Jack Morava sent me this delightful link:

Math teacher Ben Orlin writes and draws the (aptly named) blog Math With Drawings and is the author of a new book,

Math for English Majors: A Human Take on the Universal Language. To mark its publication, he devised this entertaining accompanying quiz.

A sample:

2. If you see someone write “0.33333,” how do you interpret this?(a) They clearly meant 1/3.

(b) They clearly meant 33,333/100,000.

(c) Looks like a fifth-iteration address in a generalized Cantor set. But I can’t tell the specifics without more context.

(d) It’s a coded message to my android brain. To escape the time loop, we must follow Commander Riker’s plan and decompress the main shuttle bay.

And at the end, you get to see what your results mean, e.g.:

Mostly B’s: Your style is

HYPERLITERAL.You delight in the pedantry of mathematical culture. This is a language of precision, every symbol’s meaning honed to a sharpened point—and you love wielding those sharp points to poke and prod speakers less precise than yourself.

In short, you’re a linguistic troll. Your literary equivalent is

David Foster Wallace.

As a former math major, of course I was tickled (I ended up “normcore,” but I refuse to be equated with Dan Brown); I’ll bet the book is a good read. Thanks, Jack!

oh dear. looking at it makes me update my priors, evidently I’m a troll. oi.

I am everyone except Borges. Oh well …

Oh dear, I’m afraid I’m a hyperliteral.

But as a hyperliteral, the answers above gave me pause, as the response phrase “they clearly meant” was telling me that I am supposed to read the minds of our mostly mathematically illiterate population as they attempt to do arithmetic; in which case a) would indeed be correct, as a) is most probably what most people would have meant, even if they actually said the equivalent of b).

Not mathy myself, but it occurred to me recently that e.g. “zero point one repeating six” would be clearer than the “zero point one six (one six) repeating” that you tend to hear in speech. Do mathematicians have a convention for that?

@ adam : …. that you tend to hear in speech…

I’ve been told that math is a written, not spoken, language (supposedly like classical chinese?).

Can you put “zero point one six (one six) repeating” in an example/context? There’s not much

occasion for eight or nine significant figures in conversation?

If someone were speaking the decimal value of 1/6. People tend to say “repeating” after what’s repeating, which can leave it ambiguous what exactly is repeating.

Edit: Oh, sorry, ignore that “one” in the parenthesis. I said I wasn’t mathy.

0.33333 is the price of something in a place where the currency is divided into hundred-thousandths.

Slightly ashamed to say I watched that TNG episode just a few weeks ago.

@Adam

Do mathematicians have a convention for that?Yes, several. I went to a UK school, so was taught the Dots form. This was before age 11/everybody learnt it, “mathy” or not.

That wikip buries the critical for your question

As to pi day, what’s so special about three-fourteenths?

To write 0.33333 is fine, so long as the measurement in question justified all five of those digits. Oh, it wasn’t a measurement, it was the result of a mathematical calculation and the answer is 1/3? Then the way to write it in a mathematical context is 1/3. There’s a notion that the only acceptable way to express a number is as a numeral — with a decimal point and decimal places if it isn’t an integer. This is fine in some contexts, for example science and engineering. But mathematics is about the theory behind it, so it’s useful to know what the number actually is, rather than a decimal approximation to its value.

This is fine in some contexts, for example science and engineering.Finance and economics as well. If you have data tables, people frequently enter all values to the same decimal point; if they note all values to 5 decimal points, they write 1/3 that way. They may, in that case, also write an integer that way (e.g., 5.00000).

@ adam

I did misunderstand `repeating’ and see what you mean. It’s just that (?we) don’t have much occcasion to use high-precision numbers in conversation.

[An unusual feature of math, as a language, is that it has an unbounded set of pronouns: x, y, z… , their relatives X, Y, Z, \alpha, \beta, \gammma. ASL has something similar, IIUC, in that one faces an invisible table with invisible things to point toward for reference.]

This “dots” thing must be some crazy custom of furriners. Ordinary/modal people (by which I mean those who attended American public schools some decades ago) learned the notation convention of a horizontal line (macron?) placed over the digit or digits that repeat infinitely. https://byjus.com/maths/decimal-expansion-of-rational-numbers/ illustrates this approach if you scroll down to the examples under “Theorem 3.” (That website appears to be of Indian origin FWIW, so it’s not the U.S. versus the rest of the Anglophone world.)

But I don’t know how you would say it out loud … I suppose might utter the decimal expansion of 1/6 as “zero point one six six six et cetera.”

In Germany, the line is used as well. What you say is

Periode, e.g.,zwei komma drei Periodeis 2,333333….zwei Komma eins drei Periodeis ambiguous between 2.13333… and 2.131313… To avoid that, one can insert “Periode” before the sequence that is repeated (that’s what I do) or insert a pause before the repeated sequence.There’s a notion that the only acceptable way to express a number is as a numeral — with a decimal point and decimal places if it isn’t an integer. This is fine in some contexts, for example science and engineering.As a physics teacher, I’ve never heard it. I do tell my students that decimals are usually better, but classical kinetic energy is still mv^2 / 2, not 0.5mv^2 / 2.

As for the style test, I’m the fifth kind, which I don’t think is that rare for mathy people: can’t agree with any of the answers.

Your style is

NEGATIONIST.You refuse all options. Your literary equivalent is

Bartleby.I have seriously considered adopting your suggestion, but I would prefer not to.

indeed

Not a language point but of possible interest to math enthusiasts, active or retired. I just by sheer coincidence discovered that the infinitely-repeating decimal expansion of the rational number 38/162 (or 19/81, if you like but there was a reason I was focused on the former) is quite interesting: there’s a sequence of nine-count-em-nine digits before it repeats and that sequence is 234567901. I don’t know whether there’s a different rational number where the decimal expansion has a repeating sequence of 10 digits that’s all “in order” and without repetition. But for all I know there’s some weird website where you can type a sequence of digits into a box and be told whether or not if repeated indefinitely it forms the decimal expansion of a given rational number. Or is there some proof by which essentially any string of digits you can think of must form, if infinitely repeated, the decimal expansion of *some* rational number. FWIW the expansion of 1/81 is 012345679 repeated infinitely. Again, the missing 8!

Yes, the string of digits

abcd, infinitely repeated, is the decimal expansion ofabcd/9999. If the string in the numerator hasndigits, there aren9s in the denominator. That’s not too hard to prove if you remember the formuia for the sum of a geometric series.@J. W. Brewer

There’s a straightforward method for extracting the rational number equal to -any- repeating decimal, but this blog’s margins are too narrow for me to write it out.

@ J.W.B ,

[I’m afraiid I’ll have to get back to you on that, I’m more on the knotty way of mathy]

BTW This

https://en.wikipedia.org/wiki/The_Sand_Reckoner

is really quite interesting; the original is only eight pages.

It was a serious intellectual problem in those days, as to how to write large numbers. There’s an ancient (Babylonian/Chaldean?) literature of number puzzles (\eg the Oxen of the Sun)

with enormous answers essentially impossible to speak or transcribe. Our man Archi invented a technology solving the problem and wrote it up with panache. I think it’s a significant entry in the history of human literature.

On balance, though, maybe a reasonable person standard would suffice for the spoken forms (and for the written forms too, if you’re using the dots): you could set as a norm that whatever would appear to be the most likely repeated part

mustbe the repeated part. So 1/6 would be “zero point one six six repeating”, and 16/99 would be “zero point one six one six repeating” – and “zero point one six repeating” would be written off as unsayable since it hasn’t established a pattern yet.@Jerry Friedman: Thanks for such a straightforward explanation. I feel there’s maybe a 50% chance I even knew that back in the early Eighties during my successful career competing in high school math league but then forgot it. One implication is that absolutely any prime number (other than 2 and 5, I guess?) you can pick will turn out to be a prime factor of some integer that’s nothing but a string of 9’s, although potentially a very long one. E.g., 999,999 is divisible by 7.

@jack moravia

Questions about how to write and define large numbers and rapidly increasing series are a modern problem as well— the field is called ‘googology’, and there is a wiki devoted to it:

https://googology.miraheze.org/wiki/Main_Page

A teaser for why this can be fascinating: there is a sequence (the so-called Busy Beaver) that increases faster than -any- algorithmic sequence. It was in the news recently because the fifth term in the BB sequence has been (finally) verified.

@J.W.B.: Glad it was helpful. Indeed every positive integer

nis a factor of a number consisting only of 9s, and clearly (in this case meaning “I’d have to look up the proof”) the number of 9s is less thann. In your example of 7, the number consisting of 9s has the maximum length.“zero point one six (one six) repeating”

@adam, JWB, Hans, thanks!

I always wondered how people say it in other langauges (but I simply did not know that they even write it differently).

In Russian it is 3,1234343(3) or “…one-two-three-four-three-four-three and three в периоде” (or “1234343 ten millionths and three в периоде” if you prefer).

В периоде means ‘in period’.

I have no idea if it is usually 3,(3) or 3,3(3) or 3,33(3) (3,1(3), 3,13(3), 3,133(3)) or what. I say 3,3(3), 3,13(3).

@ MattF

Thanks, I was aware of the Beaver via Scott Aaronson, but not of the weblit; thanks for the links.

The Msya and others could handle large numbers, they are said to have kept their cocoa records in beans.

For many purposes cyclic counting (\eg by days of the week or hours of the day) can be used to deal with large numbers in terms of their position in several cyclic cycles, \eg modulo 13 and 20 specifies a date in a 260-day cycle. Balinese are said to have several concurrent systems of `weeks’…

@Jerry Friedman: Shurely the vast array (50% of the whole infinite set) of *even* positive integers are not factors of any string-of-nines integer? And I’m equally dubious about 5.

WHY??? is Dan Brown the normcore guy (asks a normcore)?

@J.W.B.: Good point. I should have said all integers that are relatively prime to 10.

I have to agree with Jerry. I don’t much care for any of the answers, but then again, this is hardly the first quiz, where I’ve wondered about the sanity/expertise of the writer.

I was planning to be a math major at Princeton too but wound up being lured away by the siren song of “engineering physics”, which proved to be illusory, and I wound up doing a chemical engineering degree instead, which frankly I hated.

Anyway, nowadays, I have a subscription to watch this guy run through mathematical proofs: https://www.youtube.com/@DrBarker/videos

He seriously looks too young to have a PhD, but I’m pretty sure he does.

There’s obviously not enough input from my side of the pond here: in the UK at least the repeating(sic) digit(s) are called recurring, and have a dot above them:

https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/numeracy/decimals-and-rounding.html

I remember learning to put a dot above recurring numerals in math(s) class somewhere along the way. I never used the convention or saw it since then, as best I can recall.

https://en.wikipedia.org/wiki/Repeating_decimal#Notation

Ah, I had not dug deep enough to learn that “vinculum” was the name or at least a name (don’t think I ever heard it from a math teacher) for the U.S-among-other-places convention I learned as a boy.

@JWB, but how expansion of N/999… to arbitrary periodic sequence and divisibility of 9999…9 by any given prime number are connected?

@drasvi: For any prime number x, 1/x is by definition a rational number and for x>5 one with an infinitely-repeating decimal expansion. Thus per the one-weird-trick pointed out upthread by Jerry Packard, there must be a string-of-9’s integer of which x is one of the prime factors.

JWB, thanks!

I understand why both claims are true, but some idiotic mental block was not letting me connect two true statements:) A more parsimonious explanation is that I’m just dumb.

@J.W. Brewer: Jerry Packard is much more distinguished than I am.

My sincere apologies to all Jerrys concerned for the unfortunate mix-up! But here at the Hattic Caucus everyone is distinguished and all must have distinctions.

Clearly: The number of repeating (decimal) digits in 1/nis the order of 10 in the multiplicative group modulon, of course, which exactly dividesn-1 ifnis prime (because field; 2 and 5 are special cases), but may be smaller (like 1 repeating digit for 1/3 and 2 for 1/11; for 1/7 it happens to be 6). That applies to any integer number base greater than one. (I don’t remember how unary fractions work today; I can do Gaussian integer bases if I take a deep breath and scribble a little on a napkin [1+i is fun], but a properly rational number base sounds scary).The decomposition of the multiplicative group is funky, but you end up with a product of cyclic groups and taking the gcm of the order of 10 in those will given you the number of repeating digits. I think. And you get the larger of the exponents of 2 and 5 as the number of prefix digits that aren’t necessarily part of the repeat (though it might look as if they are), as you “fall into” smaller multiplicative groups until you reach something that’s relative prime to 10. Luckily, a lot of the funkiness is with powers of 2, but 3 is also gnarly.

And JWB meant “for any n with prime factors not either 2 or 5” not “for any n > 5”. cf 1/10.

I remember knowing a calculator game 50 years ago where no matter what number you started with, you’d eventually get 12345679 (which is 999999999/81 = 111111111/9 as JWB noted).

Is there an explanation for why this 12345679 is what it is?

“I remember knowing a calculator game 50 years ago where no matter what number you started with, you’d eventually get 12345679” – exactly the kind of stuff young people of today expect to hear (or I hope so) from people who are not as young as them.

4 Mind-Blowing Calculator Trickshas this: multiply any one-digit number, like 7, by 9, and then by 12345679. and you’ll get 777 777 777. (Wow indeed!) It was more wow when I was a high school freshman (and calculators had 9 digits), but I admit it was a bit different than what I hinted.(You create 12345679 by stringing 1s together until “casting out nines” gives you zero, i.e., 111 111 111, and then dividing, guaranteed to give a whole number. You can also get 111 111 = 7 x 15873 but it doesn’t look as magical or memorable; note that this has the same length as the repeat of 1/7. That is not a coincidence, but then nothing is ever a coincidence. [Something something 3 and 7 are mutually prime, so the repeat 142857 = 999999/7 is also divisible by 9]. Also 13*8547).

The number of repeating (decimal) digits in 1/n is the order of 10 in the multiplicative group modulo n, of course, which exactly divides n-1 if n is prime (because field; 2 and 5 are special cases), but may be smaller (like 1 repeating digit for 1/3 and 2 for 1/11; for 1/7 it happens to be 6).Clearly (this time meaning that I looked it up), the number of repeating decimal digits of 1/n is a factor of φ(n), the number of integers less than n that are relatively prime (or coprime) to n. φ(n) is also called the totient function, because reasons.

I’ll use k for the number of repeating digits. What J. W. Brewer and I pointed out above is that n is a factor of 99…9, where there are k 9’s. (Somewhere a dog is barking.) That is n divides 10^k – 1. So 10^k is 1 greater than a multiple of n, or in other symbols 10^k ≡ 1 (mod n), which for those following at home is what you meant by saying k is the order of 10 in the multiplicative group modulo n. Right?

multiply any one-digit number, like 7, by 9, and then by 12345679. and you’ll get 777 777 777If that were true, then 5 = 6.

5 × x = 6 × x ==> 5 = 6

φ(n) is also called the totient function, because reasons.#

The now-standard notation[8][11] φ(A) comes from Gauss’s 1801 treatise Disquisitiones Arithmeticae,[12][13] although Gauss did not use parentheses around the argument and wrote φA. Thus, it is often called Euler’s phi function or simply the phi function.

In 1879, J. J. Sylvester coined the term totient for this function,[14][15] so it is also referred to as Euler’s totient function, the Euler totient, or Euler’s totient.

#

Sylvester doesn’t explain his choice of “totient”.

#

where p1. (p-1) is what is commonly designated as the function of p3, the number of numbers less than pi and prime to it, (the so-called Φ function of any number I shall here and hereafter designate as its function and call its Totient)

#

A few years ago I skimmed a long work of his. His prose style is stiff and stately, crawling with cosmopolitan mix-and-match, all the accent marks being in their proper place:

#

We may, however, prove the fact in question, on a certain hypothesis to be presently stated, by availing ourselves of the knowledge that R is, to a numerical factor près, the product of the differences between the roots of ƒ and those of …

#

Of course we can’t all write like Henry James.

OED (entry from 1913) just says “irregularly < Latin

totiēs, totiens, <totso many, after quotientn.” Not very helpful.I like the lawyerly, all-encompassing “I shall here and hereafter designate”. No wriggle-room there for smart-ass temporal casuistry.

5 × x = 6 × x ==> 5 = 6Where 9 × 12345679 = x > 0

The Wikipedia article is wrong, BTW. It says the repetend is put in parentheses in Austria, but I didn’t even know until now that any convention other than dots existed – with dots on every digit of the repetend, not just the outermost one.

Said there with

periodischafter or, IIRC, occasionally withPeriodebefore the repetend.First-order oodle theory! English does have an advantage.

Because he’s mathematical: “Renowned author Dan Brown staggered through his formulaic opening sentence.”

@Stu, if you do it with 5 you get 555555555 and with 6 you get 666666666. I short circuited a little, I don’t know if Wittgenstein would have approved. I.e., your proof is correct but you cannot blame me for the antecedents. (Or rather, I will not take the blame).

Kepler’s Apostrophe (J.J. Sylvester)

Yes! on the annals of my race,

In characters of flame,

Which time shall dim not nor deface,

I’ll stamp, my deathless name.

The fire which on my vitals preys,

And inly smouldering lies,

Shall flash out to a meteor’s blaze

And stream along the skies.

Clafed as the angry ocean’s swell

My soul within me boils,

Like a chained monarch in his cell,

Or lion in the toils.

To wealth, to pride, to lofty state,

No more I’ll bend the knee,

But Fortune’s minions, meanly great,

Shall stoop their necks to me.

The God which formed me for command,

And gave me strength to rise,

Shall plant His sceptre in my hand,

His lightning in my eyes;

Shall with the thorny crown of fame

My aching temples bind,

And hail me by a mighty name

A monarch of the mind.

Me, heaven’s bright galaxy shall greet

Theirs by primordial choice,

And earth the eternal tones repeat

Of my prophetic voice.

Stung in her turn, the heartless fair

Who proudly eyes me now,

Shall weep to see some other share

The godhead of my brow;

Shall weep to see some lovelier star

Snatched to my soul’s embrace,

Ascend with me Fame’s fiery car

And claim celestial place.

Tune oh! my soul thy loftiest strain,

Exult in song and glee,

For worn has snapped each earthlier chain

And set the immortal free.

Minds destined to a glorious shape

Must first affliction feel;

Wine oozes from the trodden grape,

Iron’s blistered into steel;

So gushes from affection bruised

Ambition’s purple tide,

And steadfast faith unkindly used

Hardens to stubborn pride.

—

This is great stuff and was unfortunately never set to music. It is good that God decided to plant the sceptre in his hand and the lightning in his eyes and not vice versa.

I simply pointed out that the claim is false as stated. I don’t see the relevance of antecedents or Wittgenstein.

This is great stuff and was unfortunately never set to music.It would serve as the lyrics to a Sousa march.

So gushes from affection bruisedAmbition’s purple tide,

I take it that “purple” has the sense of “blood-red”, which was mostly poetic in English (it was still around in poetry in Sylvester’s time, according to the OED) but might be normal in some other languages, as mentioned in this recent thread.

The OED’s only sense for rhetorical “apostrophe” is the familiar one of addressing someone directly, which doesn’t fit here. I wonder what Sylvester meant. But I don’t know the Greek term for “extended whine about the foreseeable consequences of one’s decisions”.

Mathematics is not a book confined within a cover and bound between brazen clasps,

whose contents it needs only patience to ransack; it is not a mine, whose treasures may

take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze: it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

J J Sylvester, Address on Commemoration Day at Johns Hopkins University.

(telling it like it T. I. S.)

The first line (about a book and patience) reminds me my complaint about school educaton (which once drawn me into an idiotic conflict about creationism vs. evolutionism*).

That is, for me science is about questions rather than answers and presenting it as a corpus of answers which in principle can be “learned and known” makes it boring.

____

* I said that fight will creationism and desire to

convincechildren that evolution is true may negatively affect the school course of biology. I meant this above sense: make it both less factually accurate and interesting. Somehow this was taken as a support of creationism whcih led to an idiotic long scandal.It just now occurred to me to wonder what a good chatbot would do with Sylvester’s oration as a prompt, but it may be too big. I fear to try to find out.

WHY??? is Dan Brown the normcore guy (asks a normcore)?Same here.

It was in the news recently because the fifth term in the BB sequence has been (finally) verified.…and in the meantime it was discovered that we wouldn’t know the sixth term for sure until/unless the Collatz conjecture [or, rather, a different but closely related conjecture of similar-or-higher unapproachability] is proved.