Pythagorean got his knowledge from the Egyptian Priest- African. 

If I'm not mistaken they used constructions using compuses and straight edges. There are some interesting visual proofs for the pythagorean theorem in that sense.

Summing squares is arithmetic.

Very simply, they did use what we would call algebra. "They" being the ancient Egyptians were the first to find the value of Pi, creating trigonometry geometry at least a thousand years before Euclid and Pythagoras were even born.

The Pythagorean theorem is a geometric theorem, not an algebraic one. You can erect squares on the sides of a triangle. If the triangle is a right triangle, then the squares erected over the kathetes together have the same area as the one erected over the hypothenuse. You can prove this using the basic properties of areas: two areas are the same if you can cut one of them into pieces and rearrange the pieces to get the other area. Most proofs do exactly that.

I dont know what he means by "C is the vertex position". In the pythagorean theorem, its the length of the right triangle's side that is opposite the 90 degree angle, which I think is definitely different from a position which has no length. Also, all that other solving stuff could be more easily summarized as "you can solve for any side length when the two other side lengths are known". Sorry for verysmarting, I'll be on my way to sniff my own farts

Not 100% sure about this, but isn't the Pythagorean Theorem one of the first things you learn in math after plus, minus, divide and multiply? If that's the case, that makes it pretty basic knowledge.

Jesus christ, they're completely wrong on half of this. C is a side length, not a vertex, and you can't solve for anything with only one side length. This smells like a seventh grader who thinks they're clever

Ya this looks inaccurate to me.

you can only solve it 3 ways (you need to know 2 of the 3 variables). You can figure out the lengths other ways but you need trigonometry and angles to do it.

A vertex is usually a point not a line, but tbh i could have just forgot if vertex position is a thing. i majored in math but i don't remember everything.

The theorem only works on right triangles and they somehow missed that.

c is always the longest side length

a and b are interchangeable but you probably labeled them left to right out of habit so that parts accurate kind of, just written strangely.

Such a genius should know sides are marked with lowercase letters

Which proof did you look at? For me the simplest to understand is the one where you draw a square with a square inside it, exch corner of the inner square is on the perimeter of the outer square. Inner square is side c, outer square is side a + b.

From there it's just algebra to equate the area of the big square with its constituents, the inner square and the 4 right angled triangles. Probably best as an image, I'm sure there's one online.

There’s like a billion different proofs of the Pythagorean theorem. There’s geometric, algebraic, trigonometric, even using calculus. Pick your favorite.

https://en.m.wikipedia.org/wiki/Pythagorean_theorem

The reason it only works for right angled triangles is because it’s a version of the Cosine rule where the -2ab*cos(C) term disappears because C is 90 and cos(90) is 0. Getting a satisfying proof for that with a tangible explanation would probably be much more difficult, and it’s not my area of expertise anyway.

I don't have an answer, but I have the same question as you OP. It's a great question to get into first principle thinking:

https://yourbrainchild.wordpress.com/2023/12/06/modeling-first-principle-thinking/

The more fundamental explanation is actually really cool.

It turns out that the pythagorean theorem defines the distance between two points. That is, when you get more advanced, it's not a theorem but an assumption. If you change that assumption you get different geometry that still works. For example, geometry on a sphere where triangle angles add up to more than 180 degrees and the shortest path between two points is a curve (used in navigation all the time).

Have you tried these? Depending on his age, but also on how confident he is on the basics:

https://nrich.maths.org/pythagoras-theorem-and-trigonometry-short-problems

https://nrich.maths.org/pythagoras-theorem-and-trigonometry-age-11-16

Pythagoras' theorem "to the nearest 5 decimal places"? What the hell does that even mean? Pythagoras' theorem is a^2 + b^2 = c^2

the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Its got nothing to do with PI. He is a pompous idiot

yup, typical piers.

He wins the prize

People who ask these types of questions don’t understand it enough to use it in the real world.

The Pythagoean theorem is probably one of the most equations in all of math, science, engineering, data analysis, etc. It essentially tells you how to measure distance in a multidimensional space. Any time you do anything with two or more variables, this is relevant.

Indeed. To work out the distance between map co-ordinates.

I'm an engineer, so quite often.

Used in machining, looking at 3D objects usually a part that’s going to be turned(machine operation). It helps when looking at concentricity of multiple diameters along the same center line.

I understand it as a statement about area. A right triangle is a plane figure, and one of the fundamental quantities that help you understand figures in the plane is area.

Finding the length of the hypotenuse of a right triangle is hard. Finding the area of the square on the hypotenuse is almost trivial, if you know how.

Consider this square (and apologies for the roughness). Can you find its area? You could try breaking it up into four triangles and a small square. Or you could think of it as a big square minus four triangles. There's more than one way to do this. Now, once you know its area, what's its sidelength?

There's also an avenue where you can think of it as being a consequence of how a right triangle is made up of two similar copies of itself. What you want to do is rotate it so that the hypotenuse is horizontal, and drop a vertical altitude. (Or, the same picture but thought of differently: it's about how length doesn't change under rotation. So you can rotate it so that the hypotenuse you're measuring is horizontal and then drop in a new coordinate system.)

I think I've got something that can work. Like, try and ask why, for a vector x, the sqrt(x dot x) gives the length of x. It's obvious when x is on an axis. But if you show that the quantity "x dot x" is invariant under rotations then it would show this calculation should work for any x. This would lead to a proof of the Pythagorean theorem whose intuition boils down to "x dot x is invariant under rotations".

>why does the L² norm coincides with our notion of distance

Because of the Pythagorean Theorem, and because we live in flat space where the Pythagorean Theorem applies.

If you take two points in space, and draw the line segment between them - then drop lines parallel to the coordinate axes - you get a right triangle. Therefore the length of the hypotenuse is given by the Pythagorean Theorem, which is easily proven to use squares, not other powers.

For what it's worth, while there are many L^p metrics, the L^2 metric really is special: it's the only one induced by an inner product.

In order to discuss right triangles, you need a notion of angle. An angle is a number associated to two intersecting lines or vectors. This function is a bilinear form. The only exponent p for which the p-norm comes from a bilinear form is p=2. p=2 norm geometry is the only geometry with angles.

I remember doing an exercise in Axler's that asked you to prove that there is an inner product on R^2 such that the norm of (x,y) is (x^p + y^p )^(1/p) for p > 0 if and only if p = 2. So essentially, the 2 can be tracked to arising from the definition of an inner product and its associated norm, as has already been mentioned in this thread.

The Intuition is by proving it, especially with the constructions where you make squares, and move the area around.

As for "why not other powers", you can define distance using other powers, and these are called L^(p) norms, where p is the associated power. It just happens to be that, in our world, L^(2) is the observed norm.

One "reason" why that occurs is that, if you consider circles under other L^(p) norms, they will look, "not round" when drawn. Only L^(2) makes "round" circles

Don't mistake "theorem" with "theory", they are not the same, and don't mistake either for being measly.

Every time I hear "just a theory" I cringe.

A theorem, in math, is a statement that has been proved. Laws tend to be more so just observations, and in this sense a law is the lesser of the two, although the term doesn't seem to be used super consistently in math. For example, I'd classify the power law as a theorem by these definitions, but I'm also not a mathematician so I may be missing something.

To just extend on the answers: ultimately, the naming of a certain result/expected result comes down to history, not nomenclature. Mathematics has words such as theorem, lemma, proposition, corollary, tautology, triviality, rule, law, formula, (in)equality for things that are actually proven, and problem, conjecture, hypothesis, postulate, axiom for ones that are not or cannot. There are also relevant differences between those, roughly:

• Axioms are basic assumptions to work from, while conjectures are often just not yet resolved. However, a conjecture might turn out to be independent from the axioms, roughly meaning that its truth value cannot be decided. Sometimes conjectures thus become axioms.
• Laws and rules are often simple formulas (e.g. law of large numbers, law of (co)sines, ...), but not always. Equalities and inequalities are also a special type of them.
• Corollaries are "immediate" (a very subjective term) consequences of something else. However, often a theorem or lemma is a corollary, too.
• Theorem and lemma are kinds of important results, with lemmas more on the technical and theorems more on the final side. Meanwhile, propositions are often helpful intermediate results to be forgotten later. But a lot of exceptions exist.
• Tautologies and trivialities are things that follow directly from the assumptions without any thought (so this ultimately is subjective).
• Conjectures, hypotheses and problems are unresolved questions. Often a problem is more open-ended, while a conjecture proposes a clear answer; hypotheses are closer to conjectures but often the author is less sure about the correctness.
• Axioms are assumptions one bases things on. Some axioms are very basic (e.g. "there exists a set"), others are more convoluted (e.g. the continuum hypothesis).

But let me demonstrate how inconsistent this turns out to be:

• Bertrand's Postulate (proven)
• Mordell Conjecture (proven, also known as Faltings's Theorem)
• Class field axiom(s) (some rules that one often proves, not just assumes)
• Pell's equation (not just an equation, but also a statement about its solutions)
• Zorn's lemma (often an axiom, equivalent to the axiom of choice)
• Boolean prime ideal theorem (another possible axiom, weaker than the previous one)
• Diamond principle (a different and not so common axiom)
• Burnside's lemma & Polya's Enumeration Theorem (really more laws/formulas to calculate the number of orbits)
• Whitehead problem (effectively an axiom)

Best answer I have found:

Theorems are results proven from axioms, more specifically those of mathematical logic and the systems in question. Laws usually refer to axioms themselves, but can also refer to well-established and common formulas such as the law of sines and the law of cosines, which really are theorems.

In a particular context, propositions are the more trivial theorems, lemmas are intermediate results, while corollaries are results deduced easily from others. However, lemmas and corollaries may be major results on their own.

Note that a system may be given axioms in more ways than one. For example, we can use the least upper bound axiom to define the real numbers, or we can consider this axiom as a theorem if we were to construct the reals from the rationals using Dedekind cuts and prove it instead. The difference here lies in which axioms we choose to start with.

When you stated, “Given its assumptions,” it got me thinking. Can’t assumptions provide wide latitude in making theorems? For example, would the proposition that 2+2=5 always be true if we assume that everything that u/TedMerTed states is true.

In math, anything that can be proved or derived in some framework, either directly from the axioms or postulates of that framework or in combination with other theorems, is a theorem.

This is in stark contrast to how the word theorem and theory are used in, say, a scientific sense, which dies not note a form of necessary truth (where it does in math) or in a lay sense which denotes a hunch or speculation.

A law typically refers to any concise and fundamental mathematical truth. Laws may or may not be theorems themselves, and are quite often axioms or postulates.

> In maths, a "theory" is something that looks like it might be true, but hasn't been proven true in all possible cases.

No, that's a hypothesis.

A mathematical theory is just the second thing you described.

dont feel pedantic when this is genuinely used as an argument against evolution

It would be better as a proof if you enforced the angle between the green bricks as 90°.

I ran into this phenomenon recently building the grand piano. There’s a right triangle you build to brace the pedals, and the hypotenuse is 11 studs long. This blew my mind (there’s no small Pythagorean triple with hypotenuse divisible by 11), until my wife pointed out that the triangle itself has vertices in the center of the technic pins, and hence is only ten studs wide. So the triangle in the piano is 3-4-5 after all.

I’m no math whiz..lol..but I did once teach my daughter all about how manual transmissions work using one of the really big Technic cars with a visible transmission and engine🙂

being able to count the number of studs on each side really adds to the illustration. nice!

Lego Batman would be proud