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**/δ** 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 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

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 . At first glance this seems to work:

- , which is prime.
- , which is prime.
- , which is prime.

and this continues for a while… - , which is prime.

But then: - , 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:

is prime if and only if

For example,

- When then and sure enough is a prime number.
- When , then and is also prime!
- But when then and 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 this formula produces , which is the first prime. When it gives , 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

]]>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:

]]>Instead, you could call your affection for someone transcendental. Because 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 . You could do something similar for any integer.
- The number 2/3 is algebraic because it’s the solution to . You could do something similar for any fraction.
- The number is algebraic because it’s a solution to . Oooh, and now things get interesting. Because 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 , 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 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 is an algebraic number because it’s a root of , 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 has a root between 1 and 2 (since is negative, but 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 , it turns out that is not algebraic, which means that it is transcendental. The number 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 is algebraic or transcendental. Then again, we don’t even know if 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.

]]>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!

]]>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 (which is approximately 2,151,972,563.22) is very close to (which is 2,147,483,648). If you want to avoid fractions, this means that is almost the same as . 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, (which is approximately 129.7) is very close to , 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.)

]]>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: . Or . (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.

]]>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!

]]>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)

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