Pythagoras

Here’s the truth about the most famous theorem in the world: Pythagoras’ theorem isn’t really about triangles. It’s about how distance works when directions are independent of each other.

The core idea, in plain words

In a right-angled triangle with shorter sides $a$ and $b$ and longest side (the hypotenuse) $c$ ,

a^2 + b^2 = c^2.

Said plainly: walk $a$ steps east and $b$ steps north, and the straight-line distance back to where you started is $\sqrt{a^2 + b^2}$ . East and north don’t interfere with each other, they’re at right angles, and that’s exactly when their squares simply add.

The man himself, Pythagoras of Samos (c. 570–495 BCE), led a half-mathematical, half-mystical brotherhood that believed “all is number”. The Babylonians knew the rule over a thousand years earlier (the tablet Plimpton 322 lists triples like $119, 120, 169$ ), so the theorem’s name is more tribute than history. What the Greeks added was the proof: the idea that you can know it’s true for every right triangle, not just the ones you’ve measured.

An everyday analogy: the shortcut across the park

You’re at one corner of a rectangular park 30 metres by 40 metres and want the opposite corner. Walk around the edge: 70 metres. Cut across the grass: $\sqrt{30^2 + 40^2} = \sqrt{2500} = 50$ metres. The theorem is the arithmetic of the shortcut, and that worn diagonal path in every park is people doing it by instinct.

A worked example: a proof you can see

Take a big square of side $a + b$ . Inside it, place four copies of the right triangle, one in each corner, hypotenuses facing inwards. They leave a tilted square in the middle, of side $c$ . Count the area two ways:

(a + b)^2 = 4 \cdot \tfrac{1}{2}ab + c^2.

Expand the left: $a^2 + 2ab + b^2 = 2ab + c^2$ . Cancel $2ab$ :

a^2 + b^2 = c^2. \;\checkmark

Now use it. Is the triangle with sides 5, 12, 13 right-angled? $25 + 144 = 169 = 13^2$ . Yes. Whole-number solutions like this are Pythagorean triples, and there’s a recipe for all of them: for whole numbers $m > n > 0$ ,

a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2.

With $m = 3, n = 2$ : $5, 12, 13$ . With $m = 2, n = 1$ : $3, 4, 5$ . Infinitely many triangles from one formula.

A picture in words: squares on the sides

Draw a right triangle and build a real square outward on each side, like tiles. Picture the square on the hypotenuse as a pond, and the two smaller squares as buckets. The theorem says: pour both buckets into the pond and it’s exactly full. Not approximately, exactly, for every right triangle there ever was. And it doesn’t even have to be squares: semicircles, or any similar shapes on the three sides, obey the same rule.

graph LR
  P["a² + b² = c² (plane)"] --> D["Distance in n dimensions: √(x₁² + … + xₙ²)"]
  D --> I["Orthogonal vectors: ‖u+v‖² = ‖u‖² + ‖v‖²"]
  I --> R["Spacetime interval in relativity"]

Common misconceptions

“It works for any triangle.” Only right-angled ones. For others you need the law of cosines, $c^2 = a^2 + b^2 - 2ab\cos C$ , and Pythagoras is the case $C = 90°$ where the extra term vanishes.
“Pythagoras discovered it.” The Babylonians and Indians used it centuries earlier. Pythagoras’s school gets credit for the proof tradition.
“It’s just about 2D.” In 3D the diagonal of a box is $\sqrt{a^2 + b^2 + c^2}$ , and the pattern continues into any number of dimensions.
“There’s a whole-number version for every power.” $a^n + b^n = c^n$ has no positive whole-number solutions for $n \geq 3$ . That’s Fermat’s Last Theorem, finally proved by Andrew Wiles in 1995, 358 years after Fermat scribbled it in a margin.

A historical aside the Pythagoreans hated: the diagonal of a unit square is $\sqrt{2}$ , which is not a ratio of whole numbers. Legend says the disciple who revealed this was drowned. “All is number” survived, but numbers had to grow to include the irrationals.

How it connects

Sideways to Inner Product Spaces: Pythagoras holds for any two perpendicular vectors, even functions, which is why Fourier analysis works.
Up to Relativity, where the theorem gets a minus sign: the spacetime interval is $-c^2t^2 + x^2 + y^2 + z^2$ .
Down to Number Theory: Pythagorean triples and Fermat’s Last Theorem.
Out to Thales of Miletos, the earlier Greek whose geometry Pythagoras is said to have learned, and to the Philosophers to Study page.

The truth revealed: wherever two directions don’t interfere, their squares add. Pythagoras just noticed it first in a triangle.

Links, Sources & Further Reading

http://www.cut-the-knot.org/pythagoras/index.shtml

http://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml

http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html

https://betterexplained.com/articles/measure-any-distance-with-the-pythagorean-theorem/

http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L15.html

http://www.shmoop.com/squares-square-roots/pythagorean-theorem.html