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The Lost Art of Logarithms

lostartoflogarithms.com

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Re: The Lost Art of Logarithms

#151

Post by [deleted] »

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

[deleted]

Re: The Lost Art of Logarithms

#153

Post by kqr »

xelxebar wrote: post #140
kqr wrote: post #63

I can strongly recommend memorising some logarithms for use in mental maths. It's given me powers I did not expect to have! Here's what I wrote about it when I started: https://entropicthoughts.com/learning-some-logarithms

What reflections do you have putting this into action over the year since that post? BTW, your blog is one of my absolute favorites!

It's been about as useful as one would expect. I don't need it daily, but when I need it, I can usually estimate a good enough answer in the time it takes someone else to do it on a calculator.

It has also helped a little with getting a geometric appreciation for numbers, but I suspect that could be improved significantly with more active practice.

Re: The Lost Art of Logarithms

#154

Post by lkuty »

tmoertel wrote: post #64
pash wrote: post #44

Earlier quoted context omitted.

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…

For the particular case of the exponential distribution we can go further. By taking advantage of the theory of Poisson processes, we can take samples using a parallel algorithm. It even has a surprisingly succinct SQL translation: SELECT * FROM Population WHERE weight > 0 ORDER BY -LN(1.0 - RANDOM()) / weight LIMIT 100 -- Sample size. Notice our exponentially distributed random variable on prominent display in the O…

Quite off-topic, but do you know when you'll write the article about CPS, if ever?

Re: The Lost Art of Logarithms

#155

Post by cgadski »

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.

One way to understand why without writing down the CDF/PDF:

When X is an exponential variable and c is a constant, X + c has the same distribution as X after conditioning on large outcomes. In other words, these two variables have same "tail." This is true exactly for exponential distributions. (Sometimes this is called "memorylessness.")

Similarly, when U has a uniform distribution on [0, 1] and c is a constant, cU has the same distribution as U after conditioning on small outcomes.

But if cU is distributed like U near 0, then -ln(c U) is distributed like -ln(U) near infinity. But -ln(c U) = -ln(c) - ln(U), so the tail of -ln(U) doesn't change when we add a constant, meaning it must have an exponential distribution.

Re: The Lost Art of Logarithms

#156

Post by wolfi1 »

TrainedMonkey wrote: post #67
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…

All data is linear when plotted on a loglog scale with a thick marker.

the joy of an engineer is to find a straight line in a double logarithmic diagram

Re: The Lost Art of Logarithms

#157

Post by biofox »

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…

There are two books which do a fantastic job of this:

Mathematics: From the Birth of Numbers, by Jan Gullberg

and

Mathematics: A Cultural Approach, by Morris Klein

Re: The Lost Art of Logarithms

#158

Post by [deleted] »

bumbledraven wrote: post #128
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 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 .

[deleted]

Re: The Lost Art of Logarithms

#159

Post by genewitch »

kqr wrote: post #135
aquafox wrote: post #80

Earlier quoted context omitted.

But in my explanation, there is no x axis.

No but it holds more generally. Taking the log of data tends to make it look "more correct" even when it's not theoretically justified, and this can lead to very wrong conclusions.

Matt Parker says it's because that's how humans are naturally inclined to think, and used the midway point between 1 and 9 to illustrate. We'd say five but "children and others not exposed to math would say 3" and then gave some explanation with beads or coins. It didn't make sense to me but I do know that if a graph is log scale I need to actually look at it harder to make sure they're not trying to pull a fast one on us here folks.

Re: The Lost Art of Logarithms

#160

Post by mrcsd »

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 !

I really disagree with the straightforward reduction of engineering to 'math but practical', but I'm finding it hard to express exactly why I feel this way.

The history of mathmatical advancement is full of very grounded and practical motivations, and I don't believe that math can be separated from these motivations. That is because math itself is "just" a language for precise description, and it is made and used exactly to fit our descriptive needs.

Yes, there is the study of math for its own sake, seemingly detached from some practical concern. But even then, the relationships that comprise this study are still those that came about because we needed to describe something practical.

So I suppose my feeling is that, teaching math without a use case is like teaching english by only teaching sentence construction rules. It's not that there's nothing to glean from that, but it is very divorced from its real use.

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