What makes the natural log "natural"?

I enjoyed this, but I don't think you answered the question in the subject. Napier (and Briggs) popularized logarithms a half-century before Newton and Liebniz discovered Calculus, but you extensively used calculus to describe the natural-ness. Napier invented logarithms by painstakingly calculating 0.999999^n to get n for values between 0 and 1. Briggs recognized that base 10 would be more useful, and I expect by the time they got around to inventing Calculus Newton and Liebniz were using logarithms. The "natural" bit comes from just looking at the partial sums of the harmonic series: if k terms sum to s, 10×k terms sum to about s + 2.302585092994046 ( LN(10) ) if it takes k terms to sum to s, to get s+1 we will need about k×2.718281828459045 terms That was known before calculus was.

I have no idea why this is happening. When I use the YT editor to cut out the intro 5 minutes of ambient animations (as people are coming in and we make sure everything is working properly), despite what the preview indicates the end also sometimes gets cut, and it seems to do so randomly and permanently. I'm looking into it.

3blue1brown

It was interesting to hear you pronounce John Napier's name as if it were French, but he was a Scot, and Napier is a fairly common name in Scotland even today -- it rhymes with 'rapier'. Napier is credited with discovering logarithms, and with inventing a tool to aid arithmetic: https://en.wikipedia.org/wiki/John_Napier Scotland and France have historically been quite close, so that might explain how his name has become attached to natural logs there.

Frank Wales

I'm no mathematician so I might be wrong, but I think there are two tricks at play. 1. Even though the series does not converge at x=1 (it would resemble 1 - 1 + 1 - 1 + 1 - 1 + …) the expression 1/(x+1) is defined at x=1. So it can be viewed as an analytic continuation, since it as the same value everywhere in the open interval (-1;1). 2. We don't actually evaluate the series at x=1, just the integral. This might be a bit fishy and I too would be glad about a real explanation of what's going on, but I guess this might be part of what's going on.

Hey Grant, great stream as always. I enjoy them very much! I just wanted to bring to your attention that it seems like you cut of the end of it. For this and also the last stream you did on Tuesday the video just ends mid-sentence, which is quite a bummer. I would be happy if you could fix it.

I cracked up when that happened, and tried to enter it as the favorite number!

Nice lecture. Especially the explanation of \int_a^b frac{1}{x} dx = ln(x)|_a^b is fantastic. I like it a lot. If I am in the classroom, I maybe have two questions. At 53:10 I failed to follow: exp(a+b) = exp(a) exp(b) and exp(x) = exp(1)^x. But I think I should watch the previous video again... Around 1:12:30, When the common factor of *infinite* geometric series r is |r| >= 1, it does not converge. So when you plugged in the common factor (-x)'s x = 1, I would ask a question, whether is it ok to do that. This class is a joy. Thank you.

Hitoshi Yamauchi

Did I just see a gorilla walking through your set? :)

Marc Cohen

_ericBG perpendicular to what ?

Martin Embeh

The 🐒 🤣🤣🤣

Martin Embeh

gorilla

Is there a "normal" video in the pipeline too? Don't get me wrong, I love these videos, but I'm also missing the classic videos

_ericBG