Monday Morning Math: D. R. Kaprekar and his constant

Happy May Day!  It’s finals week here, and so this will be the last Monday Morning Math until sometime in September. But in the meantime, here’s a neat magic trick

• Start with a 4 digit number. The digits don’t have to all be different, but they can’t all be the same (so 1001 is OK but 1111 is not).
  
• Now look at the 4 digits, and write down both the largest number and the smallest number you can make with those digits: in my example the largest would be 1100 and the smallest 0011.   
  
• Take the difference:  I got 1100-0011, which is 1089.  (Hey, that’s pretty cool!  I’ve seen that number before.  But I digress.)
  
• Now repeat the process: take your new 4 digit number, write the largest and the smallest number you can make with those digits and take the difference.  And keep doing it.  After a while, you’ll get the same number over and over again.  And that number is……

6174

This is called Kaprekar’s constant, named after Dattatreya Ramchandra Kaprekar.  D. R. Kaprekar was born on January 17, 1905, in Dahanu, India, and enjoyed math puzzles and numbers from when he was young.  He studied math in college and in 1927, when he was 22, was awarded the R. P. Paranjpe Mathematical Prize for original work.  His discovery of the pattern above isn’t his only work, but it might be the best suited for magic tricks. 

There’s a  version with 3 digits too,  but once you get to 5 or more digits you start getting cycles instead of the same number over and over.  And that’s just Base Ten – you can explore other bases to see what happens. Enjoy!

Sources:

• Wikipedia
• MacTutor (where the picture came from)
• Hat tip to Ricardo Teixeira, whose presentation on mathemagic introduced me to this constant!

Monday Morning Math:

Despite the switch back to colder temps, the trees are exploding and it feels very much like Spring, both with the promise of summer and the bittersweetness of the upcoming graduation. 

Today’s Monday Morning Math is a reader suggestion (Thanks Q!) and has been on my whiteboard for a while. Specifically what has been on my whiteboard is this: 

That is….not intuitive. What it is is an example of a family of functions that are continuous everywhere, and differentiable nowhere.   For example, if and you get

.  

Before Karl Weierstrass gave this example in 1872, mathematicians knew that it was possible for continuous functions to have individual points where there wasn’t a derivative: one example is y=|x|, which doesn’t have a derivative at (0,0) since the function never smooths out at that point (as it were).  But most mathematicians didn’t think there could be more than a finite number of spots without a derivative.  

Karl Weierstrass was not most mathematicians. 

He described the function and proved mathematically, if not intuitively, that there was not one point with a derivative, even though the function was continuous.  The general response to this example was surprise and avoidance: my favorite reaction is from Charles Hermite, who wrote, “I turn with terror and horror from this lamentable scourge of functions with no derivatives.”  (quoted in “Math’s Beautiful Monsters” by Adam Kucharski

Over time, though, people started to see the value of this example, both in illustrating the need for definitions in Calculus to be more precise and in foreshadowing other ideas, like fractals.  And now, with the help of computers we can more easily see the monster looking like a wonderfully jagged mountain range. 

(public domain by Eeyore22)  

(and the rest of the Wikipedia article was also good!)

Monday Morning Math: New Proof of the Pythagorean Theorem

Good morning!   Spring is exploding here – we’re not done with the rains yet, but the trees are starting to bloom and the daffodils are out!  And no matter how many times you see daffodils, it is still awesome.

And so it is with proofs of the Pythagorean Theorem: no matter how many you’ve seen, it’s exciting with another one appears. It is especially exciting when the people who came up with the proof are still in high school.

Calcea Johnson and Ne’Kiya Jackson are seniors at St. Mary’s Academy in New Orleans.  Just last month they presented their proof at the American Mathematical Society meeting.  Several news outlets picked up their result, and focused on one interesting piece: that the proof used trigonometry in a novel way. As Keith McNulty of Medium.com explains:

Claims in the media that Johnson and Jackson’s proof is the first trigonometric proof of Pythagoras are overblown, but their proof could well be the most beautiful and simplest trigonometric proof we have seen to date.

You can hear them share a bit about the experience here:

Congratulations to  Calcea Johnson and Ne’Kiya Jackson! They’re going to college in math-adjacent areas (Biochemistry and Electrical Engineering) and I look forward to hearing about their next discoveries!

(There are quite a few news articles about this: in addition to the one linked above, one by Leila Sloman in Scientific American gives a good overview.)

Monday Morning Math: Einstein found

Good morning!  There’s been some exciting math in the news lately about an einstein.  Not a  person, but a shape. A tile.

First, some background.  A tessellation is a way of covering a floor (or a whole plane) with tiles so there are no gaps or overlaps.  By putting a few or a lot of restrictions on the tiles, like that the tiles have to match edge-to-edge and that you can only use one shape, you can get some interesting results.

For example, if you only want to use one shape but you want that shape to be regular (all the sides the same and all the angles the same), you might end up with something like this:

(CC by Jeff Weeks)

If you don’t require that the shape be regular, there are more options.  Here’s a tiling made up of rhombuses (where the sides are all the same, but not the angles).  This is actually a Roman mosaic so it’s a tile made up of tiles, but for the purposes of this we’re just looking at the diamond shapes, which look like they make up cubes.

(“An ancient roman geometric mosaic from the “Palazzo Massimo alle Terme” National Roman Museum of Rome, Italy”  CC by Mbellaccini)

Then there are aperiodic tessellations, which avoid repeating patterns.  Until recently people didn’t know if it was possible to have a single tile that would form an aperiodic tiling. Such a potential shape was called by the German name ein Stein, which mean “one stone” (like one tile). 

It turns out that there IS a tile that does this! This is pretty recent, so it hasn’t been peer reviewed yet, but last year David Smith found a tile that seemed to work, and with a little help from his friends wrote a paper proving it (twice, in two different ways).  

(The yellow tiles are reflections of the blue tiles.  CC by Ginger)

Cool! Even cooler, it’s part of a whole family he discovered, indicating that there are many ways to be an einstein.

Thanks to Arianna who mentioned this last week, and Joe who sent me a New York Times article about it.
Sources:

• Wikipedia
• “Elusive ‘Einstein’ Solves a Longstanding Math Problem” by Siobhan Roberts, New York Times, March 28, 2023.

Monday Morning Math: Three Number Tricks

Hello everyone!  It’s April, so time for some tricks!  Technically it’s past April Fool’s Day, but these aren’t the prank kind of tricks anyway – just a few quick fun things for a happy start to the week.

Trick 1:  Quickly guessing the number! (from thoughtco.com)
Pick a 3-digit number where all the digits are the same (like 777).
Add those digits.  Divide your original number by this sum.  Did you get…..37?  You did!

Warning:  You can probably only do this trick once before people catch on.   Or you could go fancy and do a similar trick with a 9-digit number.  In that case instead of 37 you’ll get 12345679 which I admit both surprises and delights me.  

Trick 2: Elaborate guessing the number! (from our own blog)
Pick a 3-digit number, like 360.  For this the first and last digit have to be different.
Now reverse the digits (063), and subtract the smaller number from the larger (leaving 297).
Take that new number, reverse its digits, and add those two together (297 and 792).  I got 1089 when I did that but the cool thing is, so did you!  Even if you started with a different number!!!  

Trick 3: Guessing your age  (from our notes for one of our classes.)
Pick the number of days a week that you like to go out.  Now double this number.  Add 5, and then multiply by 50.  You’ll have a pretty big number now, but you’ll need to make it even bigger. 
If you have already had your birthday this calendar year, add 1773.  If you have not had a birthday this calendar year, add 1772. 
Finally, subtract the four-digit year that you were born.   
You’ll end up with a three-digit number: The first digit of this was your original number (i.e. how many times you want to go out a week).  The second two digits are your age!!!  Pretty neat huh?  The only catch with this is that this trick changes every year – you’ll have to add a diferent amount if you try it in 2024.

Happy trickstering!

Monday Morning Math: ChatGPT “research”

Hello everyone!  Apologies for skipping a week – Naz’s Spring Break was a week ago, and I’m still catching up after accompanying a group of students to Hungary. Fortunately, the topic for this week’s Monday Morning Math fell into my lap (well, inbox – thanks Mark!) Last Monday (March 20), the BBC posted posted “The numbers that are too big to imagine”  about infinity.  Here’s a quote from the original article.

Some infinities, [Cantor] showed, are bigger than others.

How so? One of the simplest ways to understand why is to imagine the set of all the even numbers. This would be infinite, right? But it must be smaller than the set of all whole numbers, because it does not contain the odd numbers. Cantor proved that when you compare such sets, they contain numbers that do not match up, therefore there must be multiple sizes of infinity.

If you’ve taken an introduction to Proof class, you might remember the idea of sizes of infinity, but you might also remember that the whole numbers and even numbers are actually the same size of infinity.  The Cantor proof referenced in the paragraph above doesn’t exist.

The reason for the error?  Maybe Richard Fisher, the author, just made a mistake.  Or maybe the mistake was deeper than that.  At the end of the article was the quote:

The author used ChatGPT to research trusted sources and calculate parts of this story.

The error was found quickly, and corrected within a day.  The article now reads:

Some infinities, [Cantor] showed, are bigger than others.

How so? To understand why, consider the numbers as ‘sets’. If you were to compare all natural numbers (1, 2, 3, 4, and so on) in one set, and all the even numbers in another set, then every natural number could in principle be paired with a corresponding even number. This pairing suggests the two sets – both infinite – are the same size. They are ‘countably infinite’.

However, Cantor showed that you can’t do the same with the natural numbers and the ‘real’ numbers – the continuum of numbers with decimal places between 1, 2, 3, 4 (0.123, 0.1234, 0.12345 and so on.)

If you attempted to pair up the numbers within each set, you could always find a real number that does not match up with a natural number. Real numbers are uncountably infinite. Therefore, there must be multiple sizes of infinity.

Yay!  A nice explanation of the different sizes of infinity.  The explanation about ChatGPT also states, “For the sake of clarity, BBC Future does not use generative AI as a primary source or to replace the journalism needed for our articles.“

Indeed – as useful as Artificial Intelligence can be, it doesn’t replace the need for understanding and evaluating what it generates.

Monday Morning Math: 𝝅

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/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

Monday Morning Math: Primes

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 .   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

Monday Morning Math: Katherine Johnson

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:

• Visionaryproject.org
• Thehistorymakers.com
• Nasa.gov
• Space.com
• Biography.com

Monday Morning Math: Transcendental Numbers

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 : 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 , 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 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.

Monday Morning Math: Mathematically Gifted & Black

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

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  (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.)

Monday Morning Math: Happy New Year!

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: .  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?

Picture from Wikimedia showing a diagram that Gottfried Wilhelm Leibniz owned.

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 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?

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)