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  • Joseph Nebus 6:00 pm on Friday, 14 April, 2017 Permalink | Reply
    Tags: , Gottfried Wilhelm Leibniz, , ,   

    What Is The Logarithm of a Negative Number? 


    Learning of imaginary numbers, things created to be the square roots of negative numbers, inspired me. It probably inspires anyone who’s the sort of person who’d become a mathematician. The trick was great. I wondered could I do it? Could I find some other useful expansion of the number system?

    The square root of a complex-valued number sounded like the obvious way to go, until a little later that week when I learned that’s just some other complex-valued numbers. The next thing I hit on: how about the logarithm of a negative number? Couldn’t that be a useful expansion of numbers?

    No. It turns out you can make a sensible logarithm of negative, and complex-valued, numbers using complex-valued numbers. Same with trigonometric and inverse trig functions, tangents and arccosines and all that. There isn’t anything we can do with the normal mathematical operations that needs something bigger than the complex-valued numbers to play with. It’s possible to expand on the complex-valued numbers. We can make quaternions and some more elaborate constructs there. They don’t solve any particular shortcoming in complex-valued numbers, but they’ve got their uses. I never got anywhere near reinventing them. I don’t regret the time spent on that. There’s something useful in trying to invent something even if it fails.

    One problem with mathematics — with all intellectual fields, really — is that it’s easy, when teaching, to give the impression that this stuff is the Word of God, built into the nature of the universe and inarguable. It’s so not. The stuff we find interesting and how we describe those things are the results of human thought, attempts to say what is interesting about a thing and what is useful. And what best approximates our ideas of what we would like to know. So I was happy to see this come across my Twitter feed:

    The links to a 12-page paper by Deepak Bal, Leibniz, Bernoulli, and the Logarithms of Negative Numbers. It’s a review of how the idea of a logarithm of a negative number got developed over the course of the 18th century. And what great minds, like Gottfried Leibniz and John (I) Bernoulli argued about as they find problems with the implications of what they were doing. (There were a lot of Bernoullis doing great mathematics, and even multiple John Bernoullis. The (I) is among the ways we keep them sorted out.) It’s worth a read, I think, even if you’re not all that versed in how to calculate logarithms. (but if you’d like to be better-versed, here’s the tail end of some thoughts about that.) The process of how a good idea like this comes to be is worth knowing.

    Also: it turns out there’s not “the” logarithm of a complex-valued number. There’s infinitely many logarithms. But they’re a family, all strikingly similar, so we can pick one that’s convenient and just use that. Ask if you’re really interested.

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  • Joseph Nebus 6:00 pm on Monday, 5 December, 2016 Permalink | Reply
    Tags: , , , , Daffy Duck, , Gottfried Wilhelm Leibniz, , ,   

    The End 2016 Mathematics A To Z: Osculating Circle 


    I’m happy to say it’s another request today. This one’s from HowardAt58, author of the Saving School Math blog. He’s given me some great inspiration in the past.

    Osculating Circle.

    It’s right there in the name. Osculating. You know what that is from that one Daffy Duck cartoon where he cries out “Greetings, Gate, let’s osculate” while wearing a moustache. Daffy’s imitating somebody there, but goodness knows who. Someday the mystery drives the young you to a dictionary web site. Osculate means kiss. This doesn’t seem to explain the scene. Daffy was imitating Jerry Colonna. That meant something in 1943. You can find him on old-time radio recordings. I think he’s funny, in that 40s style.

    Make the substitution. A kissing circle. Suppose it’s not some playground antic one level up from the Kissing Bandit that plagues recess yet one or two levels down what we imagine we’d do in high school. It suggests a circle that comes really close to something, that touches it a moment, and then goes off its own way.

    But then touching. We know another word for that. It’s the root behind “tangent”. Tangent is a trigonometry term. But it appears in calculus too. The tangent line is a line that touches a curve at one specific point and is going in the same direction as the original curve is at that point. We like this because … well, we do. The tangent line is a good approximation of the original curve, at least at the tangent point and for some region local to that. The tangent touches the original curve, and maybe it does something else later on. What could kissing be?

    The osculating circle is about approximating an interesting thing with a well-behaved thing. So are similar things with names like “osculating curve” or “osculating sphere”. We need that a lot. Interesting things are complicated. Well-behaved things are understood. We move from what we understand to what we would like to know, often, by an approximation. This is why we have tangent lines. This is why we build polynomials that approximate an interesting function. They share the original function’s value, and its derivative’s value. A polynomial approximation can share many derivatives. If the function is nice enough, and the polynomial big enough, it can be impossible to tell the difference between the polynomial and the original function.

    The osculating circle, or sphere, isn’t so concerned with matching derivatives. I know, I’m as shocked as you are. Well, it matches the first and the second derivatives of the original curve. Anything past that, though, it matches only by luck. The osculating circle is instead about matching the curvature of the original curve. The curvature is what you think it would be: it’s how much a function curves. If you imagine looking closely at the original curve and an osculating circle they appear to be two arcs that come together. They must touch at one point. They might touch at others, but that’s incidental.

    Osculating circles, and osculating spheres, sneak out of mathematics and into practical work. This is because we often want to work with things that are almost circles. The surface of the Earth, for example, is not a sphere. But it’s only a tiny bit off. It’s off in ways that you only notice if you are doing high-precision mapping. Or taking close measurements of things in the sky. Sometimes we do this. So we map the Earth locally as if it were a perfect sphere, with curvature exactly what its curvature is at our observation post.

    Or we might be observing something moving in orbit. If the universe had only two things in it, and they were the correct two things, all orbits would be simple: they would be ellipses. They would have to be “point masses”, things that have mass without any volume. They never are. They’re always shapes. Spheres would be fine, but they’re never perfect spheres even. The slight difference between a perfect sphere and whatever the things really are affects the orbit. Or the other things in the universe tug on the orbiting things. Or the thing orbiting makes a course correction. All these things make little changes in the orbiting thing’s orbit. The actual orbit of the thing is a complicated curve. The orbit we could calculate is an osculating — well, an osculating ellipse, rather than an osculating circle. Similar idea, though. Call it an osculating orbit if you’d rather.

    That osculating circles have practical uses doesn’t mean they aren’t respectable mathematics. I’ll concede they’re not used as much as polynomials or sine curves are. I suppose that’s because polynomials and sine curves have nicer derivatives than circles do. But osculating circles do turn up as ways to try solving nonlinear differential equations. We need the help. Linear differential equations anyone can solve. Nonlinear differential equations are pretty much impossible. They also turn up in signal processing, as ways to find the frequencies of a signal from a sampling of data. This, too, we would like to know.

    We get the name “osculating circle” from Gottfried Wilhelm Leibniz. This might not surprise. Finding easy-to-understand shapes that approximate interesting shapes is why we have calculus. Isaac Newton described a way of making them in the Principia Mathematica. This also might not surprise. Of course they would on this subject come so close together without kissing.

     
  • Joseph Nebus 2:34 pm on Wednesday, 3 June, 2015 Permalink | Reply
    Tags: , , , , , , Gottfried Wilhelm Leibniz, ,   

    A Summer 2015 Mathematics A To Z: error 


    Error

    This is one of my A to Z words that everyone knows. An error is some mistake, evidence of our human failings, to be minimized at all costs. That’s … well, it’s an attitude that doesn’t let you use error as a tool.

    An error is the difference between what we would like to know and what we do know. Usually, what we would like to know is something hard to work out. Sometimes it requires complicated work. Sometimes it requires an infinite amount of work to get exactly right. Who has the time for that?

    This is how we use errors. We look for methods that approximate the thing we want, and that estimate how much of an error that method makes. Usually, the method involves doing some basic step some large number of times. And usually, if we did the step more times, the estimate of the error we make will be smaller. My essay “Calculating Pi Less Terribly” shows an example of this. If we add together more terms from that Leibniz formula we get a running total that’s closer to the actual value of π.

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    • baffledbaboon 3:13 pm on Wednesday, 3 June, 2015 Permalink | Reply

      Whenever I make an error – my partner likes to tell me that I “broke math”. This all stemming from the one time I was given all the steps to the problem and still got an answer that wasn’t even close.

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      • Joseph Nebus 10:41 pm on Friday, 5 June, 2015 Permalink | Reply

        Aw, that sort of thing happens to everybody. Mathematicians especially. There’s a bit of folklore that says to never give an arithmetic problem to a mathematician because even if you ever do get an answer from it, it won’t be anything near right.

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  • Joseph Nebus 6:49 pm on Friday, 20 June, 2014 Permalink | Reply
    Tags: , Gottfried Wilhelm Leibniz, , , ,   

    16,000 and a Square 


    I reached my 16,000th page view, sometime on Thursday. That’s a tiny bit slower than I projected based on May’s readership statistics, but May was a busy month and I’ve had a little less time to write stuff this month, so I’m not feeling bad about that.

    Meanwhile, while looking for something else, I ran across a bit about mathematical notation in Florian Cajori’s A History of Mathematical Notation which has left me with a grin since. The book is very good about telling the stories of just what the title suggests. It’s a book well worth dipping into because everything you see written down is the result of a long process of experimentation and fiddling about to find the right balance of “expressing an idea clearly” and “expressing an idea concisely” and “expressing an idea so it’s not too hard to work with”.

    The idea here is the square of a variable, which these days we’d normally write as a^2 . According to Cajori (section 304), René Descartes “preferred the notation aa to a^2 .” Cajori notes that Carl Gauss had this same preference and defended it on the grounds that doubling the symbol didn’t take any more (or less) space than the superscript 2 did. Cajori lists other great mathematicians who preferred doubling the letter for squaring, including Christiaan Huygens, Edmond Halley, Leonhard Euler, and Isaac Newton. Among mathematicians who preferred a^2 were Blaise Pascal, David Gregory (who was big in infinite series), and Wilhelm Leibniz.

    Well of course Newton and Leibniz would be on opposite sides of the aa versus a^2 debate. How could the universe be sensible otherwise?

     
  • Joseph Nebus 3:33 am on Thursday, 9 August, 2012 Permalink | Reply
    Tags: , Gottfried Wilhelm Leibniz, , , ,   

    Reblog: Kant & Leibniz on Space and Implications in Geometry 


    Mathematicians and philosophers are fairly content to share credit for Rene Descartes, possibly because he was able to provide catchy, easy-to-popularize cornerstones for both fields.

    Immanuel Kant, these days at least, is almost exclusively known as a philosopher, and that he was also a mathematician and astronomer is buried in the footnotes. If you stick to math and science popularizations you’ll probably pick up (as I did) that Kant was one of the co-founders of the nebular hypothesis, the basic idea behind our present understanding of how solar systems form, and maybe, if the book has room, that Kant had the insight that knowing gravitation falls off by an inverse-square rule implies that we live in a three-dimensional space.

    Frank DeVita here writes some about Kant (and Wilhelm Leibniz)’s model of how we understand space and geometry. It’s not technical in the mathematics sense, although I do appreciate the background in Kant’s philosophy which my Dearly Beloved has given me. In the event I’d like to offer it as a way for mathematically-minded people to understand more of an important thinker they may not have realized was in their field.

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    Frank DeVita

            

    Kant’s account of space in the Prolegomena serves as a cornerstone for his thought and comes about in a discussion of the transcendental principles of mathematics that precedes remarks on the possibility of natural science and metaphysics. Kant begins his inquiry concerning the possibility of ‘pure’ mathematics with an appeal to the nature of mathematical knowledge, asserting that it rests upon no empirical basis, and thus is a purely synthetic product of pure reason (§6). He also argues that mathematical knowledge (pure mathematics) has the unique feature of first exhibiting its concepts in a priori intuition which in turn makes judgments in mathematics ‘intuitive’ (§7.281). For Kant, intuition is prior to our sensibility and the activity of reason since the former does not grasp ‘things in themselves,’ but rather only the things that can be perceived by the senses. Thus, what we can perceive is based…

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    • BunnyHugger 7:42 pm on Thursday, 9 August, 2012 Permalink | Reply

      As Joseph knows, I have a framed portrait of Kant in my dining room (I have to stop calling things that; now it’s our dining room) that was taken from a set of prints from the Moscow Observatory — I believe from the 1940s — celebrating people who had made contributions to astronomy. I like that I have a souvenir recognizing Kant as an astronomer.

      It used to bother me that people call the nebular hypothesis “the Laplace theory” when Kant’s work on it was earlier. (I have also heard it called “the Kant-Laplace theory,” but, I think, usually by philosophers.) However, then Wikipedia told me that Kant himself may have gotten the rudiments of it from Swedenborg, and no one ever calls it “the Swedenborg-Kant-Laplace” theory, as far as I’ve heard.

      I hope that DeVita’s article called back ideas for you (regarding both Leibniz and Kant) that I tried to explain to you in our cabin on the Amsterdam-Newcastle ferry while fighting off seasickness.

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  • Joseph Nebus 12:19 am on Sunday, 22 July, 2012 Permalink | Reply
    Tags: Arabic numerals, , , Black Widowers, , , , , Girolamo Cardano, , Gottfried Wilhelm Leibniz, honeymoon, , , Inspector Danger's Crime Quiz, , , , , Kliban, , , Mac King, , Moriarty, mystery, Niccolo Targalia, , , , Union Club, Werner Wejp-Olsen,   

    Reading the Comics, July 14, 2012 


    I hope everyone’s been well. I was on honeymoon the last several weeks and I’ve finally got back to my home continent and new home so I’ll try to catch up on the mathematics-themed comics first and then plunge into new mathematics content. I’m splitting that up into at least two pieces since the comics assembled into a pretty big pile while I was out. And first, I want to offer the link to the July 2 Willy and Ethel, by Joe Martin, since even though I offered it last time I didn’t have a reasonably permanent URL for it.

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    • Chiaroscuro 1:15 am on Sunday, 22 July, 2012 Permalink | Reply

      So what’s going to happen to those two boxes of leftover cobras?

      ..on less mongoosey take, even Martin Gardner wouldn’t give a life-or-death logic puzzle to a monkey. That’s kind of messed up.

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      • Joseph Nebus 5:04 am on Sunday, 22 July, 2012 Permalink | Reply

        Yeah, the life-or-death thing is a little weird, but it’s only a little bit out of bounds considering how the guy in Magic In A Minute treats his monkey pal. (Also, logic puzzles are not, properly speaking, magic tricks, at least not without much more setup.)

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    • Donna 3:03 am on Sunday, 22 July, 2012 Permalink | Reply

      Awwww! Congratulations!! Happy Happy Joy Joy!!! I love happy weddings, happy couples, happy lives. Enjoy!

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      • Joseph Nebus 5:01 am on Sunday, 22 July, 2012 Permalink | Reply

        Thank you! The wedding came through in quite good order, and the honeymoon was grand, despite a lot of raining.

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  • Joseph Nebus 2:50 am on Tuesday, 4 October, 2011 Permalink | Reply
    Tags: , compound interest, , Gottfried Wilhelm Leibniz, ,   

    Some Names Which e Doesn’t Have 


    I’ve outlined now some of the numbers which grew important enough to earn their own names. Most of them are counting numbers; the stragglers are a handful of irrational numbers which proved themselves useful, such as π (pi), or attractive, such as φ (phi), or physically important, such as the fine structure constant. Unnamed except in the list of categories is the number whose explanation I hope to be the first movement of this blog: e.

    It’s an important number physically, and a convenient and practical number mathematically. For all that, it defies a simple explanation like π enjoys. The simplest description of which I’m aware is that it is the base of the natural logarithm, which perfectly clarifies things to people who know what logarithms are, know which one is the natural logarithm, and know what the significance of the base is. This I will explain, but not today. For now it’s enough to think of the base as a size of the measurement tool, and to know that switching between one base and another is akin to switching between measuring in centimeters and measuring in inches. What the logarithm is will also wait for explanation; for now, let me hold off on that by saying it’s, in a way, a measure of how many digits it takes to write down a number, so that “81” has a logarithm twice that of “9”, and “49” twice that of “7”, and please don’t take this description so literally as to think the logarithm of “81” is equal to that of “49”.

    I agree it’s not clear why we should be interested in the natural logarithm when there are an infinity of possible logarithms, and we can convert a logarithm base e into a logarithm base 10 just by multiplying by the correct number. That, too, will come.

    Another common explanation is to say that e describes how fast savings will grow under the influence of compound interest. A dollar invested at one-percent interest, compounded daily, for a year, will grow to just about e dollars. Compounded hourly it grows even closer; compounded by the second it grows closer still; compounded annually, it stays pretty far away. The comparison is probably perfectly clear to those who can invest in anything with interest compounded daily. For my part I note when I finally opened an individual retirement account I put a thousand dollars into an almost thoughtfully selected mutual fund, and within mere weeks had lost $15. That about finishes off compound interest to me.

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    • BunnyHugger 4:08 am on Tuesday, 4 October, 2011 Permalink | Reply

      “Gottfried Wilhelm Leibniz, otherwise famous for his work in getting the Duke of Brunswick-Lüneburg elevated to the rank of Elector for the Holy Roman Empire”

      Ahem. Did you think I wouldn’t notice this?

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      • nebusresearch 2:32 am on Thursday, 6 October, 2011 Permalink | Reply

        I was certain that you’d notice. I’ve also got a nice bit about Königsberg penciled in for sometime next week or so.

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    • MJ Howard (@random_bunny) 1:37 am on Wednesday, 5 October, 2011 Permalink | Reply

      I like e, the only think I ever confuse it with is epsilon. Unlike lambda, which among other things is the forbidden calculus.

      I find it hard to believe you didn’t take the Taylor series expansion a little further, you know, for the kids.

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      • nebusresearch 2:47 am on Thursday, 6 October, 2011 Permalink | Reply

        Yeah, the e-epsilon thing is potential trouble. It’s a shame mathematics didn’t settle on using earlier forms of the Greek alphabet so we’d have more distinct versions of letters to play with; we’re dreadfully overloaded as it is.

        Would you believe that I already have a path for getting to Taylor series mapped out? I don’t figure to get there soon, but I have a couple thoughts how to get there.

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