It's easy to take math too literally. When we readThe Tortoise and the Hare, we know it's not about zoology -- there's a deeper wisdom.
Why can't we treat equations like one of Aesop's fables, with a lesson buried inside?
看看勾股定理。好像是关于三角形的，对吧?
Well, sure. But is it only that? Seeing the equation:
has a literal interpretation "The sides of a triangle have a specific mathematical relationship."
Ok, fine. That's some nice zoology. Stepping back, what's really happening?
The sides of a triangle, which point in different directions, have a mathematical relationship.
Or better yet:
Two objects, which exist in different dimensions, can still be compared.
Here's an analogy: Who's the better athlete, Michael Jordan or Muhammad Ali?
We shouldn't ask who has a better jump shot or uppercut -- that comparison stays within a single dimension, clearly favoring one or the other. A better comparison would encompass all dimensions:Who was more dominant compared to the competition? Who held more championships? Who advanced the sport more?
Ah. We need a different type of "expansive" comparison to help each component see the other side.
In the triangle case, we have an "A direction" (East/West) and a "B direction" (North/South). Instead of comparing them directly, we square them to make area. Why?
Here's an intuition. Individual directions point in a single direction within our 2d universe. But area spanseverydirection available in our universe. By converting a single direction to its square (which points in all directions), we have a common "all directions" format that can be compared. It becomes "universal area vs. universal area" and not "pointing North vs. pointing East" (aka jump shots vs. uppercuts).
Will any type of universal self-comparison work? (Squaring, cubing, etc.)
No, unfortunately. The Pythagorean Theorem is special because it shows thespecificcomparison of squaring ($a^2 + b^2 = c^2$) keeps a simple relationship between the whole and its parts. There's probably a relationship for cubing, but it's not as clear cut.
Pythagorean TheoremFact:The sides of a right triangle follow $a^2 + b^2 = c^2$
Pythagorean TheoremWisdom:要比较不同的东西，就要找到一种通用的比较方法。(确切地说，要把自己摆成方形来创造空间。)
Equations aren't so boring when you look for the moral of the story, right?
Appendix: Visually Creating Area
In the Pythagorean Theorem, we can imagine spinning our 1d lines into area and comparing that. Here, we can spin each side into a circle:
And yowza, the area matches up!
This is a demonstration of the theorem; theproofshows that area will always combine neatly like this. (阅读更多。)
Appendix: Vector Pythagorean Theorem
We can make our "self comparison" analogy more technical with vectors.
- $\vec a$ is the vector $(a, 0)$
- $\vec b$ is a vector in a different dimension, $(0, b)$
- $\vec c$ is a vector made from both: $\vec c = \vec a + \vec b = (a, 0) + (0, b) = (a,b)$
The Pythagorean Theorem says "If we compare each item to itself, the combined self-comparisons of $\vec a$ and $\vec b$ equal the self-comparison of $\vec c$".
(c compared to itself) = (a compared to itself) + (b compared to itself)
在矢量世界里，什么是自我比较?Adot productwith yourself.
The Pythagorean Theorem is stating:
which works because:
(Since $\vec a$ and $\vec b$ are perpendicular, their dot product is zero.)
And for the parts:
This specific self-comparison maintains a simple relationship between the whole and its parts (addition).
对两个向量的简单一瞥并不能提供一种内置的方式来比较它们;instead, use a derivedscalar(single number) that allows a comparison to be made.