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The Lost Art of Logarithms

lostartoflogarithms.com

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Re: The Lost Art of Logarithms

#121

Post by b0afc375b5 »

madars wrote: post #97
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"

I always longed for a book/course on mathematics where topics are in chronological order:

1. ... (mathematical topics at the beginning of history of which I am ignorant of)

2. pythagoras theorem

3. ...

4. euclid geometry

5. ...

6. algebra

7. ...

8. calculus

9. ...

10. set theory

11. ...

12. number theory

13. etc. etc. (you get the point)

Maybe there's already something that lays out topics like this. I haven't searched too hard.

Re: The Lost Art of Logarithms

#122

Post by JackFr »

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 is how I learned them in middle school — just common logs, as an aid to doing roots, powers and multiplications of big numbers.

We were told in an off-hand way that logs could be to any base, even ‘e’, but not to worry about that for a few years.

Re: The Lost Art of Logarithms

#123

Post by avmich »

hughw wrote: post #56

I feel frustrated that we cannot conceive of numbers like 10^80 (atoms in the universe) or 10^4000 (number configurations for a system with 4000 variables having 10 states each). Maybe there are superbrains out there in the universe that can do so.

Read http://mrob.com/pub/math/largenum.html . After that you'll probably look easier to those numbers.

Re: The Lost Art of Logarithms

#124

Post by JackFr »

A 300 year old log table! What an opportunity to confirm Benford’s Law!

https://en.m.wikipedia.org/wiki/Benford's_law

“The discovery of Benford's law goes back to 1881, when the Canadian-American astronomer Simon Newcomb noticed that in logarithm tables the earlier pages (that started with 1) were much more worn than the other pages.”

Re: The Lost Art of Logarithms

#125

Post by mikewarot »

The traditional explanations of logarithms I've encountered are far too math and terminology heavy for most people to grasp.

Think of a number line.... show example..... 1..2..3..4..5.. etc

Any given move to the right, makes the value go up by 1.

But... What if we did a special number line where each time it doubled instead of adding one? 1..2..4..8..16, etc...

That line would go up way to fast to see numbers like 10, so we can expand it out a bit...show that... and start to fill in the numbers... 2^10 (1024) is almost 1000... so you can divide that distance by 3 to get 10 on the line, then move one unit left for 5... and build out a slide ruler.

Computing logarithms with a 4 function calculator isn't hard by the way, I used to do it for fun on my lunch breaks.

Re: The Lost Art of Logarithms

#126

Post by divbzero »

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…

Where can I get the meter-long slide rule the man is holding in OP?

Re: The Lost Art of Logarithms

#127

Post by DrFalkyn »

fenomas wrote: post #98
BobbyTables2 wrote: post #90

Earlier quoted context omitted.

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 memorizi…

Rules for derivatives require the least memorization

Re: The Lost Art of Logarithms

#128

Post by bumbledraven »

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 have been writing the same thing by (ab)using the existing unit of measurement known as a bel (B), which is most commonly seen with the SI prefix “deci” (d) as dB or decibel. I write the speed of light as 8.5 Bm/s (“8.5 bel meters per second”), which resembles the expression 20 dBV (“20 decibel volts”) shown at https://en.wikipedia.org/wiki/Decibel.

Re: The Lost Art of Logarithms

#129

Post by twelvechairs »

b0afc375b5 wrote: post #121
madars wrote: post #97

Earlier quoted context omitted.

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"

I always longed for a book/course on mathematics where topics are in chronological order: 1. ... (mathematical topics at the beginning of history of which I am ignorant of) 2. pythagoras theorem 3. ... 4. euclid geometry 5. ... 6. algebra 7. ... 8. calculus 9. ... 10. set theory 11. ... 12. number theory 13. etc. etc. (you get the point) Maybe there's already something that lays out topics like this. I haven't search…

I'm sympathetic but there's no clear historic chronology. For instance the ancient egyptians dealt with both algebra and calculus (at least in part) long before Pythagoras. And thats not starting on China and India which had very different chronologies.

Re: The Lost Art of Logarithms

#130

Post by kqr »

pbronez wrote: post #95
smitty1e wrote: post #92

Earlier quoted context omitted.

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.

Strictly speaking this is about dimensional analysis, not units. (When discussing curricula we should be precise!)
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