[HN Gopher] The Lost Art of Logarithms
___________________________________________________________________
The Lost Art of Logarithms
Author : ozanonay
Score : 525 points
Date : 2025-03-13 19:05 UTC (1 days ago)
(HTM) web link (www.lostartoflogarithms.com)
(TXT) w3m dump (www.lostartoflogarithms.com)
| tombert wrote:
| I started using LMAX Disruptor for some projects. One quirk with
| Disruptor is that the queue size always has to be an exponent of
| two.
|
| I wanted to make sure that I always have at least enough room for
| any size and I didn't want to manually compute, so I wrote this:
| var actualSize = Double.valueOf(Math.pow(2,
| Math.ceil(Math.log(approxSize) / Math.log(2)))).intValue();
|
| A bit much for a single line, but just using some basic log rules
| in order to the correct exponent. I learned all this in high
| school, but some of my coworkers thought I was using this
| amazing, arcane bit of math that had never been seen before. I
| guess they never use log outside of Big-O notation.
| mananaysiempre wrote:
| This is perfectly usable, of course, but I'd write
| var actualSize = Integer.highestOneBit(approxSize - 1) << 1;
|
| purely to avoid involving the horrors that live beneath the
| humble pow() and log().
|
| (Integer.highestOneBit, also known as "isolate leftmost bit",
| "most significant one", or the like, essentially has to be a
| primitive to be efficient, unlike its counterpart for the
| lowest bit, x&-x. The actual CPU instruction is usually closer
| to Integer.numberOfLeadingZeros, but that's just a bitshift
| away.)
| tombert wrote:
| That's pretty cool; I didn't even consider doing any cool
| bitwise arithmetic.
|
| I didn't particularly care about performance or anything for
| this particular case, since it runs exactly once at the start
| of the app just to initiate the Disruptor.
| markrages wrote:
| shouldn't it be
|
| var actualSize = 1 << Integer.highestOneBit(approxSize - 1);
|
| ?
| mananaysiempre wrote:
| Nope. I don't know why the Java folks decided not to use
| the fairly standard verb "isolate" for this method, but
| that's what it is[1]:
|
| > public static int highestOneBit(int i)
|
| > Returns an int value with at most a single one-bit, in
| the position of the highest-order ("leftmost") one-bit in
| the specified int value. Returns zero if the specified
| value has no one-bits in its two's complement binary
| representation, that is, if it is equal to zero.
|
| There isn't a straight floor(log2(*)) as far as I can tell,
| only Integer.numberOfLeadingZeros, and turning the former
| into the latter is annoying enough[2] that I wouldn't
| prefer it here.
|
| [1] https://docs.oracle.com/javase/8/docs/api/java/lang/Int
| eger....
|
| [2] https://docs.oracle.com/javase/8/docs/api/java/lang/Int
| eger....
| markrages wrote:
| Thanks. What a weird API.
| layer8 wrote:
| The above gives an incorrect result for _approxSize_ = 1
| (namely 0). The following works (for values up to 2^30, of
| course): var actualSize = Integer.MIN_VALUE
| >>> Integer.numberOfLeadingZeros(approxSize - 1) - 1;
|
| Or, if you want 0 to map to 0 instead of to 1:
| var actualSize = Integer.signum(approxSize) *
| Integer.MIN_VALUE >>> Integer.numberOfLeadingZeros(approxSize
| - 1) - 1;
|
| Of course, you could also use a variation of:
| var actualSize = Math.min(1,
| Math.Integer.highestOneBit(approxSize - 1) << 1);
| ajsnigrutin wrote:
| Just shift the size to the right 1 bit and count the shifts
| until the value turns to zero, and you'll get your number 2
| exponent for the size :)
| kenjackson wrote:
| Notation for writing log has always bugged me. Like I feel like
| it should be more like <10>^<527> which would be the log base 10
| of 527. That's not it, but something. The current notation just
| doesn't feel quite right.
| Jtsummers wrote:
| https://mathcenter.oxford.emory.edu/site/math108/logs/
|
| Some people have suggested the "triangle of power".
| awesome_dude wrote:
| The Triangle of power explanation of logarithms is what
| really got me across logs.
|
| It wasn't until seeing the triangle and having the
| relationships explained that I had any clue about logarithms,
| up until then logs had been some archaic number that meant
| nothing to me.
|
| Because of the triangle of power, I now rock up to B and B+
| Trees and calculate the number of disc accesses each will
| require in the worst case, depending on the number of values
| in each block (eg, log2(n), log50(n) and log100(n))
| cafeinux wrote:
| Ironically, that notation, which I just discovered,
| confuses me more than anything else. Logs clicked for me
| when someone online said "amongst all the definitions we
| have for logs, the most useful and less taught is that
| log() is just a power". At that exact instant, it's like if
| years of arcane and foreign language just disappeared in
| front of my eyes to leave only obviousness and poetry.
| awesome_dude wrote:
| That is not without humour :)
|
| I don't understand the comment about it just being a
| power, but, for me, knowing that it's filling in the
| third vertice on the triangle with exponents at the top,
| and n on the other is what makes it work for me - I now
| know in my head when I am looking for the log of n, I am
| looking for the exponent that would turn the log into n.
|
| I don't go looking for the exact log, I only look for
| whole numbers when I am calculating the value in my mind.
|
| But it makes sense when I am looking for the log2 of 8 to
| know that the answer is "what exponent will make 2 into
| 8"? and that's "3"
| cafeinux wrote:
| I said "power" as in "exponent", so we basically have the
| same understanding, I just do without the triangle.
| the__alchemist wrote:
| Yea; this owns. I used it for my own stuff.
| nh23423fefe wrote:
| I never understand why anyone thinks its good notation. The
| layout means nothing and lets you infer nothing because
| exponentiation isn't a 2d planar geometric operation. All the
| information and rules are contained in the idea "exponentials
| map the additive reals to the multiplicative reals".
|
| The notation conveys no information at all, and provides no
| means of proving anything, and even the notation for function
| composition is worse.
|
| Given the operator pow : R^2->R there are 2 possible
| inverses. root and log
|
| root isn't even interesting. its just pow with the 2nd
| argument precomposed with reciprocal. root x b = pow x 1/b
| majkinetor wrote:
| https://www.youtube.com/watch?v=sULa9Lc4pck
| empath75 wrote:
| I think it's good for explanation purposes, but not actually
| great as notation, especially for operations that are so
| common, too much room for ambiguity, especially when writing
| quickly.
| kenjackson wrote:
| I've never seen this before, and I LOVE it! I'm a new
| advocate!
| thaumasiotes wrote:
| Well, currently exponentiation has a superscript exponent to
| the right of the base, and logs have a subscript base to the
| left of the exponent. They're already very similar to your
| example, but they also include the word "log".
| Animats wrote:
| Is this the same author who wrote Win32 API books?
| Jtsummers wrote:
| Yes. https://www.lostartoflogarithms.com/author/
| fph wrote:
| https://www.charlespetzold.com/PetzoldTattoo.jpg
| ohgr wrote:
| Yes the office door stop as it was known as at our place. Top
| book though just faded in utility and no one had the heart to
| dispose of it because of the good memories.
| Dwedit wrote:
| Good book indeed, just that I wouldn't use a book to look up
| Win32 API functions.
| ohgr wrote:
| It used to be the only way!
| werdnapk wrote:
| Oh, the hours I'd spend browsing through tech books back in
| the day... good times.
| hughw wrote:
| Yeah, a good tech bookstore was a golden way to spend an
| hour.
| hyperopt wrote:
| Charles Petzold wrote one of my favorite books - "Code: The
| Hidden Language of Computer Hardware and Software". Very excited
| to see how this turns out and thanks for giving some of this
| knowledge away for free!
| tocs3 wrote:
| I would also recommend the "NAND to Tetris" book. Covers much
| the same ground (as I remember things anyway) but is a hands on
| approach. I enjoyed Code also though and is worth a look for
| those interested.
| tmoertel wrote:
| 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_ )/ _l_ has an exponential
| distribution with rate _l_.
|
| This relationship comes in handy when, for example, you want to
| draw weighted random samples. Or generating event times for
| simulations.
| zwnow wrote:
| How long do I have to study math to understand this?
| whereismyacc wrote:
| to understand what they said, or to understand a proof of why
| it would be true?
|
| any stats class would be enough to understand what they said
| pc86 wrote:
| I know what all these words mean, it "makes sense" to me in
| the sense that I read it and I think "ok.." but I wouldn't
| have the slightest idea how to use this to get weighted
| random samples or "generate event times."
|
| So I guess I "understand it" in the sense that it doesn't
| sound like a foreign language, but I can't apply it in any
| meaningful way.
| pvg wrote:
| Depends where you're starting from but from highschoolish
| maths you can probably sort this out in a few hours or days.
| jrussino wrote:
| I love that you asked this, but I think it's not _quite_ the
| right question.
|
| I've been wishing for years that someone would maintain a
| "dependency graph" for mathematical concepts. I think Khan
| Academy tried to do something like this at one point but took
| it down a long time ago. As long as we're far enough from the
| bleeding-edge of research topics, I feel like maths is the
| one field where this might be possible to do really well. You
| should be able to point at something you don't understand and
| trace backwards in a very granular fashion until you hit a
| concept you already know/understand, and then learn what you
| need to fill in the gap!
| pvg wrote:
| That's more-or-less the purpose of
| https://mathworld.wolfram.com/ or at least, it's
| significantly better at it than, say, wikipedia.
| elevation wrote:
| I had wanted to build a website like this as a hobby,
| except not limited strictly to mathematical concepts. I
| wanted to build a graph of the skills/concepts you'd need
| to understand electrical impedance, fix an air conditioning
| system, bake bread reliably.
|
| One source of information could be course syllabi, which
| describe a progression of topics and their prerequisites.
|
| Rather than be the authority on what concepts must precede
| others, I envisioned making a system that could represent
| different educational approaches: not every educator agrees
| that Calculus I ought to be a prerequisite for studying
| Physics I. Not all bread recipes use yeast.
|
| I had a hard time finding a good domain for this effort.
| "Tree of knowledge" dot TLD was taken.
| bgnn wrote:
| Depends on how much you practiced high school math. It's not
| hard but we forget it without practice.
| carabiner wrote:
| Ask chatgpt to explain it to you like you're 18. It made it
| really easy to understand.
| bobbylarrybobby wrote:
| Unless I'm missing something, this can just be directly
| verified, no "understanding" necessary. All you need to know
| is that probability distributions can be characterized by
| their probability density function (PDF).
|
| If Y=-ln(X)/lambda, then P(Y<a) = P(-ln(X)/lambda<a) =
| P(X>exp(-lambda a)) = 1-exp(-lambda a).
|
| And if Z is exponential with rate parameter lambda, then
| P(Z<a) = P(lambda exp(-lambda t)<a) = integral from 0 to a of
| lambda exp(-lambda t)dt, which can be directly computed to be
| 1-exp(-lambda a).
|
| They have the same PDF, so they're the same distribution.
| timeinput wrote:
| I mean if starting from scratch that seems like many years
| in most western education systems to get to probability,
| logarithms, exponentiation.
|
| I would say If you knew 2+2=4, and not much else you're
| years away from 'understanding', if you know ln(exp(y)) =
| y, and P(x>0.5) = 0.5 for a uniform distribution on [0, 1)
| then you don't need any additional understanding.
|
| I would bet the GP comment is somewhere inbetween the two
| extremes, but I think a random sampling of the population
| would likely result in people generally not knowing the log
| / exponentiation relation, or anything about the uniform
| distribution.
| zwnow wrote:
| Yea got many answers and I dont understand a single one.
| Good thing you barely need math in programming.
| tmoertel wrote:
| Most textbooks on probability have some discussion of the
| relationships between the various distributions commonly in
| use. If you just want a quick overview, I found John D.
| Cook's diagram to be handy:
|
| https://www.johndcook.com/blog/distribution_chart/
| dynm wrote:
| Possibly unhelpful answer: Arguably none! This is presented
| as a surprising fact, but you could easily argue that this is
| the proper _definition_ of an exponential distribution. If
| you do
|
| x = -log(rand())/lambda
|
| a bunch a times, that comes from _something_ , right? Well,
| let's call that something an exponential distribution.
|
| From this perspective, the thing that actually needs math is
| finding the density function of the exponential distribution.
| (For that, in theory you just need calc 101 and probability
| 101.)
| coliveira wrote:
| If you study calculus and introduction to probability theory,
| then you're ready to learn this. So the answer is about 2
| years after high school.
| dfawcus wrote:
| 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 did have usage
| instructions in the last few pages.
| zwnow wrote:
| Look im 30, most people I know have forgotten all of school
| math long ago, me included. Entry barrier too big now.
| dfawcus wrote:
| 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.
| mixmastamyk wrote:
| Khan academy could get you through it, in I'd guess a
| month or three, depending on your time available.
| wanderingmind wrote:
| All you need is plot -log(x) for x between 0 and 1 and you
| will see that log(x) transforms a uniform line into an
| exponential decay (towards 1). Its being said in a fancy way.
| This is also the origin of the log likelihood loss function
| in ML
| analog31 wrote:
| It seems like there's always more than one way to skin a cat,
| but I'd have turned to calculus, had I needed to derive
| something like this and didn't think to look it up (e.g.,
| before the Internet).
| pash wrote:
| 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 a standard uniform distribution and then using
| the inverse transform is the commonest way of generating random
| numbers from an arbitrary distribution.
|
| 0. https://en.m.wikipedia.org/wiki/Inverse_transform_sampling
|
| 1.
| https://en.m.wikipedia.org/wiki/Probability_integral_transfo...
|
| 2. Not every cdf is one-to-one, however, so you may need a
| generalized inverse.
| tmoertel wrote:
| 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 ORDER BY clause.
|
| If you're curious, I explore this algorithm and the theory
| behind it in
| https://blog.moertel.com/posts/2024-08-23-sampling-with-
| sql....
| lkuty wrote:
| Quite off-topic, but do you know when you'll write the
| article about CPS, if ever?
| tmoertel wrote:
| Oops. I had quite forgotten that I need to write about
| that. I said I would over a decade ago, so that's a long
| time for you to wait. Sorry about that.
|
| I mainly write for myself, so I need the time and the
| motivation. Until recently, my job at G took up my time
| and also provided an internal community where I could
| scratch the writing itch, which reduced the motivation
| for public writing on my blog. But now that I'm semi-
| retired, I'll try to write more frequently.
|
| Thanks for the accountability!
| evanb wrote:
| 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
| cgadski wrote:
| 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.
| inasio wrote:
| There used to be practical value to be able to do some basic back
| of the envelope log calculations in your head (no calculators,
| this was how you did fast multiplications/divisions or
| exponents). There's a story in Feyman's Surely you're joking book
| about Los Alamos scientists doing speed competitions for mental
| log calculations
| NoMoreNicksLeft wrote:
| If the author is in here, thank you. Been looking for a text for
| my daughter on the subject. This might just fit the bill. If
| you're just the linker, then thank you Ozanonay.
| inasio wrote:
| (I'm sure this is in the book) John Napier, the father of
| logarithms (the N in ln), basically had a sweatshop of human
| calculators making log tables over something like 20 years -
| critical for celestial navigation. There was a huge price
| attached to the person that developed a method to safely navigate
| across the oceans, also lead to the invention of the pocket watch
| dekhn wrote:
| isn't the n in ln "natural" ("logarithm natural")?
| floydnoel wrote:
| yes i was taught that "ln" stood for "natural log"
|
| would be interested to hear other definitions!
| dekhn wrote:
| I learned the multiplication using addition and a lookup table in
| a class taught by Huffman (of Huffman compression fame). You
| weren't allowed to use a calculator on the test.
|
| But my absolute favorite trick is base conversions,
| https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:...
| with some practice you can do approximate base conversions (power
| to 2 to power of 10 or e) in your head
| dkislyuk wrote:
| 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 logarithms show up
| everywhere. This is in contrast to teaching them as the inverse
| of the exponential function, which was not how the concept was
| discussed until Euler. In fact, I think learning about
| mathematics in this way is more fun -- what original problem was
| the author trying to solve, and what tools were available to them
| at the time?
| cauliflower2718 wrote:
| This follows directly from the fact that exp(x+y)=exp(x)exp(y).
| dkislyuk wrote:
| 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.
| kccqzy wrote:
| 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.
| BobaFloutist wrote:
| You see how one of those isn't like the others?
| kccqzy wrote:
| You mean we have both logarithms and roots to undo
| exponentiation? That's because exponentiation is non-
| commutative.
| saulpw wrote:
| 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
| JadeNB wrote:
| > 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.)
| saulpw wrote:
| 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.
| bumbledraven wrote:
| 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.
| kzrdude wrote:
| If logarithm sounds too advanced, just say log and logs. I
| think it could work!
| saulpw wrote:
| Mag is the inverse of log10. e.g. log10 ^6 = 6. We have no
| current shorthand for inverse log10 except "tentothe" which
| might be serviceable but is not as punchy.
| agumonkey wrote:
| 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.
| BobbyTables2 wrote:
| Math is rarely taught with practical problems in mind --
| that's engineering !
| smitty1e wrote:
| 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.
| pbronez wrote:
| 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.
| kqr wrote:
| Strictly speaking this is about dimensional analysis, not
| units. (When discussing curricula we should be precise!)
| fenomas wrote:
| 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
| DrFalkyn wrote:
| Rules for derivatives require the least memorization
| meindnoch wrote:
| I've never seen an introductory math textbook that didn't
| point out how position, velocity and acceleration are
| related by the derivative.
| mrcsd wrote:
| 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.
| agumonkey wrote:
| Well, logarithms were made from physical entities
| (celestial bodies) but not on engineering per se.
|
| I think this is already enough context to root the mental
| effort deeper.
| II2II wrote:
| I think it is a combination of factors. Mathematical pedagogy
| is legitimate if the end goal is to train mathematicians, so
| yes it is geared towards those who think in the abstract.
| (I'm going to ignore the comment about very smart, since I
| don't think mathematical ability should be used as a proxy
| for intelligence.)
|
| On the other side, I don't think those who are involved in
| curriculum development are very skilled in the applications
| of mathematics. I am often reminded of an old FoxTrot comic
| where Jason calculated the area of a farmer's field using
| calculus.
| melagonster wrote:
| Mathematicians also hate the current version of math
| education.
| BobaFloutist wrote:
| Frankly I wish I had known integral calculus going into
| geometry, I could tell there was a pattern behind formulas
| for areas and volumes but I couldn't for the life of me
| figure it out. There are worse ways to remember the formula
| for the volume of a sphere than banging out a quick
| integral!
| ForOldHack wrote:
| I had known it. Thanks Dr Steven Giavat. The geometric
| shapes gave the patterns meaning. I read 'mathematics and
| the imagination' and mathematics a human endever' while I
| was starting algebra. Also the time-life book on math.
| All very brilliant because they used the methods that
| were used to investigate it, to show how it was
| discovered. These allowed me to fly ahead in math until I
| got to trig. Which took a long year to get facile, until
| I was able to finish my degree.
|
| I had brilliant teachers.
|
| Napier's bones, were for adding exponents, hense
| multiplication. Brilliant and nessary for the development
| of the slide rule, and the foundation of modern
| engineering, until the pocket calculator.
| malshe wrote:
| We used logarithms routinely for large multiplications,
| divisions, etc. in 11th and 12th grade. No calculators were
| allowed. This was in India.
| ninalanyon wrote:
| 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.
| meta_ai_x wrote:
| I actually prefer the straightforward log is an inverse of
| exponents. It's more intuitive that way because I automatically
| can understand 10^2 * 10^3 = 10^5. Hence if you are using log
| tables, addition makes sense. I didn't need an essay to explain
| that.
|
| Take logs, add 2 + 3 = 5 and then raise it back to get 10^5.
| kqr wrote:
| This is how I've always taught logarithms to students I've
| tutored. I photocopy a table of various powers of ten, we use
| it in all sorts of ways to solve problems, and then I
| sneakily present an "inverse power" problem where they need
| to make the lookup backwards.
|
| Almost every student gets it right away, and then I tell them
| looking up things backwards in the power table is called
| taking a logarithm.
| Dunan wrote:
| That's how I mentally processed them when first learning them
| years ago. Doing operations on x and y with log(x) = y in the
| background somehow felt far less intuitive than thinking
| about 10^y = x.
|
| I really enjoyed this author's work, BTW. Just spent several
| hours reading the entire first five chapters or so. What an
| excellent refresher for high school math in general.
| analog31 wrote:
| 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.
| madars wrote:
| 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"
| b0afc375b5 wrote:
| 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.
| twelvechairs wrote:
| 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.
| PaulRobinson wrote:
| Choose a chronology that makes sense. We can see how
| Western ideas build, we have less clarity on how the
| ancient Egyptians or Chinese ideas developed, and
| therefore it's harder to explain to a learner.
|
| If you're sensitive to that singular world view warping
| the learner's prospect, you could at each point explain
| similar ideas from other cultures that pre-date that
| chronology.
|
| For example, once you've introduced calculus and helped a
| student understand it, you can then jump back and point
| out that ancient Egyptians seemed to have a take on it,
| explain it, ask the student to reason did they get there
| in the same way as the Western school of ideas did, is
| there an interesting insight to that way of thinking
| about the World?
|
| Another ideas is how ideas evolved. We know Newton and
| Leibniz couldn't have had access to direct Egyptian
| sources (hieroglyphs were a lost language in their life
| times), but Greek ideas would have been rolling around in
| their heads.
| spc476 wrote:
| 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.
| SoleilAbsolu wrote:
| 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.
| zwnow wrote:
| A book without expecting any knowledge of mathematical
| notation would be a good start. I've bought 3 math books to
| get into it and quit all of them within the first chapter.
| -__---____-ZXyw wrote:
| In a roundabout way, I wonder does this one fit what
| you're after:
|
| https://bogart.openmathbooks.org/ctgd/ctgd.html
|
| And more directly, a quick browse showed up a book
| called:
|
| "Mathematical Notation: A Guide for Engineers and
| Scientists" which looks like it addresses your issue
| directly.
| zwnow wrote:
| The issue is that I dont want to explicily learn all of
| the notation but step by step, topic related with
| usecases in the real world...
| biofox wrote:
| 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
| markstock wrote:
| Here's one that starts with the concept of a straight line
| and builds all the way to string theory. It's a monumental
| book, and it still challenges me. Roger Penrose's The Road
| To Reality.
| vonneumannstan wrote:
| This one was just discussed on HN yesterday with pretty
| good reviews: https://www.amazon.com/Math-Through-Ages-
| Teachers-Mathematic...
| twbarr wrote:
| We could really take a page from this style for teaching
| advanced computing. We try to imagine that architectures just
| kind of come out of nowhere. Starting with mechanical
| computing and unit record equipment makes so much make more
| sense.
|
| Plus, unit record equipment was cool.
| blame-troi wrote:
| Very cool. But so many of us paid not enough attention to
| the details. Only two of the people in my first shop
| attempted channel programming.
| codesuki wrote:
| I recently read mathematics for the nonmathematician. https:/
| /www.goodreads.com/book/show/281821.Mathematics_for_t...
|
| Although the math in the book is relatively basic I enjoyed
| it tremendously because it gives the historical development
| for everything and even describes the characters of different
| mathematicians, etc. The historical context helps so much
| with understanding.
| nkoren wrote:
| It's ontogeny recapitulating phylogeny, all the way down.
| coffeemug wrote:
| Where did you pick this up? Is there a book that covers it that
| way?
| II2II wrote:
| In my case, it was by chance.
|
| I had a slide rule in high school. It was more of a novelty
| item by that point in time, only one of my math teachers even
| knew what a slide rule was, but that didn't stop me from
| figuring out how it was used and how it works. It didn't take
| much to figure out that the sliding action was solving
| problems by addition, and the funky scales were logarithmic.
| In other words: it performed multiplication by adding logs.
|
| That said, I did encounter references to its original
| applications in other places. I studied astronomy and had an
| interest in the history of computation.
| dkislyuk wrote:
| Presumably the book from this thread by Charles Petzold will
| be a great canonical resource, but originally there was a
| quote by Howard Eves that I came across that got me curious:
|
| > One of the anomalies in the history of mathematics is the
| fact that logarithms were discovered before exponents were in
| use.
|
| One can treat the discovery of logarithms as the search for a
| computation tool to turn multiplication (which was difficult
| in the 17th century) into addition. There were previous
| approaches for simplifying multiplication dating back to
| antiquity (quarter square multiplication, prosthaphaeresis),
| and A Brief History of Logarithms by R. C. Pierce covers
| this, where it's framed as establishing correspondences
| between geometric and and arithmetic sequences. Playing
| around with functions that could possibly fit the functional
| equation f(ab) = f(a) + f(b) is a good, if manual, way to
| convince oneself that such functions do exist and that this
| is the defining characteristic of the logarithm (and not just
| a convenient property). For example, log probability is
| central to information theory and thus many ML topics, and
| the fundamental reason is because Claude Shannon wanted a
| transformation on top of probability (self-information) that
| would turn the probability of multiple events into an
| addition -- the aforementioned "f" is the transformation that
| fits this additive property (and a few others), hence log()
| everywhere.
|
| Interestingly, the logarithm "algorithm" was considered quite
| groundbreaking at the time; Johannes Kepler, a primary
| beneficiary of the breakthrough, dedicated one of his books
| to Napier. R. C. Pierce wrote:
|
| > Indeed, it has been postulated that logarithms literally
| lengthened the life spans of astronomers, who had formerly
| been sorely bent and often broken early by the masses of
| calculations their art required.
| JackFr wrote:
| 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.
| meindnoch wrote:
| 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))
| secondcoming wrote:
| Isn't x^2 a multiplication?
| meindnoch wrote:
| 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).
| pipes wrote:
| Coincidentally I watched this last night
| https://m.youtube.com/watch?v=7TWKSMtKCmU
|
| It gives the history / motivation behind logarithms and
| suddenly it became so much clearer to me. Pretty much
| multipling huge numbers by adding exponents , well I think I've
| understood that correctly?
|
| I think why I'm so interested in programming and computing is
| because I fascinated by the history of it all. It somehow acts
| as a motivation to understand it.
| tim333 wrote:
| I rather like Feynman's approach in the lecture Algebra from
| the Feynman Lectures
| https://www.feynmanlectures.caltech.edu/I_22.html
|
| He covers the inverse of the exponential, Henry Briggs' log
| tables and goes on to e^ix = cos x + i sin x
|
| The audio is also available
| https://www.feynmanlectures.caltech.edu/flptapes.html
| xiande04 wrote:
| 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.
| adornKey wrote:
| The Logarithmic derivative is also something that is surprisingly
| fundamental.
|
| (ln(f))' = f'/f
|
| In function theory you use it all the time. But people rarely
| notice, that it related to a logarithm.
|
| Also the functions that have nice logarithmic derivative are a
| lot more interesting than expected. Nature is full of Gompertz
| functions. Once you're familiar with it, you see it everywhere.
| sva_ wrote:
| > Charles Petzold
|
| Haven't heard that name in a while. For me he's the WinApi guy -
| learned a lot from him when I first started programming.
| mixmastamyk wrote:
| Still can! His classic book Code is fantastic and has a recent
| second addition.
| jrmg wrote:
| Code is a masterpiece.
|
| Anyone here who hasn't read it should do so - you might think
| it's 'below you', but it's so well written it's a joy to
| read, and I suspect you'll come out thinking of some things
| differently.
|
| It reminded me why I love computing.
| mixmastamyk wrote:
| ^edition.
| esafak wrote:
| PC Magazine contributor, for me.
| hughw wrote:
| 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.
| crazygringo wrote:
| I guess you have to define what you mean by "conceive".
|
| I'm not sure you can even conceive a number like 1,000, if
| you're talking about holding an intuitive visual understanding
| in your mind at once.
|
| Like, I can easily see 100 in my mind's eye as a 10x10 grid of
| circles. Even if I don't see each one clearly, I have a good
| sense of the 10 on each edge and the way it fills in. But ask
| me to imagine 10 of those side-by-side to make 1,000, and I
| don't think I can. Once I imagine the 10 groups, each one is
| just a square simplification, rather than any individual pieces
| within.
|
| But I'm totally familiar with 1,000 as a concept I can multiply
| and divide with, and I can do math with 10^80 as well. And I
| can do so fairly "intuitively" as well -- it's just all the
| numbers up to 80 digits long. Even 4,000 digits fits on a
| single page of a book.
| hughw wrote:
| My first cut at conceiving is to answer "how long would it
| take a really fast computer to count to that number". The
| answer for 10^4000 is still something like 10^3978 years. So,
| a still inconceivable time.
|
| (100 tera-ops computer) [edited to correct calculation)
| crazygringo wrote:
| 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
| abstractions we can reason about using the same types of
| mathematical methods. They're both qualitatively different
| from numbers like 5 or 10 which are undoubtedly
| "conceivable".
| hughw wrote:
| _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.
| crazygringo wrote:
| > _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 _in_ conceivable, but
| exponentially _small_ numbers are somehow _conceivable_.
| They 're just as far from 1, multiplicatively, no matter
| which direction you go in.
| hughw wrote:
| 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.
| MisterTea wrote:
| To me it's hard because we don't come across these magnitudes
| on a daily basis. Grains of sand on a beach is an off the top
| of my head way to visualize such quantities. But yeah, numbers
| that big, are so big that we have nothing to compare them with.
| kqr wrote:
| I feel like I have a much better grasp of those numbers since I
| started using logarithms for mental maths:
| https://entropicthoughts.com/learning-some-logarithms
| avmich wrote:
| Read http://mrob.com/pub/math/largenum.html . After that you'll
| probably look easier to those numbers.
| kqr wrote:
| 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
| xelxebar wrote:
| What reflections do you have putting this into action over the
| year since that post?
|
| BTW, your blog is one of my absolute favorites!
| kqr wrote:
| 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.
| nakedneuron wrote:
| Great blog!
|
| Interesting fact that memory decay also is inherently
| logarithmic/exponential.
|
| Learning logs with SRS is so meta.
| aquafox wrote:
| 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 result of a lot of
| multiplications of influential factors, we thus get a log-normal
| distribution.
| TrainedMonkey wrote:
| All data is linear when plotted on a loglog scale with a thick
| marker.
| aquafox wrote:
| But in my explanation, there is no x axis.
| kqr wrote:
| 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.
| genewitch wrote:
| 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.
| wolfi1 wrote:
| the joy of an engineer is to find a straight line in a double
| logarithmic diagram
| H8crilA wrote:
| 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.
| BinRoo wrote:
| One of my favorite tricks in elementary school was to convince
| people I can calculate any logarithm for any number of their
| choosing.
|
| > Me: Pick any number.
|
| > Friend: Ok, 149,135,151
|
| > Me: The log is 8.2
|
| Of course I'm simply counting the number of digits, using 10 as
| the base, and guessing the last decimal point, but it certainly
| impressed everyone.
| ted_dunning wrote:
| You can do even better if you memorize three numbers: 301, 477,
| 845. These are the values of 1000 _log10(n) for n = 2, 3, 7.
| From these you can quickly get the values for 4 (= 2_ 2), 5
| (=10/2), 6 (=2 _3), 8 (=2_ 2 _2) and 9 (=3_ 3).
|
| For your example 1.49 is close to 3 / 2 so the log will be very
| close 0.477 - 0.301 = 0.176.
|
| This means that your answer is near 8.176 (actual value is
| 8.173).
|
| This tiny table of logs can also let you answer parlor trick
| questions like what is the first digit of 2^1000 (the result is
| very nearly 10^301 but a bit above, so 1 is the leading digit).
| kqr wrote:
| > 149,135,151
|
| This is 8-point-something as you say.
|
| 1.49 is in between 1.2 and 1.6 and I have memorised
| log(1.2)=0.1 and log(1.6)=0.2, so I would think log(1.5) is
| close to 0.17, using sloppy linear interpolation.
|
| That would make log(149,135,151) approximately 8.17. My
| calculator also says 8.17. Your guess was good!
|
| I have found linear interpolation such an intuitive
| approximation method that the tradeoff of having to memorise
| more logarithms is worth it.
| spapas82 wrote:
| One of the best uses of logarithms is how they can be used to
| quickly calculate db (as in decibel) gains and losses mentally.
| See this older comment for more details
|
| https://news.ycombinator.com/item?id=32550539
| xelxebar wrote:
| 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/
| johnm wrote:
| Indeed, I use pen/pencil and (dot) paper. Different brain
| space.
| Rendello wrote:
| I've been doing a math course and occasionally think of picking
| up these analogue tools. Someone on Hacker News had me
| interested in the Soroban, the Japanese abacus [1], which is
| still used to train insane mental math speeds to this day [2].
|
| 1. https://en.wikipedia.org/wiki/Soroban
|
| 2. https://www.youtube.com/watch?v=s6OmqXCsYt8
| xelxebar wrote:
| We're definitely on a similar wavelength. I actually own a
| couple Japanese abaci and know the basics. Top performers
| feel near magical: double-fisters[0] and blazingly fast
| mental arithmetic [1].
|
| [0]:https://www.youtube.com/watch?v=EK6uIjjkrGE
|
| [1]:https://youtu.be/-kjUCtqSWlw?feature=shared&t=451
| jbaber wrote:
| Soroban. Japanese abacus. Every number only has one
| representation. +-*/ and other calculations.
|
| http://totton.idirect.com/
| divbzero wrote:
| Where can I get the meter-long slide rule the man is holding in
| OP?
| 7402 wrote:
| They show up on eBay. A search this minute revealed two:
|
| https://www.ebay.com/itm/205220626817
| https://www.ebay.com/itm/156686655356
|
| They go for a bit more than the original price, according to
| this:
|
| "Pricing varied by retailer, however Pickett did offer
| demonstration slide rules in 4 foot and 7 foot lengths: a 4
| foot rule sold for $15 and the 7 foot rule was $25 in 1960.
| These were available with scales to match models N4, N803,
| and N1010 with the Ln scale added. These large rules were
| available free to schools which ordered 24 or more slide
| rules!"
|
| [0] https://www.sphere.bc.ca/oldsite/test/pickett.html
| kqr wrote:
| I have several slide rules and use them daily. Especially in
| the kitchen where we deal a lot with scaling proportions they
| are _the_ best tool available: you set them for the desired
| scale, and then you can just read off any proportion you need
| in the blink of an eye.
|
| I'm honestly surprised they are not standard issue in kitchens.
| kazinator wrote:
| I recommend the classic _Introduction to Logarithms_ , by Cormen,
| Rivest, Leiserson et al.
| cuttothechase wrote:
| Charles Petzold was one of my favorite tech authors from the way
| begone era. Written in a style very similar in vein to the Lost
| of Art of Logarithms he made me fall in love with the various
| mundane tech concepts that would never jump out as a anything of
| interest, otherwise. What a treat!
| tiahura wrote:
| Programming Windows 95 was invaluable.
| westurner wrote:
| Notes from "How should logarithms be taught?" (2021)
| https://news.ycombinator.com/item?id=28519356 re: logarithms in
| the Python standard library, NumPy, SymPy, TensorFlow, PyTorch,
| Wikipedia
| alanh wrote:
| So interesting! The author doesn't, I believe, yet cover how the
| first log tables were computed (by hand), so I asked ChatGPT.
| This may be of interest:
| https://chatgpt.com/share/67d3a64d-f8a8-8012-bde3-e80813b2b4...
| JackFr wrote:
| 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."
| mikewarot wrote:
| 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.
| max_ wrote:
| I wish there was a mailing list I could subscribe to so I could
| know when the book os complete.
|
| Or a pre-order on Amazon?
| yujzgzc wrote:
| I have a few old math manuals at home, from late 19th / early
| 20th century. Many of them have a logarithm table as an appendix.
| It looked like the type of things that if you had a few extra
| sheets to print to make a booklet, you'd just add because it was
| bound to be very useful to someone.
| hansmayer wrote:
| Wow, I thought it was just some random guy, but was then quite
| surprised to see this was being authored by none other than the
| legendary Charles Petzold. I'd buy this book - just to put it
| next to my copy of "Programming Windows 95" (who remembers?) :)
| stpedgwdgfhgdd wrote:
| Well written and fun to read! (For nerds)
| jamalaramala wrote:
| There was an interesting text, by Isaac Asimov, where he
| explained in a very clear way the historical importance of
| logarithms -- they allowed Kepler to finalize his work by
| replacing tables of multiplications (which were difficult and
| error-prone) with sums.
| vismit2000 wrote:
| Realm of Algebra by Isaac Asimov:
| https://archive.org/stream/RealmOfAlgebra-English-IsaacAsimo...
| vismit2000 wrote:
| Logging the World - Oliver Johnson (Oxford Mathematics):
| https://youtu.be/UsK52iZMsxo
| stephencwelch wrote:
| Yeah love this angle - I made a video in a similar vein:
| https://www.youtube.com/watch?v=OjIwCOevUew
| sali0 wrote:
| Huge fan of your channel! Great content.
| ilija139 wrote:
| Off-topic, but anyone knows where to buy such [1] old but not
| rare (so they are cheap enough) math books? In UK and globally?
| Is e-bay and perhaps amazon the best place? How to avoid fakes?
|
| [1] https://www.lostartoflogarithms.com/chapter01/
| vanderZwan wrote:
| 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
| honest), we might as well co-opt the extra attention maths gets
| for other mathematical causes.
|
| [0] https://news.ycombinator.com/item?id=43036094
| Enginerrrd wrote:
| 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?
| anjakefala wrote:
| 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.
| vanderZwan wrote:
| 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
| sourtrident wrote:
| Funny how logarithms shaped navigation, astronomy, and music--yet
| now they're mostly a forgotten button gathering dust on our
| calculators. Hidden tech history right there.
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