Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Lost Art of Logarithms

lostartoflogarithms.com

Post reply on HN
Print view View original

81–90 of 204 posts

Re: The Lost Art of Logarithms

#81

Post by H8crilA »

aquafox wrote: post #66

Interesting insight why applying a log transform often makes data normally distributed: Pretty much all laws of nature are multiplications (F=m a, P V=n R T, etc). If you start with i.i.d random variables and multiply them, you get log-normal data by virtue of the central limit theorem (because multiplications are additions on a log scale; and the CLT is also somewhat robust to non iid-ness). Thinking of data as the…

The CLT does not require iid (independent, identically distributed) variables. Just independent and having a variance, plus some rather weak condition on slightly higher orders. Otherwise the variables can be quite different from each other.

Re: The Lost Art of Logarithms

#82

Post by analog31 »

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…

This would be an interesting thing to study: How many different ways people learned about logarithms, and how they generally fared in math. I learned about logarithms by seeing my dad use his slide rule, and studying stock charts, which tended to be semi-logarithmic.

Re: The Lost Art of Logarithms

#83

Post by crazygringo »

hughw wrote: post #78
crazygringo wrote: post #73

Earlier quoted context omitted.

But the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric, no? Your idea arbitrarily selects numbers around 2 billion as being conceivable, at least for a single core on my MacBook. But my question isn't what makes 10^4000 inconceivable -- my question is what makes 10^4000 any less conceivable than 1000. To me, they're both firmly in the realm of abstracti…

the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric But I'd just call it "instantaneous" and something I experience frequently. Whereas 10^3978 years is beyond experience and even imagination.

> But I'd just call it "instantaneous" and something I experience frequently.

Then since you're OK with all smaller numbers being equally "instantaneous", then 1/10^4000 seconds is instantaneous too. Add up enough of those to make a second, and you can conceive of your previously inconceivable number! :)

Of course, that will seem silly. I'm just illustrating I don't think there are any grounds for claiming exponentially large numbers are inconceivable, but exponentially small numbers are somehow conceivable. They're just as far from 1, multiplicatively, no matter which direction you go in.

Re: The Lost Art of Logarithms

#84

Post by dfawcus »

zwnow wrote: post #76
dfawcus wrote: post #71

Earlier quoted context omitted.

Not long. We were taught logs, and use of log tables, at Middle School. So probably around about age 11. I also vaguely recall a couple of lessons where we went over Napier's Bones, and they had us produce equivalents on pieces of paper to cut out and move around. I believe I still have my school day log tables around somewhere. I'd just have to practice for 1/2 hr to remind myself how to use them. That said, they di…

Look im 30, most people I know have forgotten all of school math long ago, me included. Entry barrier too big now.

Having scanned through his book, it makes it seem overly complex.

At middle school, this was taught after having only done simply arithmetic and learning about fractions (rational numbers) in primary school, then decimal fractions in middle school.

The use of logs from the tables was simply a set of basic rules for how to apply them to a few scenarios. I can't recall if it covered trig with those tables, but I doubt it.

I learnt as a child between 10 (when I started middle school), and say around 12 at most. I've forgotten the use, but vaguely recall some way of representing negative numbers in the table (n-bar, with a bar above the digit).

I'm way over 30. I never used log tables after high school, and have forgotten the rules for usage, but recall it didn't take long to learn the first time. *

However for simple uses (multiplication and division) I'd expect I'd be able to pick it up again in at most a weeks worth of practice. It would be made a lot easier now by being able to compare and check calculations with a computer or pocket calculator.

I'd expect any adult able to program a computer to also be able to pick it up in a similar period, or at most a month.

Remember we used to teach this to kids, and expect them to be able to pick it up (if not be accurate in application) in under a weeks worth of lessons.

* Note I didn't even know how to do long multiplication when I learnt, as due to political interference with teaching curriculum, I'd not been taught at primary school.

Re: The Lost Art of Logarithms

#85

Post by mixmastamyk »

zwnow wrote: post #76
dfawcus wrote: post #71

Earlier quoted context omitted.

Not long. We were taught logs, and use of log tables, at Middle School. So probably around about age 11. I also vaguely recall a couple of lessons where we went over Napier's Bones, and they had us produce equivalents on pieces of paper to cut out and move around. I believe I still have my school day log tables around somewhere. I'd just have to practice for 1/2 hr to remind myself how to use them. That said, they di…

Look im 30, most people I know have forgotten all of school math long ago, me included. Entry barrier too big now.

Khan academy could get you through it, in I'd guess a month or three, depending on your time available.

Re: The Lost Art of Logarithms

#86

Post by hughw »

crazygringo wrote: post #83
hughw wrote: post #78

Earlier quoted context omitted.

the length of time it takes a modern computer to count to 10 or 1,000 is perhaps inconceivably small by your metric But I'd just call it "instantaneous" and something I experience frequently. Whereas 10^3978 years is beyond experience and even imagination.

> But I'd just call it "instantaneous" and something I experience frequently. Then since you're OK with all smaller numbers being equally "instantaneous", then 1/10^4000 seconds is instantaneous too. Add up enough of those to make a second, and you can conceive of your previously inconceivable number! :) Of course, that will seem silly. I'm just illustrating I don't think there are any grounds for claiming exponentia…

All right! I cannot count to any fractional number, even conceivable ones, so using the counting time to assess conceivability fails for all fractions.

For small numbers, groping for any experience in the macro world fails once you get below the Planck scale.

I'm afraid you might have broken the fabric of the universe by even typing such a small number.

Re: The Lost Art of Logarithms

#87

Post by JadeNB »

saulpw wrote: post #57
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…

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 logarithms without saying the word, if it is more useful it, too, will probably be used by mathematicians, and then it will similarly come to sound like advanced math. So what's the point of running away from a name for what we're doing that fits with what it's actually called, if eventually we'll just have to make up a new, even less threatening name for it?

(I'd argue that "logarithm" is frightening less because it sounds like advanced math than because it's an unfamiliar and old-fashioned-sounding word. I'm not completely sure that "magnitude" avoids both these issues, but it's at least arguable that it suffers less from them.)

Re: The Lost Art of Logarithms

#89

Post by kccqzy »

dkislyuk wrote: post #60
cauliflower2718 wrote: post #51

Earlier quoted context omitted.

This follows directly from the fact that exp(x+y)=exp(x)exp(y).

Yes, but such a property was not available to Napier, and from a teaching perspective, it requires understanding exponentials and their characterizations first. Starting from the original problem of how to simplify large multiplications seems like a more grounded way to introduce the concept.

From a teaching perspective it goes like this: first we learn additions, and to undo additions we have subtractions; then we learn repeated additions i.e. multiplications, and to undo multiplications we have divisions; finally we learn repeated multiplications, i.e. exponentiation, and to undo exponentiation we have logarithms and roots.

Re: The Lost Art of Logarithms

#90

Post by BobbyTables2 »

agumonkey wrote: post #61
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…

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 !
Post reply on HN