## Author Archive

### Monday Morning Math: 𝝅

March 13, 2023

Tomorrow, March 14, is 3/14 [in month/year form] and so we celebrate Pi Day!  And since I’m teaching History of Math for the first time in rather a lot of years, I’m thinking that the perfect topic is the history of the symbol.

But first, what is 𝝅?  The idea is that no matter how big or small a circle, the circumference is always a little more than 3 times as large as the diameter:  that ratio is about 3.14 in decimal terms.  Because it’s a ratio, the first symbols were also written as ratios:  William Oughtred called it 𝝅 in 1647, and while he didn’t explain what either of the terms meant, since 𝝅 is the Greek p it likely stood for periphery (according to my source, though I think of perimeter myself when I see it); likewise, δ is the Greek d and likely stood for diameter.  Several other mathematicians adopted this notation.

The first person to use a single symbol to represent this ratio was Johann Christoph Sturm, who in 1689 referred to it with the letter e.  Wait, what?  (Double check.)  Well that is something I didn’t know before.  Cool!  But using e didn’t catch on, and less than twenty years later, in 1706, William Jones used the symbol 𝝅 for this same ratio. No explanation as to why, and also no consistency – he used the same symbol to mean other things earlier in the same book.  This use of 𝝅 also didn’t catch on: other mathematicians continued to use other symbols for the ratio of a circumference to the diameter, and 𝝅 itself continued to be used for different mathematical numbers.  But eventually, over the 1700s, its use caught on and so we have the well known symbol today.

Source:  A History of Mathematical Notations by Florian Cajori

### Sofia Kovalevskya

March 6, 2023

Good morning!  In honor of Women’s History Month we are featuring Sonia Kovalevskya, the first woman to receive a (modern) doctorate in mathematics.

Sofia Vasilyevna Korvin-Krukovskaya was born on January 3, 1850 in Moscow, Russia, although her birthday is sometimes listed as January 15 (the equivalent date in the Gregorian Calendar, which Russia adopted in 1918).  Her name, too, is written many ways: Sophie, Sofia, Sofya, Sonia.  She grew up literally surrounded by mathematics: the walls of her room were covered in her dad’s Calculus notes from when he was a student. She wrote later:

The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals.

Her parents were well off and she and her siblings had private tutors, but Sofia liked math so much that she ignored her other subjects and her dad put stop to the math studying.  Or tried to, at least – she studied on the sly after her parents had gone to bed.

Sofie wasn’t able to go to university in Russia, what with being female and all, so she married Vladimir Kowalevski (aka Kovalevskij or Kovalevsky) and they moved first to Austria and then to Germany where, because of her continuing femaleness, she still couldn’t take classes. She was, however, able to take private lessons from the mathematician Karl Weierstrass. She wrote several papers and with Weierstrass’s support [and influence at the University of Göttingen] was granted a well-earned doctorate.

Sophie had a daughter Sofia (who was called Fufa), moved multiple times, and after the death of her husband, became a professor at the University of Stockholm in Sweden. She wrote mathematical papers, non-mathematical works, and was recognized for her contributions even in her lifetime.  Short as it was – she was only 41 years old when she passed away from pneumonia, a complication of the flu.  Many schools have hosted Sonia Kovalevsky Days in her honor, bringing her love of mathematics to new generations.

Sources:

• The Potential to Inspire by Laura P. Schaposnick (written in verse in both English and Spanish)
• Wikipedia
• MacTutor, which itself referenced her autobiography: A Russian childhood: Sofya Kovalevskaya
• SK Days at the Association for Women in Mathematics

### Monday Morning Math: Primes

February 27, 2023

Good morning everyone!

I thought that there was no formula for prime numbers.  There are things that look like they will generate prime numbers, but don’t – for example, let $f(n)=n^2+n+41$.   At first glance this seems to work:

• $f(0)=0^2+0+41=41$, which is prime.
• $f(1)=1^2+1+41=43$, which is prime.
• $f(2)=2^2+2+41=47$, which is prime.
and this continues for a while…
• $f(39)=39^2+39+41=1601$, which is prime.
But then:
• $f(40)=40^2+40+41=1681=41^2$, which is not prime.

Bummer!  This formula appears to be due to Leonard Euler, a Swiss mathematician who lived in the 1700s and is the namesake for one of our cats.

But it turns out the world of prime formulas is more complicated than I’d realized, and there are formulas that work!  Sort of. The first is Wilson’s formula, named after John Wilson, a mathematician and judge who lived in England in the 1700s, although in doing a quick reference check I just saw that it was used 700 years earlier by Abu Ali al-Hasan ibn al-Haytham, who should really get his own Monday Morning Math. (Note made for the future.)  This formula can be written several ways, but the way I first saw it – earlier this month, when Q came and wrote it on my white board and said this would make a great topic for Monday Morning Math (Thanks Q!) is this:

$n+1$ is prime if and only if $n!\equiv n \mod (n+1)$

For example,

• When $n=1$ then $1!=1\mod 2$ and sure enough $1+1=2$ is a prime number.
• When $n=2$, then $2!=2\mod 3$ and $2+1=3$ is also prime!
• But when $n=3$ then $3!=6\mod 4=2\mod4 \neq 3\mod 4$ and $3+1=4$ is not prime.

This is a formula, but it’s perhaps more of a test for prime numbers than a formula for generating them.  For that we’ll turn to Willans’ formula, found by C.P. Willans in 1964:

When $n=1$ this formula produces $p_1=2$, which is the first prime.  When $n=2$ it gives $p_n=3$, which is the second prime.  And so on – you get all the primes this way!!!!  Pretty amazing.  You can watch a youtube video all about it here:

Happy priming!

Sources:  Wikipedia and Q

### Monday Morning Math: Katherine Johnson

February 20, 2023

Good morning! This week, on February 24, marks three years since Katherine Johnson passed away, and it seems a good opportunity to write about her.

Katherine Coleman was born on August 26, 1918, in White Sulphur Springs, West Virginia.  Her mother, Joylette, was a teacher and her father, Joshua, a farmer; she also had three older siblings: Charles, Margaret and Horace.  The school for African Americans in White Sulphur Springs only went through the 8th grade, so the family moved to where the kids could get more schooling.

Katherine’s father had been good at math and Katherine was too.  Very good. Indeed, she was quite good in many subjects. She skipped a few grades, started high school when she was ten, and was supported and encouraged by her family and teachers.  She graduated summa cum laude from West Virginia State College in 1937 with degrees in French and mathematics, and began teaching school herself at the age of 19.   Two years later West Virginia University began to integrate its graduate school, and Katherine attended classes for a time, and then married James Francis Goble.  They had three children: Constance, Joylette and Kathy, and after several years Katherine returned to teaching.

In 1953 Katherine and her family moved to Newport News, Virginia, so she could work in nearby Hampton at the Langley Research Center, which was part of the National Advisory Committee for Aeronautics (NACA) [which eventually became NASA]. She started as a computer, a person who performs mathematical calculations, but she and a colleague were soon assigned to what was supposed to be a temporary assignment with the (then all male) flight research team, where she worked for several years.

James Goble passed away in 1956 after a several-years battle with cancer.  Three years later Katherine married James A. Johnson, whom she had met through her minister. Throughout all this Katherine continued her work, performing calculations for Alan Shepard’s Mercury mission in 1961, John Glenn’s orbit around the earth in 1966, and the moon landing in 1969.  She continued working at NASA for more than 30 years, and during that time she co-authored a book on space and dozens of research articles, and continued work in many areas such as .  She also worked on the Space Shuttle program and an eventual mission to Mars.

Katherine Johnson retired in 1986 and in 2015 was awarded the Presidential Medal of Freedom by President Barack Obama. Around the same time the book Hidden Figures by Margot Lee Shetterly was published and became a major film, allowing many people to learn of all that Katherine Johnson had accomplished.  Her own daughters, too, followed in her footsteps: Constance and Kathy became educators, and Joylette a computer analyst at Lockheed Martin.

Katherine Johnson passed away on February 24, 2020, at the age of 101.  In addition to Hidden Figures there are several other biographies of her, including one that she herself wrote for children and young adults: Reaching for the Moon.

Sources:

### Monday Morning Math: Transcendental Numbers

February 13, 2023

Good morning!  Last week the Math Club made paper flowers and valentines for residents of a local nursing home.  Many of the valentines had math on them, like the graph of r=1-sin(θ) (seen here). If you wanted to make a mathy valentine for someone, you might write, “My love for you is like $\pi$ : neverending.”  But then you might get distracted thinking, well, technically all numbers are neverending.  Even a number like 1 could be written as 1.000000….  So you’d try to pick a different property of $\pi$, like the fact that it can’t be written as a fraction.  But calling your love irrational might not be what you’re going for.

Instead, you could call your affection for someone transcendental.  Because $\pi$ is indeed transcendental, a fact that was proved in 1882 by Ferdinand von Lindemann.  So let’s talk about transcendental numbers!  The actual definition is sort of a definition by exclusion – they are numbers that are not algebraic. OK, so what are algebraic numbers?  These are numbers that are the roots of polynomials where the coefficients are integers.  Here are some examples:

• The number 14 is algebraic because it’s the solution to $x-14=0$.  You could do something similar for any integer.
• The number 2/3 is algebraic because it’s the solution to $3x-2=0$.  You could do something similar for any fraction.
• The number $\sqrt{2}$ is algebraic because it’s a solution to $x^2-2=0$.  Oooh, and now things get interesting.  Because $\sqrt{2}$ can’t be written as a fraction, so this shows that a number that is irrational might still be algebraic.

Based on that last example you might guess that not just square roots but cube roots, fourth roots, etc. of integers are also algebraic.  And you’d be right.   And combinations of those are also algebraic, like the golden ratio $\frac{1+\sqrt{5}}{2}$ , because it turns out that the set of algebraic numbers is closed under addition, subtraction, multiplication, and even division (as long as you don’t divide by 0).  That means that if you add, subtract, multiply, or divide algebraic numbers, you get another algebraic number. It’s also closed under square roots, cube roots, etc.   This means if you write a number like $\sqrt{\frac43-\sqrt{\frac{2+\sqrt[5]{3}}{4+\sqrt{7}}}}$ it’s going to be an algebraic number and you don’t have to figure out what polynomial it’s a root of (although you can if you want).

But these aren’t the only algebraic numbers!  You could go complex and show that the imaginary number $i$ is an algebraic number because it’s a root of $x^2+1$, but even if you stick with real numbers there are algebraic numbers that can’t be written with the symbols described above.  For example, the polynomial $x^5-x-1$ has a root between 1 and 2 (since $1^5-1-1$ is negative, but $2^5-2-1$ is positive), and that root will automatically be algebraic, but it turns out that root can’t be written in a closed form, meaning it can’t be written just with +,-,x,/, and roots.  Isn’t that wild?

So going back to $\pi$, it turns out that $\pi$ is not algebraic, which means that it is transcendental.  The number $e$ is also transcendental.  In fact, most numbers are transcendental, in the sense that the set of algebraic numbers is countable but the set of transcendental numbers is not.  But it can be really hard to tell if a particular number is transcendental: we still don’t know, for example, if $\pi+e$ is algebraic or transcendental.  Then again, we don’t even know if $\pi+e$ is rational or irrational.  And with that, I suppose, you could even create a valentine for someone you’re not sure how you feel about.

### Monday Morning Math: Mathematically Gifted & Black

February 6, 2023

This post is a complication, about several mathematicians!  Every day during the month of February the site Mathematically Gifted & Black honors a mathematician.  For example,

February 1 featured Ruthmae Sears, an Associate Professor of Mathematics Education and the Associate Director for the Coalition of Science Literacy at the University of South Florida.  An article from the University of South Florida contains additional information about her:

As a mathematics educator, Ruthmae Sears has a true flair for problem solving. Her work extends far beyond using formulas and finding solutions to abstract mathematical problems. Using mathematical reasoning to examine social disparities such as poverty, literacy and structural racism, Sears develops community-centric solutions. Her work emphasizes inclusivity in all spaces, stemming from her belief that schools are microcosms of a community.

February 2 highlighted Clarence W Johnson, a Math Professor at Cuyahoga Community College in Ohio.  In that article he offers the following words of inspiration:

Never let other people convince you that you are incapable of succeeding. Draw strength from both the positive and negative actions of others. However, most of your success will be drawn from qualities within yourself.

February 3 recognizes Ayanna Perry the Associate Director for the Teaching Fellows Program at Knowles Teacher Initiativewith in New Jersey.  There is more information in this article, including a link to an article in Mathematics Teacher entitled “7 Features of Equitable Classroom Spaces”  that I just requested through InterLibrary Loan.

You can come back each day to learn about a new person, or browse previous years to learn even more, including the 2018 Nominee Clarence Francis Stephens, who for several years taught just down the road at SUNY Geneseo.

Happy Black History Month!

### Monday Morning Math: More Math and Music

January 30, 2023

Good morning! This week we have a bit more info about music (thanks TwoPi!):   Last week I mentioned a “just fifth”, which is when the frequencies of two notes are in a 3:2 ratio: A is 440 Hz (on a piano at least) and the note E above that is at 660Hz.  These two notes are a perfect fifth, and the ratio 660:440 is a 3:2 ratio.

The frequencies in an octave, on the other hand, are in a 2:1 ratio:  The A above the one with 440 Hz has 880 Hz.  So the work that Jing Fang did, showing that 53 fifths was almost exactly 31 octaves, amounted to showing that $\left(\frac32\right)^{53}$ (which is approximately 2,151,972,563.22) is very close to  $2^{31}$ (which is 2,147,483,648).  If you want to avoid fractions, this means that $3^{53}$ is almost the same as $2^{84}$.  They won’t be exactly the same, but it’s pretty close.

But let’s go a little more in depth!  The savvy reader (that would be you) might realize that we can’t ever have a power of 3 exactly equal to a power of 2, because they are different prime factors.  This means we’ll never have a power of 3/2 equal to a power of 2.  This is a fundamental problem with piano tuning:  if you prioritize fifths then the octaves don’t exactly match up, and if you prioritize the octaves then the fifths don’t exactly match up.  Piano tuning today prioritizes the octaves, and uses the fact that 12 fifths is approximately the same as 7 octaves.  That is, $\left(\frac32\right)^{12}$ (which is approximately 129.7) is very close to $2^7$, which is 128.

Isn’t that cool?  If you want to hear some of it, here’s a 3-minute video of mathematician and musician Eugenia Cheng giving more detail and connecting it to Bach and to Category Theory:

And if you want even more music, here’s some music played on a Mobius strip, just for fun:

(And there are more videos on the AMS page on Math and Music.)

### Monday Morning Math: Happy New Year!

January 23, 2023

Good morning everyone! Happy 2023!    Today’s post is in honor of two new years.

The first is the new calendar year: 2023.  If you want to do something fun (which of course you do) then you can see if you can use exactly the digits 2, 0, 2 and 3  and different math configurations to write the numbers 1-100.  For example: $1=\frac22+0\cdot3$.  Or  $1=2\cdot0+3-2$.  (There are examples all over the internet, so a quick search reveals many solutions should you wish.)

The other is the Lunar New Year, which began yesterday.  This is also known as the Spring Festival, and is observed around the world, including China, Indonesia, Japan, Malaysia, the Philippines, Singapore, South Korea, Taiwan, Thailand, and Vietnam: we are now in the Year of the Water Rabbit in many countries, and the Year of the Cat in Vietnam.

In honor of the New Year we’ll talk about the mathematician Jing Fang  京房.  He was born in China 2100 years ago (78 BCE, during the Han Dynasty).  He was a mathematician and, appropriately enough for this post, he described astronomy – the solar and lunar eclipses.

But these weren’t his only accomplishments.  He was also very good at making predictions using the Yijing, or I Ching. There are 64 hexagrams, each made up of 6 rows that have one long or two short marks in each row.  While this isn’t mathematics, it does lead to the math question to ponder: namely, can you explain why there are exactly 64 configurations?

And his math-connection doesn’t end there either – he used mathematics to describe music theory, particularly that 53 fifths (technically “just fifths”, which may or may not be the same as a perfect fifth – but you can here one here) was almost exactly 31 octaves.  It took more than 1600 years for anyone (said anyone being Nicholas Mercator) to caculate the difference between the two more exactly than Jing Fang.

Math Moons and Music – a good way to start the year.

Sources: Wikipedia and…just that, because the few other sources I found had the same information.

### Monday Morning Math: the Winter Solstice

December 26, 2022

December 12 was going to be the final Monday Morning Math of the semester, but finals had started two days earlier and with one thing and another (well, really just one thing – the aforementioned final exams), it didn’t happen.

It felt a bit odd to take a break without announcement, however, so here is one final MMM for 2022.

(Arriving on a Monday at least, even though it’s not quite morning anymore.) And the timing is perhaps good for a math-adjacent topic: the Winter Solstice, which happened on Wednesday, December 21.  This is the shortest day of the year in the northern hemisphere, with just under 9 hours of daylight (technically 8 hours, 59 minutes, and 10 seconds) here in Rochester.  But there are two things about the solstice that I find interesting mathematically.

The first is that if you google “When is the Winter Solstice?” you get not just a day but a time: 4:47pm here.  This feels a little weird to me if I think about it being a day, but it has to do with northern hemisphere being tilted as far away as possible from the rays of the sun, as in this tweet from NASA below:

Or, if you want to envision the Earth with its axis vertical, it’s moving along a plane that is not horizontal, as explained on NASA’s blog

This is the image that makes the idea of a moment for the solstice make sense to me: it’s at the very peak of the ellipse, and that happens at one particular moment rather than a full day.

The second math adjacent thing is about subtraction.  You might think that the shortest day must have the latest sunrise and the earliest sunset, but actually neither is true: the lastest sunrise doesn’t happen until January 3, 2023, at 7:42am, which is about 3 minutes later than it rose on Dec 21.  So that’s kind of a bummer, in terms of how dark the mornings are.  But that’s compensated by the fact that the earliest sunset happened several weeks ago, back on December 9.  The sun set at 4:35pm that day, about 3 minutes earlier than it set on the solstice.  The sunrises and sunsets don’t quite change symmetrically, and that’s why the shortest day is about halfway in between.

Happy Solstice and Happy Holidays to everyone! We’ll start again in about a month – see y’all in 2023!

### Monday Morning Math: Whose sequence?

December 5, 2022

Today’s post is a follow-up to last week’s about Leonardo Pisano, also known as Fibonacci.  And it is specifically about the sequence that bears his name: 0, 1, 1, 2, 3, 5, 8, … where each number is the sum of the two before it.  This appears as the answer to a puzzles that he proposed in his book Liber Abaci:

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?

The sequence was given the name “the Fibonacci Sequence” by Edouard Lucas (1842-1891), a French mathematician for whom a similar sequence: 1, 3, 4, 7, 11, … is named.  Blog commenter S shared that leading up to that time “it was rarely known and when used was called by many names including “the sequence of Lamé” (as Gabriel Lamé had used the sequence for analysis of the Euclidean algorithm)” (Thanks S!)

I recently got the book Finding Fibonacci by Keith Devlin, and while I haven’t read the whole book yet (something for the break!), he does talk briefly about the history of these numbers, which poetically were used in prosody (the study of rhythm and sound/stress in poetry):

They first appeared, it seems, in the Chandahshastra (The Art of Prosody) written by the Sanskrit grammarian Pingala sometime between 450 and 200 BCE.  Prosody was important in ancient Indian ritual.  In the sixth century, the Indian mathematician Virahanka showed how the sequence arises in the analysis of meters with long and short syllables.  Subsequently, the Jain philosopher Hemachandra (ca. 1150) composed a text about them.

S, too, linked to some photos of the sequence appearing in these sources.  I am wondering if the long and short analysis is similar to some problems we’ve given in our classes, which essentially boil down to the number of ways that you can add to 1, 2, 3, 4, 5, 6, etc. using just the numbers 1 and/or 2, where order matters: for 1 there is only 1, for 2 it can be 2 or 1+1, for 3 it can be 2+1, 1+2, or 1+1+1, and for 4 it can be 2+2, 2+1+1, 1+2+1, 1+1+2, or 1+1+1+1.

The idea that the sequence comes up in poetry makes me happy.  The idea of continuing to call it the Fibonacci sequence doesn’t, however, since despite what may have been Lucas’s good intentions, it ignores these earlier contributions.  I’ve seen reference to calling it the Fibonacci-Hemachandra numbers, but even that would bypass Pingala and Virahanka, as well as some other mathematicians, as noted in the abstract of “The so-called fibonacci numbers in ancient and medieval India” by Parmanand Singh

What are generally referred to as the Fibonacci numbers and the method for their formation were given by Virahṅka (between a.d. 600 and 800), Gopla (prior to a.d. 1135) and Hemacandra (c. a.d. 1150), all prior to L. Fibonacci (c. a.d. 1202). Nryana Paita (a.d. 1356) established a relation between his smasika-paṅkti, which contains Fibonacci numbers as a particular case, and “the multinomial coefficients.”

I will admit to still feeling like I don’t have a handle on the history of the sequence – just enough to be uncertain how to call it.  I’ll close with a photo of a bust of Acharya Hemachandra at Hemchandracharya North Gujarat University (CC license)

### Monday Morning Math: Leonardo Pisano (Fibonacci)

November 28, 2022

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:

### Happy Thanksgiving!

November 21, 2022

(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:

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

November 14, 2022

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.

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### Monday Morning Math: Code Talkers

November 7, 2022

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.

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### Monday Morning Math: the witch of Agnesi

October 31, 2022

Happy Halloween!  For this day, it seems appropriate to talk about the Witch of Agnesi. Oooohhhhh!  Spooky!

But this witch is a curve.  It’s described with the algebraic formula:   $y = \frac{64}{x^2+16}$ (or, more generally, as $y = \frac{8a^3}{x^2+4a^2}$ – in the previous equation I used $a=2$).

The original construction is described geometrically, starting with a circle of radius $a$.  Here’s a short video of how it is constructed

So how did this curve get its name?   Maria Gaetana Agnesi was a wicked smart woman who was born in Milan, Italy, in 1718, about 15 years after this curve was first studied by Pierre de Fermat and Guido Grandi.  By the time she entered her teens she spoke 7 languages, and by the time she entered her twenties she was also accomplished at philosophy and mathematics, which she discussed with her father’s visitors, part of an intellectual salon.  After her mother died Maria Agnesi took over running the house and wrote a calculus book for some of her 20 younger siblings.  This book was published when she was 30, when the study of Calculus itself was only decades old.  It was over one thousand pages, and Agnesi was granted an honorary appointment at the University of Bologna.  Agnesi spent most of her adult life focused on theology and serving others, particularly people who were poor or sick.

In 1901, John Colson translated it into English, and here is where a significant mistranslation occurred.  Agnesi had called the above curve averisera (related to the word for “turning”), but the word for “witch” is similar, avversiera, and that’s what Colson used.

(Public domain from Wikimedia)

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