When your publisher makes you look foolish

I once wrote a book chapter and when I opened the hardcopy, I discovered to my horror that the word “Mathematical” was misspelt twice in the opening sentence. Come on, my English is far from perfect but I wouldn’t spell Mathematical wrong! One was rendered “Mmatical” and the other “Maathethematical”, so somebody cut 5 letters from one and inserted them into the other. I checked that these errors were not in my submitted manuscript.

This morning I was reading an article on Mathematics Education and did a double-take when I saw this quote.
null
64 + 64 = 128, so how did the 545 come about? I got very distracted as I try to figure out what went wrong. I was thinking this might yet become fodder for the age old “math ed researchers can’t do math” argument. I then decided to find the original source, perhaps there was something mentioned prior to when the quote was extracted. But for all the power of google, I made no headway when I searched for 545 or anything with 545. Finally I thought that maybe the original source can be found in google books. Thankfully it is and it can be found here. The 545 was never in the original quote.

However, the mathematics still does not seem right. [tex] 128 \times 10 [/tex] is still off by a factor of one thousand. Common sense tells me that detectors does not add to the creation of the data points. What about the rotating rate? I found a white paper on LiDAR which is the system being described that contained the 1.3 million statistic. Again it does not explain how the number is calculated but I found another fact that says the lasers fire at the repletion rate of 20,000 Hz. So the data generated should be
[tex] 64 \times 20,000 \approx 1.3 \times 10^6 [/tex].
Now I am at peace.

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Hardy, Ramanujan and MacMahon

I came across a very well written article named Partnership, Partition, and Proof: The Path to the Hardy Ramanujan Partition Formula. One of the author Adrian Rice’s Major point (pun intended) was that MacMahon’s manual computation of p(200) which concurred with Hardy and Ramanujan’s estimate was a big driving force behind the completion of their work. Hardy and Ramanujan’s acknowledgement is quoted below.

To Major MacMahon in particular we owe many thanks for the amount of trouble he has taken over very tedious calculations. It is certain that, without the encouragement given by the results of these calculations, we should never have attempted to prove theoretical results at all comparable in precision with those which we have enunciated.

Nowadays instead of the services of Major MacMahons, we use Maple MatheMAtica. Interestingly, if you have watched “The Man Who Knew Infinity”, the relation between Hardy and Ramanujan, and MacMahon was portrayed as adversarial. Someone actually put the whole movie up on youtube. This part can be found here from 21:30 onwards to about 27:00. From another paper by Rice, Partnership and Partition: A Case Study of Mathematical Exchange‪, it seems that although MacMahon was made a member of St John’s college in 1904 but he only moved to Cambridge in 1922. In retrospect, perhaps this was the reason that Ramanujan was only introduced to MacMahon at such a late stage in the movie. Since we are discussing Ramanujan versus MacMahon (real or reel) it is fitting to read Andrews’ review of the movie in the notices of AMS, where he discusses the meeting in the movie between the two protagonists and gives a nice quote of Rota.

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e-pic day

Today, the Second Day of July in the year 2018, is a good day. No, an epic day if you write the date this way: 2/7/18. Yes, my erudite readers, the date does remind you of the mathematical constant often known as Euler’s number, e=2.718… . Around the world and in some local schools, 14th of March, is celebrated annually as Pi Day because the mathematical constant pi, usually denoted by the Greek letter [tex]\pi[/tex], equals 3.14… . Readers will no doubt recall having learnt in schools that [tex]\pi[/tex] is the ratio of the circumference of a circle to its diameter and appears in formulas like circumference, [tex]c=2\pi r[/tex] and the area enclosed by the circle, [tex]A=\pi r^2[/tex]. At this point, readers are urged to refrain from using the tired pun “this looks like Greek to me.” Pi Day is fairly well known, see for example the website www.piday.org, and we have another 256 days to prepare for the next Pi Day, so let us focus on [tex]\pi[/tex]’s less illustrious cousin e.
It is highly unlikely that many of us can afford to wait a hundred years for the date 2/7/18 to reappear, so let us make the most of this once in a lifetime opportunity to celebrate e. First, let us familiarize ourselves with e. Like [tex]\pi[/tex], it is also an irrational number. This does not mean that e behave unreasonably but rather, it means that e, as well as [tex]\pi[/tex], cannot be expressed as a ratio of integers. For [tex]\pi[/tex] in particular, this can potentially cause much confusion since primary school children are taught in mathematics classes to use the ratio 22/7 for [tex]\pi[/tex], which actually contradicts the very definition of an irrational number when that concept is introduced in secondary schools. It is generally problematic to define an object by what it is not. When children asks “what is a cat?” Please do not tell them it is not a dog. So perhaps a better way to describe an irrational number is this: if you write any irrational number in decimal form, the digits goes on indefinitely without any repeating patterns.
Now, you may ask where does e come from? The following example should be of interest. (Pun definitely intended.) Just last month, the US Federal Reserve raised interests rate which means that it is time to think about putting our hard earned money into banks. To simplify calculations, let us assume that our friendly neighbourhood bank promises us a staggering 100% interest per annum. So if we put a deposit of $1000, we will expect in one year’s time to get $1000 in interest, giving us a handsome sum of $2000. Suppose a competing bank claims they can do it better. They still pay the same interest rate per annum but they pay out interest every half a year. How much more would that be? After six months, your original $1000 earn $500 in interest. Things then get better for you because for the next six months, you should be earning interest based on the principal amount of $1500 and not $1000. Now 50% of $1500 equals $750, so you will earn a total of $2250 after one year if the bank pays interest every six months, albeit at the same annual rate. You earn an extra $250. This is the power of compound interest. What if a third bank offers the same annual rate with quarterly payouts? After three months, you get $1250 in total. In the next three months, you get 25% of $1250 which is $312.50 giving a total $1562.50. The calculations is getting complicated but you should persevere on since you are getting rich. Let us tabulate this. The interest rate is 100% or 1 times of the principal. So if the payout is every quarter, the interest rate is 25% or 0.25 = ¼ times of the principal.

[tex]\begin{array}{llll}
Period &Principal (\$) &Interest (\$) &Total Amount (\$) \\
\hline
Q1 & 1000 & 250 & 1000*(1+1/4) = 1250\\
Q2 & 1250 & 312.5 & 1250*(1+1/4) = 1562.5\\
Q3 & 1562.5 & 390.625 & 1562.5*(1+1/4) =1953.125\\
Q4 & 1952.125 & 488.28125 & 1953.125*(1+1/4) = 2411.40625
\end{array}
[/tex]

So we observe that even though the interest rate remains constant, by paying out twice as frequently, we increase our wealth from $2000 to $2250, and further to $2411.41 if the pay out is four times a year. A natural question would be how much more can we earn if the pay out is more frequent, say every month, week, day or hour! Before you start dreaming about what you can do with your new found wealth, you should know that even if a crazy bank does promise to pay you interest every hour or every minute, it will not quite break the bank. There is a limit to how much you will get. We can observe from the pattern in our previous tabulation that if the bank pays out [tex]n[/tex] times a year, the interest rate is [tex]1/n[/tex]. You can also expect to be paid a total of [tex]n[/tex] times and the final amount can be computed as:
[tex]1000(1+1/n) (1+1/n) … (1+1/n) = 1000 (1+1/n)^n[/tex]
The value of [tex](1+1/n)^n[/tex] gets larger as [tex]n[/tex] gets larger but there is a limit to how much it grows. It can never get beyond a special number, and … yes, you have guessed it, that number is e. Mathematically, as [tex]n[/tex] gets indefinitely large, the value [tex](1+1/n)^n[/tex] approaches e, i.e.
[tex]\displaystyle \lim_{n \rightarrow \infty} (1+1/n)^n = e.[/tex]
So no matter how frequent the bank pays interest, at 100% per annum you can never get more than $2719 in a year. If a bank is willingly to pay out 4822 times a year, then with compounded interest you will earn $2718.00 at the end of the year.

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5.714825714

I finally caught the Avengers: Infinity War on the big screen more than a month after it was released, in which time it had already became the fourth highest grossing films of all time. As of writing, it has surpassed 2 billion USD gross which, if we simplistically divided by 10, means about 200 million people (counting multiplicity) have viewed it. After watching the movie, I remembered how thrilled I was about the first Avengers movie. In the intervening years, I watched much fewer movies, only making it a point to always go to cinema for every star wars film. (Yes, I watched Han Solo the day before I watched Infinity War.) I caught Age of Ultron on a small screen and that was not so good. Dr Strange on a plane and missed Civil War and Thor 3 completely. It was only the thought that Infinity War was a significant movie where some superheroes die that and so should be a spectacle for the big screen that compelled me to spend $12 to watch it. The movie did not blow me away as there were literally too many things going on and the ending was sort of a downer with everybody going down too easily. I really thought that they would succeed in resisting Thanos but perhaps I was too naive and forgot that there would be a fourth avengers movie next year. Anyhow, I came back and started reading all sorts of commentaries and binge-watched youtube videos on all things Marvel and was reminded of how interesting and intricately connected everything was. The post titled is related to an article that ranked all 19 movies in the MCU.

This is probably the most interesting development of the entire rankings. Avengers: Infinity War and Black Panther ended up in a dead tie at number five. I need you to understand that when I say “tie,” I don’t mean that they were close and I rounded or anything like that. After doing the math, Infinity War and Black Panther both received the exact same average ranking of 5.714285714. Completely level after nine decimal places.

In fact, the average ranking would not only agree to 9 decimal places. The two numbers should be equal. A careful look should allow one to notice that 5714 is repeated and it is quite conceivable that the number is a repeating decimal with these digits 571428 going on indefinitely. A quick guess and a confirmation calculation tells me that the rank is in fact 40/7. This makes perfect sense and when I went back to the top of the article to find more information, I noticed that the poll consisted of ordinal rankings of the movies by 21 staff members. So the two movies must have received the same total rank of 120, which is not so Marvel-ous as a coincidence.

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Darwin on learning mathematics

During the three years which I spent at Cambridge my time was wasted, as far as the academical studies were concerned, as completely as at Edinburgh and at school. I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense

From the life and letters of Charles Darwin, also available here.

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Mathematics is a performance art,

but one whose only audience is fellow performers.

The above quote according to Hymen Bass in Mathematics and Teaching (2015) is due to Samuel Eilenberg.
The next quote, due to Bass himself, is on the approach taken by some mathematics educators to counter the “content only” type of teaching.

This approach poposes offering tangible and somehow “real-world” related mathematical activities with which to engage learners but leaving the development of mathematical ideas to largely unrestrained student imagination and invention. In such cases, the discipline of mathematical ideas may be softened to the point of dissipation.

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Google Books Ngram

Pak’s history of the Catalan numbers mentioned an Ngram chart. But perhaps because I wasn’t reading carefully or possibly due to the fact that I was reading the print copy and did not access url pointing to the said chart, I did not find out about this tool until today.

Here’s the link

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Faraday

Compared to mathematics, physics is extremely well-funded. For example, I have attended many talks by Nobel Laureates compared to Fields Medalists. (Locally, I have attended talks by Serre, S.T. Yau and Pierre Louis Lions. I somehow missed talks by Smale and Ngo Bao Chao. Overseas, I have heard Bhargava and Tao lecture.) But back to physics, in addition to the Nobel Lauretes, there have been many public talks on quantum mechanics, cosmology and particle physics, in relation to the LHC.

The latest was on 25th August by David Gross. He mentioned that Faraday, as an answer to the question what was his discovery good for, was said to have answered “you will be able to tax it.” A quick search on the web would reveal that this story is probably apocrypha, but still a good story. It was only a day later via a tweeter feed (it’s GMT +8 here) that I noted Faraday passed away on 25th August 1867, 150 years ago to the day!

It was certainly not the first time I heard the Faraday joke and conceivably I might have heard it from the same speaker but it also triggered my memory of another similar story. Through the modern marvels of ipad-photography (I took a quick snap of the slide and the photo had the date), google calendar and the internet I tracked it down to a talk by John Ellis on 19th Jan 2012 on the LHC. (A few months before the Higgs boson was observed.) The conversion with M. Thatcher and Ellis as follows:

Thatcher: “What do you do?”
Ellis: “Think of things for the experiments to look for, and hope they find something different.”
Thatcher: “Wouldn’t it be better if they found what you predicted?”
Ellis: “Then we would not learn how to go further!”

This rings well with David Gross’ own take;”The most important product of knowledge is ignorance.”

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What are the chances that a bird will poop on my car?

This question was posed as a scenario of mathematics in everyday life. Some googling led to Randall “XKCD” Munroe’s post here. Borrowing similar ideas leads to my own computation below.
I park at an open air carpark for work. The easy assumptions are my car has a 4.3m x 1.75m footprint and I park for 8 hours a day. For data on birds, after some searching I found a 2016 article that estimates (provided I interpreted correctly) a total of 377 birds in 113 hectares of built-up area at another university in the country. So if we assume a similar bird population here and that birds poop uniformly everywhere once every hour,
[tex] \frac{377}{113 \times 10,000} \times 8 \times 4.3 \times 1.75 \approx 2\% [/tex]

In my opinion, the estimate is on the low side. In practical terms, it appears birds spend a large amount on time perched on trees. I also read that birds tend to clear their vowels to lighten their loads before they fly. So it is a catch-22 for me. I either park in the shade under trees and risk being bombarded or I park in the open and find myself a 40 degrees oven when I leave work on a typical sunny afternoon.

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Queen of Mathematics

Number theory has been called the Queen of Mathematics. Until some fifty years ago, it did not occur to anyone that number theory, especially the study of prime numbers, would have any immediate applications to business. More recently, the Queen has been relegated to be the object of a courtship, inspired by material gains, rather than awe. As a result, progress has been made in unexpected directions, which have required deeper investigations. — Papa Paulo

A.K.A. Paulo Ribenboim from his pseudo-novel-number theory text “Prime Numbers, Friedns Who Give Problems: A Trialogue with Papa Paulo” (p. 50).

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Triangular numbers modulo powers of 2 and its generalizations

Someone discussed with me an interesting problem that he was working on with his students. They found that the congruence
[tex] \frac{1}{2}X(X+1) \equiv a \pmod{n} [/tex]
has a solution for every [tex] 0 \le a < n [/tex] if and only if [tex] n =2^k[/tex].

My first instinct of course was to complete the squares for triangular numbers and reduce the problem to [tex] X^2 \equiv a \pmod{n} [/tex].
This turn out to work well for odd modulus and the solutions for triangular numbers and squares correspond. But when the modulus was a power of 2, completing the square would not work. A simple search found a few websites where the phenomenon was recorded and it seems a (perhaps original?) source is Knuth’s the Art of Computer Programming Volume 3, Section 6.4, Exercise 20. Knuth was talking about hashing but essentially the exercise is the above problem. I prefer to rephrase it as for every positive integer k, these two sets are identical:
[tex] \{ \frac{x(x-1)}{2} \pmod{2^k} \} = \{0, 1, \ldots, 2^k-1 \} [/tex]

Knuth gave a slick proof which can be easily adapted and generalized to the following:
For a prime [tex] p, 1 \le m < p, k \ge 1,[/tex]
[tex] \{ px^2+mx \pmod{p^k} \} = \{0, 1, \ldots, p^k-1 \} [/tex]

Proof: Suppose
[tex] px^2+mx \equiv py^2+my \pmod{p^k} [/tex]
[tex] (x-y) ( p(x+y)+m) \equiv 0 \pmod{p^k}[/tex]
Since [tex] p \nmid p(x+y)+m [/tex] we can conclude that
[tex] x \equiv y \pmod{p^k}[/tex].

So each of [tex]x = 0, 1, \ldots p^k-1 [/tex] gives rise to a different value of [tex] px^2+mx \pmod{p^k}[/tex].

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If there is some good inflammable stuff it will catch fire

Do not try to satisfy your vanity by teaching them great many things. Awake their curiosity. It is enough to open the minds, do not overload them. Put there just a spark. If there is some good inflammable stuff it will catch fire.

The quote appears at the end of chapter 14 of Polya’s Mathematical Discovery. Polya attributes the quote to Anatole France from Le jardin d’Epicure. Perhaps he translated the French into English. He further adds: There is a great temptation to paraphrase this passage: “Do not try to satisfy your vanity by teaching high school kids a lot of … just because you wish to make people believer that you understand it yourself …” Yet les us resist temptation.

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Catalan numbers

I really enjoyed reading Federico Ardila’s article in the Mathematical Intelligencer. Apparently there was a vote of 3030 members at an assembly of CUP (Not Cambridge University Press but the Candidatura d’Unitat Popular). The vote had to do with forming an alliance with another party and ultimately related to the independence of Catalonia. The amazing thing that happened was that the vote came out 1515 Yes and 1515 No.

The probability that a YES-NO vote of 2m persons ends up in a tie is [tex] \binom{2m}{m}/2^{2m} [/tex], closely related to the Catalan number [tex] \binom{2m}{m}/(m+1) [/tex]. I love the dual connections and of course Ardila did not fail to mention Stanley’s Enumerative Combinatorics. What I did not know was that Stanley even included a joke. Made my day.

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Truncatable Primes

A colleague asked about sequences of primes a(n) such that a(n+1) is obtained by appending a single digit (in base 10) to the right of a(n).
For example: 3, 31, 311 …

Some thinking lead to the conjecture that such sequences are of finite length and that it is possible to use an exhaustive search to find all of them. A natural question would be what is the longest possible sequence but I was unable to find any conclusive answer on the web. So I decided to write a simple (and not very efficient) recursion in maple to search for all such primes. Here’s my ugly code:

cat3prime:= proc(n)
local d, s, i; s:=n; d:=irem(n,10);
if isprime(s) then print(s); return(cat3prime(10*s+1));
else for i from d to 7 by 2 do
if isprime(s-d+i+2) then s:=s-d+i+2; print(s); return cat3prime(10*s+1); fi; od;
while (irem(s,10)=9) do s:=(s-9)/10; od;
if s=0 then return print(“search complete”);
else return cat3prime(s+2); fi; fi;
end proc;

The search yielded five sequences of length 8 and no other longer sequences:
2,23,233,2339,23399,233993,2339933,23399339
2,29,293,2939,29399,293999,2939999,29399999
3,37,373,3733,37337,373379,3733799,37337999
5,59,593,5939,59393,593933,5939333,59393339
7,73,739,7393,73939,739391,7393913,73939133

Only one of the above sequences appears as is on OEIS but with a more careful search, I found a sequence called right-truncatable primes.

https://oeis.org/A024770

Which contains 83 primes, where successively truncating one digit from the right still results in a prime. I verified that my own maple search also yielded 83 primes and I guess the two lists must be identical. See the wikipedia entry on truncatable primes https://en.wikipedia.org/wiki/Truncatable_prime

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Twin corrections

Today is the 20th anniversary of the passing of Erdős and I would like to make two corrections. I had always thought the accent on Erdős’ name was ö , html code &#246 but it is actually Hungarian, html code &#337. The second is the coffee quote which I had attributed to him. I realised my mistake a number of years ago but never got a chance to correct it online. Both errors were perpetuated in this post from 2004. Here is a quote from Erdős’ paper “Child Prodigies”

In Hungary, many mathematicians drink strong coffee, in fact Rényi once said “a mathematician is a machine which turns coffee into theorems.”

Correction done but sadly I am still not quite sure how to pronounce his name correctly.

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Bollobas on solving problems

What you should be terrified of is a blank sheet in front of you after having thought about a problem for a little while. If after a session your wastepaper basket is full of notes of failed attempts, you may still be doing very well. Avoid pedestrian approaches, but always be happy to put in work. In particular, doing the simplest cases of a problem is unlikely to be a waste of time and may well turn out to be very useful.

Bela Bollobas, from Advice to Young Mathematicians

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Logical order

The most efficient logical order for a subject is usually different from the best psychological order in which to learn it

William Thurston from his book Three dimensional geometry and topology.

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One-seventh ellipse

Fun fact. It is well known that
[tex] \frac{1}{7} = 0.\overline{142857} [/tex].

It turns out that if the repeating digits are taken in sequence as (x,y) pairs in the following manner to form six points:
(1,4), (4,2), (2,8), (8,5), (5,7), (7,1),
then these six points actually lie on an ellipse defined by the following equation.
[tex] 19x^2+36xy+41y^2-333x-531y+1638=0 [/tex]

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Lakatos on Discovery

Discovery does not go up or down, but follows a zig-zag path: prodded by counterexamples, it moves from the naïve conjecture to the premises and the turns back again to delete the naïve conjecture and replace it by the theorem. Naïve conjecture and counterexamples do not appear in the fully fledged deductive structure: the zig-zag of discovery cannot be discerned in the end-product.

From his Proofs and Refutations.

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Notes from ICME13

Gila Hanna mentioned the carpet proof of the irrationality of [tex]\sqrt{2}[/tex]. A little digging reveals that it was due to Tennenbaum (1950s) and popularized by Conway (1990s). The original proof appeared in a book but the simple idea is described in this paper “Picturing Irrationality” by Steven Miller and David Montague in the Mathematics Magazine (2012). What is more interesting is the expanded version of their paper on arXiv called “Irrationality from the Book”. One can only infer the amount of editing and changes that took place from submission to publication.

This is a quote from Guershon Harel that I am trying to piece together from memory.

Some of us are constructing open neighbourhoods* but these neighbourhoods are in no way compact.

By neighbourhoods, he was referring to the impact of mathematics education on the curriculum.

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