Why Logarithms Appear in This Integral

Рет қаралды 2,184

Жыл бұрын

Before the days of Calculus, one Pierre de Fermat wanted to find the area under the function f(x)=x^n. This problem we now call "integration" was then called "quadrature" or "squaring". Fermat was able to square every function f(x)=x^n for any rational n except for one case: n=-1 (that is, the hyperbola). It turns out that this unique nature of the hyperbola was tied to logarithms and Euler's number e. But why? Why does the area under the hyperbola have anything to do with logarithms or e?
Other resources
e: The Story of a Number by Eli Maor
www.amazon.com/Story-Number-P...
Images
upload.wikimedia.org/wikipedi...
Music
patriciataxxon.bandcamp.com/a...
Starboard by Patricia Taxxon

Пікірлер: 12

@jHan Жыл бұрын

Mistake at 7:17: log(1)=0, log(10)=1, log(100)=2, log(1000)=3

@comic4relief Ай бұрын

Glaring mistake.

@mohammedbelgoumri Жыл бұрын

In his video on exponentials and logarithms, Grant from @3b1b asks if anyone knows of a beautiful geometric interpretation of logs. Here it is.

@Zosso-1618 6 ай бұрын

This is amazing! I remember reading various proofs that the function describing the area under a hyperbola has logarithmic properties. But in one sentence, you made that connection deeper than my reading did! Bravo!

@guillaume5313 Жыл бұрын

Great content as always !

@TheMemesofDestruction Жыл бұрын

Fantastic decomposition!

@matiasvergara5069 Жыл бұрын

Very nice video jHan!

@DavyCDiamondback Жыл бұрын

Really, it's ln(|x|)+C, coincidentally, no matter what C you choose, if this function intercepts y=x^2, it intercepts perpendicularly

@eduardoguimaraes5226 8 ай бұрын

Do you use manim?

@Avighna Жыл бұрын

The video is amazing, but the title isn't as great as it could be. Something like "How Descartes discovered the power rule without calculus" might be better, anyway amazing job man!

@davidwright8432 7 ай бұрын

True, but that drags Descartes into it, and in terms of this explanation, he's irrelevant. Interesting guy, tho!

@Zosso-1618 6 ай бұрын

Did you watch the video? Descartes isn’t mentioned once, that was Fermat. And most of the video is about the area under the hyperbola, not the power rule.