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  • Joseph Nebus 3:21 pm on Thursday, 2 July, 2015 Permalink | Reply
    Tags: Big Ben, , , , , , user interface, web pages,   

    My Mathematics Blog Abbreviated Statistics, June 2015 

    So, that was a fairly successful month. For June this blog managed a record 1,051 pages viewed. That’s just above April’s high of 1,047, and is a nice rebound from May’s 936. I feel comfortable crediting this mostly to the number of articles I published in the month. Between the Mathematics A To Z and the rush of Reading The Comics posts, and a couple of reblogged or miscellaneous bits, June was my most prolific month: I had 28 articles. If I’d known how busy it was going to be I wouldn’t have skipped the first two Sundays. And i start the month at 25,871 total views.

    It’s quite gratifying to get back above 1,000 for more than the obvious reasons. I’ve heard rumors — and I’m not sure where because most of my notes are on my not-yet-returned main computer — that WordPress somehow changed its statistics reporting so that mobile devices aren’t counted. That would explain a sudden drop in both my mathematics and humor blogs, and drops I heard reported from other readership-watching friends. It also implies many more readers out there, which is a happy thought.

    Unfortunately because of my computer problems I can’t give reports on things like the number of visitors, or the views per visitor. I can get at WordPress’s old Dashboard statistics page, and that had been showing the number of unique visitors and views per visitor and all that. But on Firefox 3.6.16, and on Safari 5.0.6, this information isn’t displayed. I don’t know if they’ve removed it altogether from the Dashboard Statistics page in the hopes of driving people to their new, awful, statistics page or what. I also can’t find things like the number of likes, because that’s on the New Statistics page, which is inaccessible on browsers this old.

    Worse, I can’t find the roster of countries that sent me viewers. I trust that when I get my main computer back, and can look at the horrible new statistics page, I’ll be able to fill that in, but for now — nothing. I’m sorry. I will provide these popular lists when I’m able.

    I can say what the most popular posts were in June. As you might expect for a month dominated by the A-To-Z project, the five most popular posts were all Reading The Comics entries:

    Finally after that some of the A To Z posts appear, with fallacy, and graph, and n-tuple the most popular of that collection.

    Among the search terms bringing people here were:

    • real life problems involving laws of exponents comic strip (three people wanted them!)
    • if the circumference is 40,000,000 then what is the radius (why, one Earth-radius, of course) (approximately)
    • poster on mathematical diagram in the form of cartoon for ,7th class student
    • how to figure out what you need on a final to pass (you need to start sooner in the term)
    • how to count fish (count all the things which are not fish, and subtract that from the total number of all things, and there you go)
    • einstein vs pythagoras formula (do I have to take a side?)
    • origin is the gateway to your entire gaming universe.
    • big ben anthropomorphized (it’s not actually Big Ben, you know. The text is clear that it’s Big Ben’s Creature.)
    • wyoming rectangular most dilbert (are we having Zippy the Pinhead fanfiction yet?)

    Sorry it’s an abbreviated report. Or, sry is abbrev rept, anyway. I’ll fill in what I can, when I can, and isn’t that true of all of us?

  • Joseph Nebus 2:49 pm on Wednesday, 1 July, 2015 Permalink | Reply
    Tags: , , , median, , quintiles, , word counts   

    A Summer 2015 Mathematics A To Z: quintile 


    Why is there statistics?

    There are many reasons statistics got organized as a field of study mostly in the late 19th and early 20th century. Mostly they reflect wanting to be able to say something about big collections of data. People can only keep track of so much information at once. Even if we could keep track of more information, we’re usually interested in relationships between pieces of data. When there’s enough data there are so many possible relationships that we can’t see what’s interesting.

    One of the things statistics gives us is a way of representing lots of data with fewer numbers. We trust there’ll be few enough numbers we can understand them all simultaneously, and so understand something about the whole data.

    Quintiles are one of the tools we have. They’re a lesser tool, I admit, but that makes them sound more exotic. They’re descriptions of how the values of a set of data are distributed. Distributions are interesting. They tell us what kinds of values are likely and which are rare. They tell us also how variable the data is, or how reliably we are measuring data. These are things we often want to know: what is normal for the thing we’re measuring, and what’s a normal range?

    We get quintiles from imagining the data set placed in ascending order. There’s some value that one-fifth of the data points are smaller than, and four-fifths are greater than. That’s your first quintile. Suppose we had the values 269, 444, 525, 745, and 1284 as our data set. The first quintile would be the arithmetic mean of the 269 and 444, that is, 356.5.

    The second quintile is some value that two-fifths of your data points are smaller than, and that three-fifths are greater than. With that data set we started with that would be the mean of 444 and 525, or 484.5.

    The third quintile is a value that three-fifths of the data set is less than, and two-fifths greater than; in this case, that’s 635.

    And the fourth quintile is a value that four-fifths of the data set is less than, and one-fifth greater than. That’s the mean of 745 and 1284, or 1014.5.

    From looking at the quintiles we can say … well, not much, because this is a silly made-up problem that demonstrates how quintiles are calculated rather instead of why we’d want to do anything with them. At least the numbers come from real data. They’re the word counts of my first five A-to-Z definitions. But the existence of the quintiles at 365.5, 484.5, 635, and 1014.5, along with the minimum and maximum data points at 269 and 1284, tells us something. Mostly that numbers are bunched up in the three and four hundreds, but there could be some weird high numbers. If we had a bigger data set the results would be less obvious.

    If the calculating of quintiles sounds much like the way we work out the median, that’s because it is. The median is the value that half the data is less than, and half the data is greater than. There are other ways of breaking down distributions. The first quartile is the value one-quarter of the data is less than. The second quartile a value two-quarters of the data is less than (so, yes, that’s the median all over again). The third quartile is a value three-quarters of the data is less than.

    Percentiles are another variation on this. The (say) 26th percentile is a value that 26 percent — 26 hundredths — of the data is less than. The 72nd percentile a value greater than 72 percent of the data.

    Are quintiles useful? Well, that’s a loaded question. They are used less than quartiles are. And I’m not sure knowing them is better than looking at a spreadsheet’s plot of the data. A plot of the data with the quintiles, or quartiles if you prefer, drawn in is better than either separately. But these are among the tools we have to tell what data values are likely, and how tightly bunched-up they are.

  • Joseph Nebus 1:20 pm on Tuesday, 30 June, 2015 Permalink | Reply
    Tags: , , , , , , , gambler's fallacy   

    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!


    • 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. :)


    • 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 3:04 pm on Monday, 29 June, 2015 Permalink | Reply
    Tags: , , , math jokes, proper, subsets, triviality   

    A Summer 2015 Mathematics A To Z: proper 


    So there’s this family of mathematical jokes. They run about like this:

    A couple people are in a hot air balloon that’s drifted off course. They’re floating towards a hill, and they can barely make out a person on the hill. They cry out, “Where are we?” And the person stares at them, and thinks, and watches the balloon sail aimlessly on. Just as the balloon is about to leave shouting range, the person cries out, “You are in a balloon!” And one of the balloonists says, “Great, we would have to get a mathematician.” “How do you know that was a mathematician?” “The person gave us an answer that’s perfectly true, is completely useless, and took a long time to produce.”

    (There are equivalent jokes told about lawyers and consultants and many other sorts of people.)

    A lot of mathematical questions have multiple answers. Factoring is a nice familiar example. If I ask “what’s a factor of 5,280”, you can answer “1” or “2” or “55” or “1,320” or some 44 other answers, each of them right. But some answers are boring. For example, 1 is a factor of every whole number. And any number is a factor of itself; you can divide 5,280 by 5,280 and get 1. The answers are right, yes, but they don’t tell you anything interesting. You know these two answers before you’ve even heard the question. So a boring answer like that we often write off as trivial.

    A proper solution, then, is one that isn’t boring. The word runs through mathematics, attaching to many concepts. What exactly it means depends on the concept, but the general idea is the same: it means “not one of the obvious, useless answers”. A proper factor, for example, excludes the original number. Sometimes it excludes “1”, sometimes not. Depends on who’s writing the textbook. For another example, consider sets, which are collections of things. A subset is a collection of things all of which are already in a set. Every set is therefore a subset of itself. To be a proper subset, there has to be at least one thing in the original set that isn’t in the proper subset.

  • Joseph Nebus 5:12 pm on Sunday, 28 June, 2015 Permalink | Reply
    Tags: , , John Allen Paulos, ,   

    The Alice in Wonderland Sesquicentennial 

    I did not realize it was the 150th anniversary of the publication of Alice in Wonderland, which is probably the best-liked piece of writing by any mathematician. At least it’s the only one I can think of that’s clearly inspired a Betty Boop cartoon. I’ve had cause to talk about Carroll’s writing about logic and some other topics in the past. (One was just a day short of three years ago, by chance.)

    As mentioned in his tweet John Allen Paulos reviewed a book entirely about Lewis Carroll/Charles Dodgson’s mathematical and logic writing. I was unaware of the book before, but am interested now.

    • howardat58 7:17 pm on Sunday, 28 June, 2015 Permalink | Reply

      Guess what ! I was in the same maths class as Robin Wilson, Oxford, Keble college ’62 to ’65, but I never wanted to become a “famous mathematician”!!!


  • Joseph Nebus 5:40 pm on Saturday, 27 June, 2015 Permalink | Reply
    Tags: , vertex   

    Vertex of a parabola – language in math again 

    Joseph Nebus:

    I don’t want folks thinking I’m claiming a monopoly on the mathematics-glossary front. HowardAt58 is happy to explain words too. Here he talks about one of the definitions of “vertex”, in this case the one that relates to parabolas and other polynomial curves. As a bonus, there’s osculating circles.


    Originally posted on Saving school math:

    Here are some definitions of the vertex of a parabola.

    One is complete garbage, one is correct  though put rather chattily.

    The rest are not definitions, though very popular (this is just a selection). But they are true statements

    Mathwarehouse: The vertex of a parabola is the highest or lowest point, also known as the maximum or minimum of a
    Mathopenref: A parabola is the shape defined by a quadratic equation. The vertex is the peak in the curve as shown on
    the right. The peak will be pointing either downwards or upwards depending on the sign of the x2 term.
    Virtualnerd: Each quadratic equation has either a maximum or minimum, but did you that this point has a special name?
    In a quadratic equation, this point is called the vertex!
    Mathwords: Vertex of a Parabola: The point at which a parabola makes its sharpest turn.
    Purplemath: The…

    View original 419 more words

  • Joseph Nebus 3:18 pm on Friday, 26 June, 2015 Permalink | Reply
    Tags: , , directions, , orthogonal, perpendicular, vector spaces   

    A Summer 2015 Mathematics A To Z: orthogonal 


    Orthogonal is another word for perpendicular. So why do we need another word for that?

    It helps to think about why “perpendicular” is a useful way to organize things. For example, we can describe the directions to a place in terms of how far it is north-south and how far it is east-west, and talk about how fast it’s travelling in terms of its speed heading north or south and its speed heading east or west. We can separate the north-south motion from the east-west motion. If we’re lucky these motions separate entirely, and we turn a complicated two- or three-dimensional problem into two or three simpler problems. If they can’t be fully separated, they can often be largely separated. We turn a complicated problem into a set of simpler problems with a nice and easy part plus an annoying yet small hard part.

    And this is why we like perpendicular directions. We can often turn a problem into several simpler ones describing each direction separately, or nearly so.

    And now the amazing thing. We can separate these motions because the north-south and the east-west directions are at right angles to one another. But we can describe something that works like an angle between things that aren’t necessarily directions. For example, we can describe an angle between things like functions that have the same domain. And once we can describe the angle between two functions, we can describe functions that make right angles between each other.

    This means we can describe functions as being perpendicular to one another. An example. On the domain of real numbers from -1 to 1, the function f(x) = x is perpendicular to the function g(x) = x^2 . And when we want to study a more complicated function we can separate the part that’s in the “direction” of f(x) from the part that’s in the “direction” of g(x). We can treat functions, even functions we don’t know, as if they were locations in space. And we can study and even solve for the different parts of the function as if we were pinning down the north-south and the east-west movements of a thing.

    So if we want to study, say, how heat flows through a body, we can work out a series of “direction” for functions, and work out the flow in each of those “directions”. These don’t have anything to do with left-right or up-down directions, but the concepts and the convenience is similar.

    I’ve spoken about this in terms of functions. But we can define the “angle” between things for many kinds of mathematical structures. Once we can do that, we can have “perpendicular” pairs of things. I’ve spoken only about functions, but that’s because functions are more familiar than many of the mathematical structures that have orthogonality.

    Ah, but why call it “orthogonal” rather than “perpendicular”? And I don’t know. The best I can work out is that it feels weird to speak of, say, the cosine function being “perpendicular” to the sine function when you can’t really say either is in any particular direction. “Orthogonal” seems to appeal less directly to physical intuition while still meaning something. But that’s my guess, rather than the verdict of a skilled etymologist.

  • Joseph Nebus 8:45 pm on Thursday, 25 June, 2015 Permalink | Reply
    Tags: , , grad school, , , , , , wordplay   

    Reading the Comics, June 25, 2015: Not Making A Habit Of This Edition 

    I admit I did this recently, and am doing it again. But I don’t mean to make it a habit. I ran across a few comic strips that I can’t, even with a stretch, call mathematically-themed, but I liked them too much to ignore them either. So they’re at the end of this post. I really don’t intend to make this a regular thing in Reading the Comics posts.

    Justin Boyd’s engagingly silly Invisible Bread (June 22) names the tuning “two steps below A”. He dubs this “negative C#”. This is probably an even funnier joke if you know music theory. The repetition of the notes in a musical scale could be used as an example of cyclic or modular arithmetic. Really, that the note above G is A of the next higher octave, and the note below A is G of the next lower octave, probably explains the idea already.

    If we felt like, we could match the notes of a scale to the counting numbers. Match A to 0, B to 1, C to 2 and so on. Work out sharps and flats as you like. Then we could think of transposing a note from one key to another as adding or subtracting numbers. (Warning: do not try to pass your music theory class using this information! Transposition of keys is a much more subtle process than I am describing.) If the number gets above some maximum, it wraps back around to 0; if the number would go below zero, it wraps back around to that maximum. Relabeling the things in a group might make them easier or harder to understand. But it doesn’t change the way the things relate to one another. And that’s why we might call something F or negative C#, as we like and as we hope to be understood.

    After a blackboard full of work the mathematician must conclude 'the solution is not in this piece of chalk'.

    Hilary Price’s Rhymes With Orange for the 23rd of June, 2015.

    Hilary Price’s Rhymes With Orange (June 23) reminds us how important it is to pick the correct piece of chalk. The mathematical symbols on the board don’t mean anything. A couple of the odder bits of notation might be meant as shorthand. Often in the rush of working out a problem some of the details will get written as borderline nonsense. The mathematician is probably more interested in getting the insight down. She’ll leave the details for later reflection.

    Jason Poland’s Robbie and Bobby (June 23) uses “calculating obscure digits of pi” as computer fun. Calculating digits of pi is hard, at least in decimals, which is all anyone cares about. If you wish to know the 5,673,299,925th decimal digit of pi, you need to work out all 5,673,299,924 digits that go before it. There are formulas to work out a binary (or hexadecimal) digit of pi without working out all the digits that go before. This saves quite some time if you need to explore the nether-realms of pi’s digits.

    The comic strip also uses Stephen Hawking as the icon for most-incredibly-smart-person. It’s the role that Albert Einstein used to have, and still shares. I am curious whether Hawking is going to permanently displace Einstein as the go-to reference for incredible brilliance. His pop culture celebrity might be a transient thing. I suspect it’s going to last, though. Hawking’s life has a tortured-genius edge to it that gives it Romantic appeal, likely to stay popular.

    Paul Trap’s Thatababy (June 23) presents confusing brand-new letters and numbers. Letters are obviously human inventions though. They’ve been added to and removed from alphabets for thousands of years. It’s only a few centuries since “i” and “j” became (in English) understood as separate letters. They had been seen as different ways of writing the same letter, or the vowel and consonant forms of the same letter. If enough people found a proposed letter useful it would work its way into the alphabet. Occasionally the ampersand & has come near being a letter. (The ampersand has a fascinating history. Honestly.) And conversely, if we collectively found cause to toss one aside we could remove it from the alphabet. English hasn’t lost any letters since yogh (the Old English letter that looks like a 3 written half a line off) was dropped in favor of “gh”, about five centuries ago, but there’s no reason that it couldn’t shed another.

    Numbers are less obviously human inventions. But the numbers we use are, or at least work like they are. Arabic numerals are barely eight centuries old in Western European use. Their introduction was controversial. People feared shopkeepers and moneylenders could easily cheat people unfamiliar with these crazy new symbols. Decimals, instead of fractions, were similarly suspect. Negative numbers took centuries to understand and to accept as numbers. Irrational numbers too. Imaginary numbers also. Indeed, look at the connotations of those names: negative numbers. Irrational numbers. Imaginary numbers. We can add complex numbers to that roster. Each name at least sounds suspicious of the innovation.

    There are more kinds of numbers. In the 19th century William Rowan Hamilton developed quaternions. These are 4-tuples of numbers that work kind of like complex numbers. They’re strange creatures, admittedly, not very popular these days. Their greatest strength is in representing rotations in three-dimensional space well. There are also octonions, 8-tuples of numbers. They’re more exotic than quaternions and have fewer good uses. We might find more, in time.

    Beside an Escher-esque house a woman says, 'The extra dimensions are wonderful, but oy, the property taxes.'

    Rina Piccolo’s entry in Six Chix for the 24th of June, 2015.

    Rina Piccolo’s entry in Six Chix this week (June 24) draws a house with extra dimensions. An extra dimension is a great way to add volume, or hypervolume, to a place. A cube that’s 20 feet on a side has a volume of 203 or 8,000 cubic feet, after all. A four-dimensional hypercube 20 feet on each side has a hypervolume of 160,000 hybercubic feet. This seems like it should be enough for people who don’t collect books.

    Morrie Turner’s Wee Pals (June 24, rerun) is just a bit of wordplay. It’s built on the idea kids might not understand the difference between the words “ratio” and “racial”.

    Tom Toles’s Randolph Itch, 2 am (June 25, rerun) inspires me to wonder if anybody’s ever sold novelty 4-D glasses. Probably they have, sometime.

    Now for the comics that I just can’t really make mathematics but that I like anyway:

    Phil Dunlap’s Ink Pen (June 23, rerun) is aimed at the folks still lingering in grad school. Please be advised that most doctoral theses do not, in fact, end in supervillainy.

    Darby Conley’s Get Fuzzy (June 25, rerun) tickles me. But Albert Einstein did after all say many things in his life, and not everything was as punchy as that line about God and dice.

    • sheldonk2014 1:53 am on Friday, 26 June, 2015 Permalink | Reply

      I know theory that # are in everything,I guess I need to read more about this theory


    • ivasallay 4:30 pm on Monday, 29 June, 2015 Permalink | Reply

      The comic about ratio and racial made me laugh a lot. Thank you so much for sharing that one!


  • Joseph Nebus 2:42 pm on Wednesday, 24 June, 2015 Permalink | Reply
    Tags: , , , ordering, pronunciation, ,   

    A Summer 2015 Mathematics A To Z: n-tuple 


    We use numbers to represent things we want to think about. Sometimes the numbers represent real-world things: the area of our backyard, the number of pets we have, the time until we have to go back to work. Sometimes the numbers mean something more abstract: an index of all the stuff we’re tracking, or how its importance compares to other things we worry about.

    Often we’ll want to group together several numbers. Each of these numbers may measure a different kind of thing, but we want to keep straight what kind of thing it is. For example, we might want to keep track of how many people are in each house on the block. The houses have an obvious index number — the street number — and the number of people in each house is just what it says. So instead of just keeping track of, say, “32” and “34” and “36”, and “3” and “2” and “3”, we would keep track of pairs: “32, 3”, and “34, 2”, and “36, 3”. These are called ordered pairs.

    They’re not called ordered because the numbers are in order. They’re called ordered because the order in which the numbers are recorded contains information about what the numbers mean. In this case, the first number is the street address, and the second number is the count of people in the house, and woe to our data set if we get that mixed up.

    And there’s no reason the ordering has to stop at pairs of numbers. You can have ordered triplets of numbers — (32, 3, 2), say, giving the house number, the number of people in the house, and the number of bathrooms. Or you can have ordered quadruplets — (32, 3, 2, 6), say, house number, number of people, bathroom count, room count. And so on.

    An n-tuple is an ordered set of some collection of numbers. How many? We don’t care, or we don’t care to say right now. There are two popular ways to pronounce it. One is to say it the way you say “multiple” only with the first syllable changed to “enn”. Others say it about the same, but with a long u vowel, so, “enn-too-pull”. I believe everyone worries that everyone else says it the other way and that they sound like they’re the weird ones.

    You might care to specify what your n is for your n-tuple. In that case you can plug in a value for that n right in the symbol: a 3-tuple is an ordered triplet. A 4-tuple is that ordered quadruplet. A 26-tuple seems like rather a lot but I’ll trust that you know what you’re trying to study. A 1-tuple is just a number. We might use that if we’re trying to make our notation consistent with something else in the discussion.

    If you’re familiar with vectors you might ask: so, an n-tuple is just a vector? It’s not quite. A vector is an n-tuple, but in the same way a square is a rectangle. It has to meet some extra requirements. To be a vector we have to be able to add corresponding numbers together and get something meaningful out of it. The ordered pair (32, 3) representing “32 blocks north and 3 blocks east” can be a vector. (32, 3) plus (34, 2) can give us us (66, 5). This makes sense because we can say, “32 blocks north, 3 blocks east, 34 more blocks north, 2 more blocks east gives us 66 blocks north, 5 blocks east.” At least it makes sense if we don’t run out of city. But to add together (32, 3) plus (34, 2) meaning “house number 32 with 3 people plus house number 34 with 2 people gives us house number 66 with 5 people”? That’s not good, whatever town you’re in.

    I think the commonest use of n-tuples is to talk about vectors, though. Vectors are such useful things.

    • howardat58 3:29 pm on Wednesday, 24 June, 2015 Permalink | Reply

      Now I’m waiting for V


      • Joseph Nebus 3:38 am on Thursday, 25 June, 2015 Permalink | Reply

        I haven’t written it yet. I’m open to suggestions or to bribes, if there’s enough money in the Vandermonde Identity community.


  • Joseph Nebus 3:00 pm on Tuesday, 23 June, 2015 Permalink | Reply
    Tags: 5280, , factors, , miles,   

    Missed A Mile 

    I’m honestly annoyed with myself. It’s only a little annoyed, though. I didn’t notice when I made my 5,280th tweet on @Nebusj. It’s one of those numbers — the count of feet in a mile — that fascinated the young me. It seemed to come from nowhere — why not 5,300? Why not 5,250? Heck, why not 5,000? — and the most I heard about why it was that was that 5,280 was equal to eight furlongs. What’s a furlong, I might wonder? 5,280 divided by eight is 660, which doesn’t clear things up much.

    Yes, yes, I know now why it’s 5,280. It was me at age seven that couldn’t sort out why this rather than that.

    But what a number. It had that compelling mix of precision and mystery. And so divisible! When you’ve learned how to do division and think it’s fun, a number like 5,280 with so many divisors is a joy. There’s 48 of them, all told. All the numbers you see on a times table except for 7 and 9 go into it. It’s practically teasing the mathematically-inclined kid to find all these factors. 5,279 and 5,281 are mere primes; 5,278 and 5,282 aren’t nearly so divisor-rich as 5,280. Even 1,760, which I knew well as the number of yards in a mile, isn’t so interesting. And compared to piddling little numbers like 12 or 144 — well!

    5,280 is not why I’m a mathematician. I credit a Berenstain Bears book that clearly illustrated what mathematicians do is “add up sums in an observatory on the moon”. But 5,280 is one of those sparkling lights that attracted me to the subject. I imagine having something like this, accessible but mysterious, is key to getting someone hooked on a field. And while I agree the metric system is best for most applications, it’s also true 1,000 isn’t so interesting a number to stare at. You can find plenty of factors of it, but they’ll all follow too-easy patterns. You won’t see a surprising number like 55 or 352 or 1,056 or 1,320 among them.

    So, I’m sorry to miss an interesting number like that for my 5,280th tweet. I hope I remember to make some fuss for my 5,280th blog post.

    • Ken Dowell 4:02 pm on Tuesday, 23 June, 2015 Permalink | Reply

      You had me going to my Twitter page to see if I also missed the one milestone. But alas I’m only around 2700. Since that’s about 5 years worth I might need a reminder around five years from now to look out for 5280.


      • Joseph Nebus 3:25 am on Thursday, 25 June, 2015 Permalink | Reply

        It’s a shame there’s no setting warnings about how many tweets you’ve made, or alarms to make sure the significant ones are credited so. Maybe if I ever make my silly little Twitter app …


    • dianafesmire 4:42 pm on Tuesday, 23 June, 2015 Permalink | Reply

      I was also thinking about 5280 this week while I was at the horse races with friends. We were trying to figure out how long the races were. They were listed in the program in number of furlongs. I had no idea that 5280 had so many factors and the numbers on both sides of it are prime – Wow! I think I will pose the idea to my sixth graders of finding all the factors the first few weeks of school this year while we are in our Prime Time unit. Thanks for the math! Got my morning off to a great start!


      • Joseph Nebus 3:29 am on Thursday, 25 June, 2015 Permalink | Reply

        I’m delighted you enjoyed, and glad I could give a fun puzzle to play with. It’s not exactly coincidence that 5,280 should have so many factors. The length of “a mile” can be a pretty arbitrary thing — see how much variation there is in the length of “miles” across different unit systems — so why not pick a unit that can be divided lots of ways as need be?


    • sheldonk2014 5:58 pm on Tuesday, 23 June, 2015 Permalink | Reply

      I know I get excited in a low level way,like when I get a bag if new socks but a number you got this one Joseph


      • Joseph Nebus 3:30 am on Thursday, 25 June, 2015 Permalink | Reply

        Well, now … how many socks? Six — a perfect number — or maybe twelve — an abundant number — perhaps?


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