This morning I was reading an article on Mathematics Education and did a double-take when I saw this quote.

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.

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.

]]>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.

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.

]]>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.

]]>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.

Here’s the link

]]>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.”

]]>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.

]]>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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