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The Lost Art of Logarithms

lostartoflogarithms.com

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Re: The Lost Art of Logarithms

#181

Post by Enginerrrd »

vanderZwan wrote: post #179

Related: in a reaction to a comment I wrote about logarithms about a month ago[0], saulpw recently linked his own idea of making logarithms more "accessible" to the masses by introducing magnitude-based notation: https://saul.pw/mag/ I think it is a really nice idea that should be spread more widely. It might be Pi day, and while I traditionally complain that Tau is better for contrarian reasons (hey at least I'm hon…

For what it's worth, I'm really not a fan.

There's a reason we use scientific notation, and it's actually partly because in the era of slide-rules, it was INCREDIBLY helpful notation that makes it trivial to estimate things like order of magnitude. People performed all manner of operations and kept the magnitude part in their head. It MADE people more magnitude aware.

This magnitude-only based notation is the one that's actually more needlessly complex and error prone. There's no sensible way to manage significant figures or rounding error in simple operations like addition and subtraction. And simple operations, like adding/subtracting two numbers are really non-trivial. If you have to start each operation by converting to a useful format and then converting back, what have you gained exactly by using the notation?

Re: The Lost Art of Logarithms

#182

Post by anjakefala »

Enginerrrd wrote: post #181
vanderZwan wrote: post #179

Related: in a reaction to a comment I wrote about logarithms about a month ago[0], saulpw recently linked his own idea of making logarithms more "accessible" to the masses by introducing magnitude-based notation: https://saul.pw/mag/ I think it is a really nice idea that should be spread more widely. It might be Pi day, and while I traditionally complain that Tau is better for contrarian reasons (hey at least I'm hon…

For what it's worth, I'm really not a fan. There's a reason we use scientific notation, and it's actually partly because in the era of slide-rules, it was INCREDIBLY helpful notation that makes it trivial to estimate things like order of magnitude. People performed all manner of operations and kept the magnitude part in their head. It MADE people more magnitude aware. This magnitude-only based notation is the one tha…

I think the argument is that for non-scientific usecases, folks don't really need to think about error or significant digits. By focusing on the significand too much laypeople aren't grasping how large and small these numbers are relative to each other.

It's not being put forward as a recommended tool for scientists when reasoning about precise values. It's put forward for laypeople when trying to understand the vastness of the universe.

Re: The Lost Art of Logarithms

#183

Post by xiande04 »

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…

If you like this approach, I highly recommend Mathematics: It's Content, Methods, and Meaning by Kolmogorov. He uses this same approach, but applies it to many more concepts in math (about 1,000 pages!). In fact, I think I actually heard about that book on this site, so I guess I'm paying it forward.

This approach was to align with the Soviet philosophy of dialectical materialism, which claims that all things arise from a material need. Not sure I'm fully onboard with the philosophy as a whole, but Kolmogorov's book was really eye opening.

Re: The Lost Art of Logarithms

#184

Post by ninalanyon »

malshe wrote: post #74
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…

We used logarithms routinely for large multiplications, divisions, etc. in 11th and 12th grade. No calculators were allowed. This was in India.

Same here when I was at school in the late 1960s and early 1970s. No one had a calculator.

So we were taught logarithms as a tool first.

Re: The Lost Art of Logarithms

#185

Post by SoleilAbsolu »

spc476 wrote: post #138
b0afc375b5 wrote: post #121

Earlier quoted context omitted.

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…

There is Mathematics for the Million by Lancelot Hogben, which not only covers math, but the history of math and why it was developed over the centuries. It starts with numbers, then geometry, arithmetic, trig, algebra, logarithms and calculus, in that order. It's a very cool book.

I was going to say the same! I got it years ago, it's hard to top a math book with a quote from a certain Al Einstein on the back cover singing its praises! Morris Kline's "Mathematics for the Nonmathematician" takes a similar approach, as I believe other books by the author do. Can also recommend "Code" by Charles Petzold and "The Information" by James Gleick, while not comprehensive they do cover the development of key mathematical insights over time.

Re: The Lost Art of Logarithms

#186

Post by vanderZwan »

anjakefala wrote: post #182
Enginerrrd wrote: post #181

Earlier quoted context omitted.

For what it's worth, I'm really not a fan. There's a reason we use scientific notation, and it's actually partly because in the era of slide-rules, it was INCREDIBLY helpful notation that makes it trivial to estimate things like order of magnitude. People performed all manner of operations and kept the magnitude part in their head. It MADE people more magnitude aware. This magnitude-only based notation is the one tha…

I think the argument is that for non-scientific usecases, folks don't really need to think about error or significant digits. By focusing on the significand too much laypeople aren't grasping how large and small these numbers are relative to each other. It's not being put forward as a recommended tool for scientists when reasoning about precise values. It's put forward for laypeople when trying to understand the vast…

Yes, that was my take-away too. I suspect it's a wonderful way of teaching intuition for differences in scale, which is something that's a lot more important to us today than it was a few centuries ago. Easy mental guesstimates should not be underestimated as a valuable tool

Re: The Lost Art of Logarithms

#187

Post by secondcoming »

meindnoch wrote: post #162
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…

By the way, there's another function that can be used to turn multiplication into addition: f(x) = x^2 / 2 a * b = f(a + b) - (f(a) + f(b))

Isn’t x^2 a multiplication?

Re: The Lost Art of Logarithms

#188

Post by meindnoch »

secondcoming wrote: post #187
meindnoch wrote: post #162

Earlier quoted context omitted.

By the way, there's another function that can be used to turn multiplication into addition: f(x) = x^2 / 2 a * b = f(a + b) - (f(a) + f(b))

Isn’t x^2 a multiplication?

No, you misunderstood what I meant.

Normally, a sliderule at distance x has the value of log(x) written on it, which allows doing multiplications by moving along the sliderule, since log(ab) = log(a) + log(b).

Now imagine a sliderule onto which values of x^2/2 are written. This also allows you to multiply two numbers, because ab = (a+b)^2/2 - (a^2/2 + b^2/2).

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