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  • Joseph Nebus 4:00 pm on Monday, 31 July, 2017 Permalink | Reply
    Tags: , , , , , , summer,   

    The Summer 2017 Mathematics A To Z: Arithmetic 

    And now as summer (United States edition) reaches its closing months I plunge into the fourth of my A To Z mathematics-glossary sequences. I hope I know what I’m doing! Today’s request is one of several from Gaurish, who’s got to be my top requester for mathematical terms and whom I thank for it. It’s a lot easier writing these things when I don’t have to think up topics. Gaurish hosts a fine blog, For the love of Mathematics, which you might consider reading.


    Arithmetic is what people who aren’t mathematicians figure mathematicians do all day. I remember in my childhood a Berenstain Bears book about people’s jobs. Its mathematician was an adorable little bear adding up sums on the chalkboard, in an observatory, on the Moon. I liked every part of this. I wouldn’t say it’s the whole reason I became a mathematician but it did made the prospect look good early on.

    People who aren’t mathematicians are right. At least, the bulk of what mathematics people do is arithmetic. If we work by volume. Arithmetic is about the calculations we do to evaluate or solve polynomials. And polynomials are everything that humans find interesting. Arithmetic is adding and subtracting, of multiplication and division, of taking powers and taking roots. Arithmetic is changing the units of a thing, and of breaking something into several smaller units, or of merging several smaller units into one big one. Arithmetic’s role in commerce and in finance must overwhelm the higher mathematics. Higher mathematics offers cohomologies and Ricci tensors. Arithmetic offers a budget.

    This is old mathematics. There’s evidence of humans twenty thousands of years ago recording their arithmetic computations. My understanding is the evidence is ambiguous and interpretations vary. This seems fair. I assume that humans did such arithmetic then, granting that I do not know how to interpret archeological evidence. The thing is that arithmetic is older than humans. Animals are able to count, to do addition and subtraction, perhaps to do harder computations. (I crib this from The Number Sense:
    How the Mind Creates Mathematics
    , by Stanislas Daehaene.) We learn it first, refining our rough instinctively developed sense to something rigorous. At least we learn it at the same time we learn geometry, the other branch of mathematics that must predate human existence.

    The primality of arithmetic governs how it becomes an adjective. We will have, for example, the “arithmetic progression” of terms in a sequence. This is a sequence of numbers such as 1, 3, 5, 7, 9, and so on. Or 4, 9, 14, 19, 24, 29, and so on. The difference between one term and its successor is the same as the difference between the predecessor and this term. Or we speak of the “arithmetic mean”. This is the one found by adding together all the numbers of a sample and dividing by the number of terms in the sample. These are important concepts, useful concepts. They are among the first concepts we have when we think of a thing. Their familiarity makes them easy tools to overlook.

    Consider the Fundamental Theorem of Arithmetic. There are many Fundamental Theorems; that of Algebra guarantees us the number of roots of a polynomial equation. That of Calculus guarantees us that derivatives and integrals are joined concepts. The Fundamental Theorem of Arithmetic tells us that every whole number greater than one is equal to one and only one product of prime numbers. If a number is equal to (say) two times two times thirteen times nineteen, it cannot also be equal to (say) five times eleven times seventeen. This may seem uncontroversial. The budding mathematician will convince herself it’s so by trying to work out all the ways to write 60 as the product of prime numbers. It’s hard to imagine mathematics for which it isn’t true.

    But it needn’t be true. As we study why arithmetic works we discover many strange things. This mathematics that we know even without learning is sophisticated. To build a logical justification for it requires a theory of sets and hundreds of pages of tight reasoning. Or a theory of categories and I don’t even know how much reasoning. The thing that is obvious from putting a couple objects on a table and then a couple more is hard to prove.

    As we continue studying arithmetic we start to ponder things like Goldbach’s Conjecture, about even numbers (other than two) being the sum of exactly two prime numbers. This brings us into number theory, a land of fascinating problems. Many of them are so accessible you could pose them to a person while waiting in a fast-food line. This befits a field that grows out of such simple stuff. Many of those are so hard to answer that no person knows whether they are true, or are false, or are even answerable.

    And it splits off other ideas. Arithmetic starts, at least, with the counting numbers. It moves into the whole numbers and soon all the integers. With division we soon get rational numbers. With roots we soon get certain irrational numbers. A close study of this implies there must be irrational numbers that must exist, at least as much as “four” exists. Yet they can’t be reached by studying polynomials. Not polynomials that don’t already use these exotic irrational numbers. These are transcendental numbers. If we were to say the transcendental numbers were the only real numbers we would be making only a very slight mistake. We learn they exist by thinking long enough and deep enough about arithmetic to realize there must be more there than we realized.

    Thought compounds thought. The integers and the rational numbers and the real numbers have a structure. They interact in certain ways. We can look for things that are not numbers, but which follow rules like that for addition and for multiplication. Sometimes even for powers and for roots. Some of these can be strange: polynomials themselves, for example, follow rules like those of arithmetic. Matrices, which we can represent as grids of numbers, can have powers and even something like roots. Arithmetic is inspiration to finding mathematical structures that look little like our arithmetic. We can find things that follow mathematical operations but which don’t have a Fundamental Theorem of Arithmetic.

    And there are more related ideas. These are often very useful. There’s modular arithmetic, in which we adjust the rules of addition and multiplication so that we can work with a finite set of numbers. There’s floating point arithmetic, in which we set machines to do our calculations. These calculations are no longer precise. But they are fast, and reliable, and that is often what we need.

    So arithmetic is what people who aren’t mathematicians figure mathematicians do all day. And they are mistaken, but not by much. Arithmetic gives us an idea of what mathematics we can hope to understand. So it structures the way we think about mathematics.

    • ivasallay 5:34 pm on Monday, 31 July, 2017 Permalink | Reply

      I think you covered arithmetic in a very clear, scholarly way.

      When I was in the early elementary grades, we didn’t study math. We studied arithmetic.

      Here’s a couple more things some people might not know about arithmetic:
      1) How to remember the proper spelling of arithmetic: A Rat In The House May Eat The Ice Cream.
      2) How to pronounce arithmetic: https://www.quora.com/Why-does-the-pronunciation-of-arithmetic-depend-on-context


      • Joseph Nebus 6:27 pm on Wednesday, 2 August, 2017 Permalink | Reply

        Thanks! … My recollection is that in elementary school we called it mathematics (or just math), but the teachers were pretty clear about whether we were doing arithmetic or geometry. If that was clear, since I grew up on the tail end of the New Math wave and we could do stuff that was more playful than multiplication tables were.

        I hadn’t thought about the shifting pronunciations of ‘arithmetic’ as a word. I suppose it’s not different from many multi-syllable words in doing that, though. My suspicion is that the distinction between ‘arithmetic’ as an adjective and as a noun is spurious, though. My hunch is people shift the emphasis based on the structure of the whole sentence, with the words coming after ‘arithmetic’ having a big role to play. I’d expect that an important word immediately follows ‘arithmetic’ often if it’s being used as an adjective (like, ‘arithmetic mean’), but that’s not infallible. As opposed to those many rules of English grammar and pronunciation that are infallible.

        Liked by 1 person

    • gaurish 9:48 am on Saturday, 12 August, 2017 Permalink | Reply

      A Beautiful introduction to Arithmetic!


    • Jayeesha 12:06 pm on Thursday, 31 August, 2017 Permalink | Reply

      Mental Arithmetic, I like it


      • Joseph Nebus 1:14 am on Friday, 8 September, 2017 Permalink | Reply

        It’s a fun pastime. Also a great way to find yourself reassuring the cashier that yes, you meant to give $20.17.


  • Joseph Nebus 4:00 pm on Tuesday, 25 July, 2017 Permalink | Reply
    Tags: , , , , summer   

    There’s Still Time To Ask For Things For The Mathematics A To Z 

    I’m figuring to begin my Summer 2017 Mathematics A To Z next week. And I’ve got the first several letters pinned down, in part by a healthy number of requests by Gaurish, a lover of mathematics. Partly by some things I wanted to talk about.

    There are many letters not yet spoken for, though. If you’ve got something you’d like me to talk about, please head over to my first appeal and add a comment. The letters crossed out have been committed, but many are free. And the challenges are so much fun.

  • Joseph Nebus 4:00 pm on Thursday, 13 July, 2017 Permalink | Reply
    Tags: , , , , summer   

    What Would You Like In The Summer 2017 Mathematics A To Z? 

    I would like to now announce exactly what everyone with the ability to draw conclusions expected after I listed the things covered in previous Mathematics A To Z summaries. I’m hoping to write essays about another 26 topics, one for each of the major letters of the alphabet. And, as ever, I’d like your requests. It’s great fun to be tossed out a subject and either know enough about it, or learn enough about it in a hurry, to write a couple hundred words about it.

    So that’s what this is for. Please, in comments, list something you’d like to see explained.

    For the most part, I’ll do a letter on a first-come, first-serve basis. I’ll try to keep this page updated so that people know which letters have already been taken. I might try rewording or rephrasing a request if I can’t do it under the original letter if I can think of a legitimate way to cover it under another. I’m open to taking another try at something I’ve already defined in the three A To Z runs I’ve previously done, especially since many of the terms have different meanings in different contexts.

    I’m always in need of requests for letters such as X and Y. But you knew that if you looked at how sparse Mathworld’s list of words for those letters are.

    Letters To Request:

    • A
    • B
    • C
    • D
    • E
    • F
    • G
    • H
    • I
    • J
    • K
    • L
    • M
    • N
    • O
    • P
    • Q
    • R
    • S
    • T
    • U
    • V
    • W
    • X
    • Y
    • Z

    I’m flexible about what I mean by “a word” or “a term” in requesting something, especially if it gives me a good subject to write about. And if you think of a clever way to get a particular word covered under a letter that’s really inappropriate, then, good. I like cleverness. I’m not sure what makes for the best kinds of glossary terms. Sometimes a broad topic is good because I can talk about how an idea expresses itself across multiple fields. Sometimes a narrow topic is good because I can dig in to a particular way of thinking. I’m just hoping I’m not going to commit myself to three 2500-word essays a week. Those are fun, but they’re exhausting, as the time between Why Stuff Can Orbit essays may have hinted.

    And finally, I’d like to thank Thomas K Dye for creating banner art for this sequence. He’s the creator of the longrunning web comic Newshounds. He’s also got the book version, Newshounds: The Complete Story freshly published, a Patreon to support his comics habit, and plans to resume his Infinity Refugees spinoff strip shortly.

    • gaurish 2:12 pm on Monday, 17 July, 2017 Permalink | Reply

      A – Arithmetic
      C – Cohomology
      D – Diophantine Equations
      E – Elliptic curves
      F – Functor
      G – Gaussian primes/integers/distribution
      H – Height function (elliptic curves)
      I – integration
      L – L-function
      P – Prime number
      Z – zeta function

      I will tell more later. The banner art is very nice.

      Liked by 1 person

      • Joseph Nebus 5:37 pm on Tuesday, 18 July, 2017 Permalink | Reply

        Thank you! That’s a great set of topics to start on.

        And thanks for the kind words about the art. I’m quite happy with it and hope to get more for other projects. And, as ever, do hope people consider Thomas K Dye’s comic strips and Patreon.


      • gaurish 4:57 am on Wednesday, 26 July, 2017 Permalink | Reply

        J – Jordan Canonical Form
        K – Klien Bottle
        M – Meromorphic function
        N – Nine point circle
        O – Open set
        Q – Quasirandom numbers
        R – Real number
        S – Sárkőzy’s Theorem
        T – Torus
        U – Ulam Spiral
        V – Venn diagram
        W – Well ordering principle
        X – <I couldn’t find a word with x, but can discuss the importance of x as a variable>
        Y- Young tableau

        Liked by 2 people

        • elkement (Elke Stangl) 7:06 am on Wednesday, 26 July, 2017 Permalink | Reply

          Ha – I also suggested Open Set yesterday, see my comments below ;-) (from July 25 – on letters M N O R T V).


        • Joseph Nebus 12:39 pm on Thursday, 27 July, 2017 Permalink | Reply

          And thank you again! This gets the alphabet a good bit more done.

          And, yeah, ‘x’ is hard. But it’ll all be worth it in the end, I hope.


    • The Chaos Realm 4:53 pm on Monday, 17 July, 2017 Permalink | Reply

      I used the Riemann Tensor definition/explanation to front one of my sub-chapter pages in my poetry book (courtesy the guidance of a teacher I know). :-)

      Liked by 1 person

      • Joseph Nebus 5:35 pm on Tuesday, 18 July, 2017 Permalink | Reply

        Ah, that’s wonderful! There is this beauty in the way mathematical concepts are expressed — not the structure of the ideas, but the way we write them out, especially when we get a good idea of what we want to express. I’d like if more people could appreciate that without worrying that they don’t know, say, what a Ricci Flow would be.

        Liked by 1 person

        • The Chaos Realm 5:51 pm on Tuesday, 18 July, 2017 Permalink | Reply

          Thanks! I know there’s a really poetic beauty about astrophysics that I have loved for years. I may not understand all the equations, but I do feel I “get” physics in a way. looks up Ricci Flow. It’s definitely one of my major forms of inspirations…one of my most used muses!


          • Joseph Nebus 6:18 pm on Sunday, 23 July, 2017 Permalink | Reply

            I’m glad you do enjoy. There’s a lot about physics and mathematics that can’t be understood without great equations, but then there’s a lot about architecture that can’t be understood without a lot of mathematics and legal analyses. Nevertheless anyone can appreciate a beautiful building, and surely people can be told interesting enough stories about mathematics to appreciate the beauty there. Ideally, anyway.

            Liked by 2 people

    • mathtuition88 5:11 pm on Monday, 24 July, 2017 Permalink | Reply

      V for Voronoi diagram would be nice


    • mathtuition88 4:12 pm on Tuesday, 25 July, 2017 Permalink | Reply

      How about D for discrete Morse theory and M for Morse theory? These are subjects I am not familiar with myself.. it would be great to have an article describing the gist of it :)


      • Joseph Nebus 12:35 pm on Thursday, 27 July, 2017 Permalink | Reply

        I’m thoroughly unfamiliar with either myself, but I’m excited to give them a try! ‘M’ had been free, at least.

        Liked by 1 person

    • elkement (Elke Stangl) 4:14 pm on Tuesday, 25 July, 2017 Permalink | Reply

      N – N-Sphere or N-Ball
      O – Open Set
      R – Riemann Tensor


      • elkement (Elke Stangl) 4:35 pm on Tuesday, 25 July, 2017 Permalink | Reply

        Ah, the Riemann Tensor has already been claimed :-) Sorry, I did not read the other comments carefully. So then:
        R – Ricci Tensor


        • Joseph Nebus 12:38 pm on Thursday, 27 July, 2017 Permalink | Reply

          You know, deep down, I worried I was making trouble for myself mentioning the Ricci Tensor (or was it the Ricci Flow I mentioned? Ricci something, anyway) but what’s the fun of this without making trouble for myself?

          Liked by 1 person

      • Joseph Nebus 12:35 pm on Thursday, 27 July, 2017 Permalink | Reply

        Thank you, that’s getting the alphabet filled out a bit more.

        Liked by 1 person

    • elkement (Elke Stangl) 4:16 pm on Tuesday, 25 July, 2017 Permalink | Reply

      BTW – service for the other readers: Here is the neat table showing what Joseph has already covered in previous A-Z series: https://nebusresearch.wordpress.com/2017/06/29/a-listing-of-mathematics-subjects-i-have-covered-in-a-to-z-sequences-of-the-past/


    • elkement (Elke Stangl) 4:26 pm on Tuesday, 25 July, 2017 Permalink | Reply

      M – Manifold
      T – Topology (or Topological Manifold). Alternative: Tangent Bundle
      V – Volume Forms or Vector Bundle

      So you see, I am still in awe of the math that underpins General Relativity :-) But please, totally explain it from a ‘pure math’ perspective … I am most interested in if and how your perspective may differ from how such things are introduced in theoretical physics.

      Liked by 1 person

    • mathtuition88 9:20 am on Wednesday, 26 July, 2017 Permalink | Reply

      Reblogged this on Singapore Maths Tuition.


    • sheldonk2014 11:22 pm on Tuesday, 8 August, 2017 Permalink | Reply

      Hey Joseph
      How’s the pinball playing
      Great to see you


  • Joseph Nebus 1:20 pm on Tuesday, 30 June, 2015 Permalink | Reply
    Tags: , , , , , , summer,   

    Reading the Comics, June 30, 2015: Fumigating The Theater Edition 

    One of my favorite ever episodes of The Muppet Show when I was a kid had the premise the Muppet Theater was being fumigated and so they had to put on a show from the train station instead. (It was the Loretta Lynn episode, third season, number eight.) I loved seeing them try to carry on as normal when not a single thing was as it should be. Since then — probably before, too, but I don’t remember that — I’ve loved seeing stuff trying to carry on in adverse circumstances.

    Why this is mentioned here is that Sunday night my computer had a nasty freeze and some video card mishaps. I discovered that my early-2011 MacBook Pro might be among those recalled earlier this year for a service glitch. My computer is in for what I hope is a simple, free, and quick repair. But obviously I’m not at my best right now. I might be even longer than usual answering people and goodness knows how the statistics survey of June will go.

    Anyway. Rick Kirkman and Jerry Scott’s Baby Blues (June 26) is a joke about motivating kids to do mathematics. And about how you can’t do mathematics over summer vacation.

    Ruben Bolling’s Tom The Dancing Bug (June 26) features a return appearance of Chaos Butterfly. Chaos Butterfly does what Chaos Butterfly does best.

    Charles Schulz’s Peanuts Begins (June 26; actually just the Peanuts of March 23, 1951) uses arithmetic as a test of smartness. And as an example of something impractical.

    Alex Hallatt’s Arctic Circle (June 28) is a riff on the Good Will Hunting premise. That movie’s particular premise — the janitor solves an impossible problem left on the board — is, so far as I know, something that hasn’t happened. But it’s not impossible. Training will help one develop reasoning ability. Training will provide context and definitions and models to work from. But that’s not essential. All that’s essential is the ability to reason. Everyone has that ability; everyone can do mathematics. Someone coming from outside the academy could do first-rate work. However, I’d bet on the person with the advanced degree in mathematics. There is value in training.

    The penguin-janitor offers a solution to the unsolved mathematics problem on the blackboard. It's a smiley face. It wasn't what they were looking for.

    Alex Hallatt’s Arctic Circle for the 28th of June, 2015.

    But as many note, the Good Will Hunting premise has got a kernel of truth in it. In 1939, George Dantzig, a grad student in mathematics at University of California/Berkeley, came in late to class. He didn’t know that two problems on the board were examples of unproven theorems, and assumed them to be homework. So he did them, though he apologized for taking so long to do them. Before you draw too much inspiration from this, though, remember that Dantzig was a graduate student almost ready to start work on a PhD thesis. And the problems were not thought unsolvable, just conjectures not yet proven. Snopes, as ever, provides some explanation of the legend and some of the variant ways the story is told.

    Mac King and Bill King’s Magic In A Minute (June 28) shows off a magic trick that you could recast as a permutations problem. If you’ve been studying group theory, and many of my Mathematics A To Z terms have readied you for group theory, you can prove why this trick works.

    Guy Gilchrist’s Nancy (June 28) carries on Baby Blues‘s theme of mathematics during summer vacation being simply undoable.

    As only fifty percent of the population is happy, and one person is in a great mood, what must the other one be in?

    Piers Baker’s Ollie and Quentin for December 28, 2014, and repeated on June 28, 2015.

    Piers Baker’s Ollie and Quentin (June 28) is a gambler’s fallacy-themed joke. It was run — on ComicsKingdom, back then — back in December, and I talked some more about it then.

    Mike Twohy’s That’s Life (June 28) is about the perils of putting too much attention into mental arithmetic. It’s also about how perilously hypnotic decimals are: if the pitcher had realized “fourteen million over three years” must be “four and two-thirds million per year” he’d surely have been less distracted.

    • Thumbup 2:42 pm on Tuesday, 30 June, 2015 Permalink | Reply

      Smart kid. It definitely doesn’t mix!


      • Joseph Nebus 6:44 pm on Saturday, 4 July, 2015 Permalink | Reply

        I’m actually a little surprised Hammy didn’t think mathematics and candy couldn’t mix. The use seems quite dear to a kid’s heart. At least I would’ve thought it dear when I was a kid.

        Liked by 1 person

        • Thumbup 7:33 pm on Saturday, 4 July, 2015 Permalink | Reply

          Joseph Nebus, Really?! Cool. Mathematics rather an amazing thing. Yeah. You have a good fourth.


    • scifihammy 3:41 pm on Tuesday, 30 June, 2015 Permalink | Reply

      Love the cartoons – especially the “smiley face” answer and the 50%! So many people I know don’t get that either. :)


      • Joseph Nebus 6:47 pm on Saturday, 4 July, 2015 Permalink | Reply

        Well, the 50 percent thing is a tricky point. I mean, all probabilities are tricky; that’s why you should never, ever, ever trust your instinctive response to a probability question. But applying a probability is even more tricky. I imagine that’s because we have a feeling for how things “ought” to be, but they never actually are that way. It’s very hard to feel confident in the application of something when every example seems wrong in one way or another.

        Liked by 1 person

        • scifihammy 7:08 am on Sunday, 5 July, 2015 Permalink | Reply

          That’s it exactly! Took me a while to grasp that you can toss a coin 99 times and get 99 Heads in a row, but that still doesn’t mean that the next toss will be Tails; probably more likely the coin is weighted! ;) Still, as you say, it is hard to ignore your instinct on an outcome.


          • Joseph Nebus 6:38 pm on Sunday, 5 July, 2015 Permalink | Reply

            At 99 heads in a row I’d certainly bet on the coin being weighted.

            Still, yeah; I forget where (possibly one of John D Cook’s Twitter feeds) I saw the warning from. But it’s good to remember that there are entire fields of psychology dedicated to studying how bad people’s intuitive feeling for probability problems are.

            Liked by 1 person

    • sheldonk2014 10:35 pm on Tuesday, 30 June, 2015 Permalink | Reply

      I love thus as a theory things being dine under adverse circumstances
      Didn’t even think that was a actual theory
      That’s great


      • Joseph Nebus 6:50 pm on Saturday, 4 July, 2015 Permalink | Reply

        I’m not so sure it’s a theory so much as it is making sure the show goes on. But it did produce a great Muppet Show episode. Of course, aren’t they all great? Even the bad ones still have Fozzie Bear.


    • ivasallay 8:58 am on Saturday, 4 July, 2015 Permalink | Reply

      I thought Baby Blues was cute, but Nancy offended me.


      • Joseph Nebus 6:53 pm on Saturday, 4 July, 2015 Permalink | Reply

        That’s an interesting split of feelings, given they’re basically the same joke. What do you suppose makes the difference?

        Liked by 1 person

    • ivasallay 6:24 am on Wednesday, 8 July, 2015 Permalink | Reply

      The kid in Baby Blues was only speaking for himself. Nancy implied that the whole world should feel that way.


  • Joseph Nebus 4:51 pm on Friday, 18 July, 2014 Permalink | Reply
    Tags: , cursive, , , , , summer   

    Reading the Comics, July 18, 2014: Summer Doldrums Edition 

    Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.

    Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.

    Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.

    Richard Thompson’s Richard’s Poor Almanac (July 7, rerun) brings back our favorite set of infinite monkeys, here, to discuss their ambitious book set at the Museum of Natural History.

    Tom Thaves’s Frank and Ernest (July 16) builds on the (true) point that the ancient Greeks had no symbol for zero, and would probably have had a fair number of objections to the concept.

    'The day Einstein got the wind knocked out of his sails': Einstein tells his wife he's discovered the theory of relativity.

    Joe Martin’s _Mr Boffo_ strip for the 18th of July, 2014.

    Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.

    Charles Schulz’s Peanuts (July 18) — a rerun, of course, from the 21st of July, 1967 — mentions Sally going to Summer School and learning all about the astronomical details of summertime. Astronomy has always been one of the things driving mathematical discovery, but I admit, thinking mostly this would be a good chance to point out Dr Helmer Aslaksen’s page describing the relationship between the solstices and the times of earliest and latest sunrise (and sunset). It’s not quite as easy as finding when the days are longest and shortest. Dr Aslaksen has a number of fascinating astronomy- and calendar-based pages which I think worth reading, so, I hope you enjoy.

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