Monday Morning Math: Leonardo Pisano (Fibonacci)

Good morning!  I hope you all enjoyed your Thanksgiving Holidays. Both boys came home for the weekend, which was a treat. =)

On Wednesday I got a text wishing me a Happy Fibonacci Day.  I had to think a minute: Nov 23, so 1123, from the sequence 1, 1, 2, 3, 5, 8, 13,….  And this inspired this week’s post about Fibonacci.  

Leonardo Pisano was born around 1170 in Italy, probably Pisa — hence the “Pisano” part of his name.  He was born to the Bonacci family — hence the  “filuis Bonacci” (abbreviated to “Fibonacci”) part of his name.  His father was a diplomat, and as a result of his father’s post Leonardo was educated in North Africa and traveled widely, which meant he was exposed to different number systems, including the base ten number system that we use today.  Indeed, it is likely that Leonardo himself is the reason we use it: he found it to be much better for calculation than the Roman number system (which would have used XXIII for a number like 23).  He returned to Pisa around 1200 and wrote several books that illustrated this system, the most famous of which is Liber Abaci (Book of Calculation – abaci is related to abacus).  Here’s a statue, by Giovanni Paganucci, of Fibonacci holding a book (CC license).

Although Fibonacci’s most significant mathematical contributions are related to his books sharing the decimal number system and methods of calculation with western Europe, he has become most famous because of a single problem that was in the book:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

This problem leads to the number system 1, 1, 2, 3, 5, 8, …, where each number is the sum of the previous two, which now bears the name the Fibonacci sequence in his honor.  Although maybe it shouldn’t – the sequence was known in India well before Fibonacci. By whom, you might wonder?  I started to write a brief summary, but realized I didn’t know enough about the history myself to do it justice so that will have to wait for next week…

Sources: 

• http://www-groups.dcs.st-and.ac.uk/history/Biographies/Fibonacci.html
• “The so-called fibonacci numbers in ancient and medieval India” by ParmanandSingh, Historia Mathematica, Vol. 12, Issue 3, August 1985, pp. 229-244 https://www.sciencedirect.com/science/article/pii/0315086085900217?via%3Dihub

Happy Thanksgiving!

(This was originally published on November 27, 2008, but I ran across it in looking up Thanksgiving Math and thought it might be good to update the links and repost!)

While you’re baking your turkey and pondering why the temperature continues to rise even after the bird comes out of the oven or thinking about how Game Theory showed that tryptophan affects trust and cooperation, you might also be wondering, “What kind of math did the folk at the first Thanksgiving study?”

The answer is…I don’t know. So I looked around at what textbooks would have been available. I couldn’t find any math texts of the Wampanoags from around 1621, and while I was enticed by a reference to “our earliest native American arithmetic, the Greenwood book of 1729” from David Smith’s History of Mathematics Vol.2, in looking further it seems that the phrase “native American” was used only in contrast to having been originally published in England, Spain, or other non-American country. So as far as the Wampanoags are concerned, I’ll admit to remaining in the dark about what specific kinds of math they would have been learning (formally or informally).

Then I looked into the Pilgrims. As they hailed from England, it seems likely that they’d be using a an English text. And for this they had few choices: there was a 1537 book entitled An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym [that last word is Algorithm], which was a translation of a book in Latin by Luca Pacioli. Wikipedia mentions that another textbook was published in 1539, but I couldn’t find reference to the title or author (though I suppose I could add one myself).

A more likely candidate would be Arithmetick, or, The ground of arts by Robert Recorde, first published in 1543 and listed in several places as “The first really popular arithmetic in English”. The title is short for The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions. Here’s a woodcut from the 1543 version

and here’s the title page from the 1658 edition, with its fancy modern spelling:

(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)

I couldn’t track down any copies of what exactly was in the book, but this article about medicine in English texts quotes quite a bit from Recorde’s book. For example:

In 1552, Record amplified his arithmetical text with, among other material, a “second part touchyng fractions”. This new section included additional chapters on various rules of proportion, including “The Rule of Alligation”, so-called “for that by it there are divers parcels of sundry pieces, and sundry quantities alligate, bounde, or myxed togyther”.

The rule of alligation, we are told, “hath great use in composition of medicines, and also in myxtures of mettalles, and some use it hath in myxtures of wynes, but I wyshe it were lesse used therein than it is now a daies”. Despite Record’s regret about the adulteration of wine, the first problem the Master uses to exemplify his discussion of alligation in fact deals with mixing wines; the second involves a merchant mixing spices; and the remainder involve the mixing of metals. None of the examples involve medicine, although the merchant’s spices are once called “drugges”.

A third part was added in 1582 by John Mellis. Here’s a page (from Newcastle University) showing one of the pictures in the margin (possibly from that year, or possibly a later edition; I couldn’t tell for certain).

Robert Recorde is actually better known for a later book: The Whetstone of Witte, which was published in 1557 and so is another candidate for An Original Thanksgiving Math Text.  It was called because, according to the poem on the front, “Here, if you list your wittes to whette, Moche sharpenesse therby shall you gette,” and appears (using Google books) to have been well-known enough to be referenced by Shakespeare in Act 1 Scene 2 of As You Like It (written around 1600) when Celia says, “…for always the dulness of the fool is the whetstone of the wits. How now, Witte! whither wander you?”

The Whetstone is best known for its introduction of the equal sign =, which Recorde explains below:

Which reads as

Nowbeit, for easy alteration of equations.  I will pro(?provide?) a few examples, because the extraction of their roots, may the more aptly be wrought.  And to avoid the tedious repetition of these words “is equal to”, I will set as I do often in work use a pair of parallels or Gemowe lines of one length, thus: ======, because no two things can be more equal….

So there you have it: math that some of the first Thanksgiving folk could possibly have studied.

Monday Morning Math: Goldbach’s Conjecture

Good morning!  Today we’ll talk about something we don’t know.  Or, rather, something we don’t know *for sure*, because it hasn’t been proven (yet?)

It’s possible to write numbers (bigger than 1)  as a sum of prime numbers. For example, 14 is equal to 3+11. The question is, what’s the smallest number of prime numbers that you need?  Well, 280 years ago the German mathematician Christian Goldbach wrote to the Swiss mathematician Leonard Euler about this very thing:

He wrote in this letter – specifically, in the margin sideways – that he thought you needed at most three primes.  Euler replied and mentioned an earlier conversation where Goldbach had thought that the even numbers needed only two primes, which would imply what Goldbach mentioned in this letter. 

There’s a little bit of squirreliness in terms of what a prime number is, since Goldbach and Euler considered 1 to be prime (which is why there are all those 1s in the letter), but with a little bit of modification the two questions are:

• Can every odd integer [>=9] be written as the sum of three odd primes?
• Can every even integer   [>=6] be written as the sum of two odd primes?

The answer to the first question was proved to be YES less than ten years ago, by the Peruvian mathematician Harald Andrés Helfgott.  Yay!  But the answer to the second question remains “Probably”, so we will have to wait for another breakthrough.

Sources: 

• Wikipedia
• Wolfram MathWorld

Monday Morning Math: Code Talkers

Good morning!  November is Native American History Month: the website https://indigenousmathematicians.org/ highlights many mathematicians and was updated in 2021.  

Today’s math topic is codes – the kind where messages are encoded for secrecy.  During World War I, 19 Choctaw soldiers used the Choctaw language as a code for sending military messages in secret.  As described in the website for the Choctaw Nation of Oklahoma:

During the first world war, with the tapping of the American Army’s phone lines, the Germans were able to learn the location of where the Allied Forces were stationed, as well as where supplies were kept. When the Choctaw men were put on the phones and talked in their Native speech, the Germans couldn’t effectively spy on the transmissions.

Native Americans did not receive nationwide citizenship until 1924, yet the Choctaws were both patriotic and valiant, with a desire to serve in the war effort. Many Choctaw men volunteered in WWI to fight for our country. Choctaw Code Talkers of WWI were instrumental in ending war. Members of Choctaw and other Tribal Nations also served with distinction using Native languages in World War II, Korea and Vietnam.

The Choctaw Code Talkers

In World War II the Marine Corp recruited 29  Diné (Navajo) men to develop a much more complicated code based on their language:

The code primarily used word association by assigning a Navajo word to key phrases and military tactics. This system enabled the Code Talkers to translate three lines of English in 20 seconds, not 30 minutes as was common with existing code-breaking machines. The Code Talkers participated in every major Marine operation in the Pacific theater, giving the Marines a critical advantage throughout the war. During the nearly month-long battle for Iwo Jima, for example, six Navajo Code Talker Marines successfully transmitted more than 800 messages without error. Marine leadership noted after the battle that the Code Talkers were critical to the victory at Iwo Jima. At the end of the war, the Navajo Code remained unbroken. (from intelligence.gov)

Some of the Diné Code Talkers

In 2000 the Code Talkers were awarded with Congressional Gold Medals.

Sources:

• https://www.choctawnation.com/about/history/code-talkers/
• https://www.nationalww2museum.org/war/articles/american-indian-code-talkers
• https://www.cia.gov/stories/story/navajo-code-talkers-and-the-unbreakable-code/