames. This is because in a moving frame of reference, the lengths of the sides of a triangle may appear different due to the relative motion. However, the Pythagorean theorem remains true regardless of the frame of reference, ensuring that the laws of physics remain consistent.

In special relativity, the concept of spacetime is introduced, where time and distance are not independent but are combined into one entity. The Pythagorean theorem is extended to include time, and through the use of Lorentz transformations, it is shown that the speed of light remains constant in all inertial reference frames, maintaining the equivalence of these frames.

In conclusion, the Pythagorean theorem is a fundamental mathematical concept that has many practical applications and is crucial in maintaining the equivalence of inertial reference frames in physics. Its representation of the absolute distance as sqrt(x^2 + y^2) is based on the relationship between the sides of a right triangle and is essential in understanding the laws of physics in different frames of reference.