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The Lost Art of Logarithms

lostartoflogarithms.com

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Re: The Lost Art of Logarithms

#91

Post by evanb »

pash wrote: post #44
tmoertel wrote: post #20

Here's an logarithmic fact that I've made use of frequently: If X is a random variable having a uniform distribution between zero and one, then –ln( X )/ λ has an exponential distribution with rate λ . This relationship comes in handy when, for example, you want to draw weighted random samples. Or generating event times for simulations.

The general version of this is called inverse transform sampling [0], which uses the fact that for the cdf F of any random variable X the random variable Y = F(X) has a standard uniform distribution [1]. Since every cdf increases monotonically on the unit interval, every cdf is invertible [2]. So apply the inverse cdf to both sides of the previous equation and you get F^-1(Y) = X is distributed like X . Sampling from…

Inverse transform sampling is a special case of normalizing flow where we don't need to learn anythin.g

https://en.wikipedia.org/wiki/Flow-based_generative_model

Re: The Lost Art of Logarithms

#92

Post by smitty1e »

BobbyTables2 wrote: post #90
agumonkey wrote: post #61

Earlier quoted context omitted.

I often wonder about this. I also believe that mathematical pedagogy strive to attract people that are very smart and think in the abstract like euler, and not operationally, meaning they will get it intuitively. For other people, you need to swim in the original problem for a while to see the light.

Math is rarely taught with practical problems in mind — that’s engineering !

Therein lies the rub. Treating abstract and the concrete in isolation was always tough sledding for me.

Bouncing between the two is where the action is.

And units: if I had it all to do over, I would pore over the units sooner rather than later.

Re: The Lost Art of Logarithms

#94

Post by xelxebar »

How timely! I just learned how to use a slide rule yesterday. Looking to pick one up, and a bit overwhelmed by the plethora of choices, I went down a small rabbit hole[0]. Some slide rules produced are pure works of art!

Lately, I've been rediscovering the surprising niceties that analog tools can provide over our everything-is-a-panel-of-glass interfaces these days. Recently, I have been enjoying pen and paper as my editor for initial drafts of projects I'm coding.

Does HN have love for any analog tools in particular?

[0]:https://sliderulemuseum.com/

Re: The Lost Art of Logarithms

#95

Post by pbronez »

smitty1e wrote: post #92
BobbyTables2 wrote: post #90

Earlier quoted context omitted.

Math is rarely taught with practical problems in mind — that’s engineering !

Therein lies the rub. Treating abstract and the concrete in isolation was always tough sledding for me. Bouncing between the two is where the action is. And units: if I had it all to do over, I would pore over the units sooner rather than later.

Absolutely. Units are such a useful idea.

I was recently struggling to model a financial process and solved it with Units. Once I started talking about colors of money as units, it became much easier to reason about which operations were valid.

Re: The Lost Art of Logarithms

#96

Post by johnm »

xelxebar wrote: post #94

How timely! I just learned how to use a slide rule yesterday. Looking to pick one up, and a bit overwhelmed by the plethora of choices, I went down a small rabbit hole[0]. Some slide rules produced are pure works of art! Lately, I've been rediscovering the surprising niceties that analog tools can provide over our everything-is-a-panel-of-glass interfaces these days. Recently, I have been enjoying pen and paper as my…

Indeed, I use pen/pencil and (dot) paper. Different brain space.

Re: The Lost Art of Logarithms

#97

Post by madars »

dkislyuk wrote: post #45

I found that looking at the original motivation of logarithms has been more elucidating than the way the topic is presented in grade-school. Thinking through the functional form that can solve the multiplication problem that Napier was facing (how to simplify multiplying large astronomical observations), f(ab) = f(a) + f(b), and why that leads to a unique family of functions, resonates a lot better with me for why lo…

Toeplitz wrote "Calculus: The Genetic Approach" and his approach of explaining math via its historical development is apparently more widely used: https://en.wikipedia.org/wiki/Genetic_method . Felix Klein remarked: "on a small scale, a learner naturally and always has to repeat the same developments that the sciences went through on a large scale"

Re: The Lost Art of Logarithms

#98

Post by fenomas »

BobbyTables2 wrote: post #90
agumonkey wrote: post #61

Earlier quoted context omitted.

I often wonder about this. I also believe that mathematical pedagogy strive to attract people that are very smart and think in the abstract like euler, and not operationally, meaning they will get it intuitively. For other people, you need to swim in the original problem for a while to see the light.

Math is rarely taught with practical problems in mind — that’s engineering !

Agreed, and it's such a shame! A kid goes to math class and learns, say, derivatives as this weird set of transformations that have to be memorized, and it's only later in in physics class that they start to see why the transformations are useful.

I mean, imagine a programming course where students spend the whole first year studying OpenGL, and then in the second year they learn that those APIs they've been memorizing can be used to draw pictures :D

Re: The Lost Art of Logarithms

#99

Post by saulpw »

JadeNB wrote: post #87
saulpw wrote: post #57

Earlier quoted context omitted.

I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). https://saul.pw/mag

> I think this should be front and center. To that end I propose "magnitude notation"[0] (and I don't think we should use the word logarithm, which sounds like advanced math and turns people away from the basic concept, which does make math easier and more fun). The only reason that "logarithm" sounds like advanced math is because it was so useful that mathematicians, well, used it. Since this terminology is just log…

It's written like ^6 and said like "mag 6", which sounds like an earthquake (and this is basically the Richter scale writ large). One syllable, sounds cool, easy to type/spell, evokes largeness. "Logarithm" is 3-4 syllables, hard to pronounce, hard to spell, sounds jargon-y.
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