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Good morning! It’s bright and sunny, flowers are in bloom, and spring is here.  Today – 5/11 – is also a doubly prime day since both 5 and 11 are prime numbers [although 511 isn’t since it is divisible by 7 and 73]. There is no sure way to find prime numbers – in fact, the way that data is encoded depends on that – but there are some interesting patterns that happen with prime numbers.  Here are three.  (Because 3 is prime!)

First, there’s the equation f(n)=n² +n+41.  This generates prime numbers for quite a while:

• f(0)=41, which is prime
• f(1)=43, which is prime
• f(2)=47, which is prime 

and so on.

But when n=40, f(40)=40² +40+41=1681, which is 41² and so not a prime number.

Second, there’s the Ulam spiral.  If you start writing numbers in a spiral like this:

Then highlight the prime numbers:​

it turns out that a lot of the prime numbers fall on diagonals.   Stanisław Ulam noticed this in 1963, and Martin Gardner shared it a year later in his “Mathematical Games” column in Scientific American.  It remains noticeable even as the spirals increase. [There are some pictures on wikipedia.]

And finally, even though we can’t predict exactly when they will occur, the number of primes less than N is roughly N/ln(N).    For example,

• There are approximately 10/ln(10)≈4.3 primes less than 10 
  [In fact, there are 4:  2, 3, 5, and 7]
  
• There are approximately 20/ln(20)≈6.7 primes less than 20 
  [In fact, there are 8:  2, 3, 5, 7 above plus 11, 13, 17, 19]
  
• There are approximately 100/ln(100)≈21.7 primes less than 100 
  [In fact, there are 25 of them.]

So it’s not exact, but it’s not nothing, and that’s pretty good.

I hope you all have a prime day!  This will be the last Monday Morning Math until mid-September, so have a wonderful summer/winter as well!

Thanks to TL for suggesting the Ulam spiral!
​

Quick, think of a number.  Is it rational?  It might be – a lot of numbers are, and back in the olden olden days Pythagoras and his buddies thought that every number was rational: there’s a story that when someone proved that √2 was irrational, they were put to death.  Well, that’s the story anyway, although pretty much everything we know about Pythagoras is uncertain.

But most numbers are not rational.  This alone is really weird, because between every two rational numbers there is an irrational number, and between every two irrational numbers there is a rational number, so it seems like there would be the same number of rationals and irrationals.  And yet almost every number is irrational.  The set of rational numbers is countable, meaning there is a way to line them up, so that you’d be guaranteed to reach every one (something like 0, 1, -1, 2, -2, 1/2, -1/2, 3, -3,, 1/3, -1/3, 3/2, -3/2, 2/3, -2/3, 4, -4, etc.).  The set of irrational numbers is uncountable, so there’s no way to line them up and label every one – you’d end up skipping some, that’s how many there are.

The numbers π≈3.14 and e≈2.7 are irrational.  But their sum π+e?  Maybe irrational, maybe rational.  Same with the product πe and the quotient π/e.

They’re probably irrationals, thinks me.  But only probably – every once in a while a number that people think is irrational turns out to be rational, which is the exact opposite of what happened with √2.  Here’s one example (which I’m saying like I know a bunch, though it would be more accurate to say that here’s the one and only example I know):

In the early 1800s the mathematician Adrien-Marie Legendre wanted to find an approximation for the prime counting function π(x), which counts how many prime numbers are less than or equal to x.  We don’t have a formula for  π(x), but an approximation would still be good.  Legendre thought that a good approximation would be x/(ln(x)-B), where B is a particular constant.  More precisely, B is the limit as n approaches infinity of (ln(n)-n/π(x)) [which is kind of circular, it seems to me], and Legendre thought that B would end up being around 1.08 but not necessarily rational. This mysterious and possibly irrational number B became known as Legendre’s Constant.   However, in 1849 Pafnuty Chebyshev showed that not only was B a rational number, but that it was equal to exactly 1.

Thanks Q for this inspiration! 

Good morning! On the drive in this morning, passing trees newly in bloom, I saw white petals everywhere. It turns out they were actually snowflakes – it’s spring here! So in this mixed-up weather time, I will talk about mixed up word problems. I don’t actually mind word problems in general – earlier this semester I was talking about different kinds of averages with students, and shared one that I still enjoy thinking about:

If you drive 50 miles at 50 miles per hour and then 50 miles at 100 miles per hour, what is your average speed?

Yes, fine, that’s probably too fast to drive. But still. I like this problem because the answer isn’t actually 75 miles per hour, but just under 67 mph, since you are going 100 miles in 1.5 hours. Looking at it another way, since you’re going the same distance each way, you spend more time at the slower speed so your average speed is closer to 50 than to 100.

The trouble with how-long-would-it-take problems is that there isn’t just one way to solve them. The problem above uses the harmonic mean (take the reciprocal of 50 and 100, average them, and take the reciprocal of that), but other problems don’t use a mean at all. Here’s one that I used to use in our introduction-to-problem-solving class:

If a chicken and a half can lay an egg and a half in a day in a half, how many eggs can three chickens lay in three days?

This problem can be solved1 but opinions as to whether problems like this are fun puzzle solving or malicious trickery are divided (much like that poor half a chicken). And that opens the door to problems like the ones below, which I admit are just a bit tempting to put on a quiz. I shall resist, though.

• An orchestra of 120 players takes 75 minutes to play Beethoven’s 9th symphony. How long would it take for 60 players to play the symphony? (This dates back to at least 2017.)
  
• An orchestra takes 75 minutes to play Beethoven’s 9th symphony. How long would it take for the orchestra to play Beethoven’s 3rd symphony?
  
• Avery takes 40 minutes to drive to work. Charlie takes 45 minutes to drive to work. How long will it take them if they drive together? [Credit to TwoPi for this one.]
  
• Quinn can juggle 4 saws for 2 minutes. How long can Quinn juggle 6 saws?

That’s it for now, but feel free to share your own!

1. The answer is 6 eggs: just doubling the number of chickens would double the number of eggs to 3, so also doubling the number of days would again double the number of eggs. ︎

Good morning!  There was no Monday Morning Math last week because April was full of Aprilness [a synonym for busyness, here with snow and flowers].  But we’re back now and get to celebrate a birthday!

Specifically, today is the 1073rd birthday of Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji, who was born on April 13, 953, in Baghdad, Iraq.  That is, he was probably born in Baghdad, although the Al-Karaji part of his name, if accurately passed down to us, would indicate his family was from Iran, and there is some thought that this part of his name is actually al-Karkhi, from a suburb of Baghdad.  But no matter which version, if either, is correct, he was a mathematician and engineer who spent most of his life in Baghdad.  To add to the questions about him, there is also some disagreement about how much he contributed originally to mathematics as opposed to organizing it differently than before, but honestly, organization is huge for understanding, so his results are significant regardless.  

I’ll mention two aspects of his legacy in particular.  First, Al-Karaji described the terms x2, x3, x4 , … and even 1/x, 1/x2, 1/x3, …. generally, as opposed to specifically tying them to geometric properties.   He talked about how to multiply and divide monomials, putting abstractness into algebra in a way that is still significant for how it is taught today.

Second, he proved some results about sums (although the results themselves were already known):

• If you’re going to add up a bunch of numbers  1+2+3+…+n then the sum is the last number, plus the next, divided by 2.  That is, the sum n(n + 1)/2 .)
• If you’re going to add up a bunch of squares  12 +22 +32 +…+n2 , then the result is what you’d get by adding up all the numbers (as in the first bullet), and then adding on the product of each number by the one before it: 1·0+2·1+3·2+…+n · (n+1).
• And, finally, if you’re going to add up a bunch of  cubes 13 +23 +33 +…+n3 then the result is the square of what you’d get by just adding up the numbers themselves: (1+2+3+…+n)2

His proof about the sums of cubes was new, but he didn’t actually prove it for a generic number n. He instead proved it when n is equal to 10, in a way that generalizes to other numbers.  This is one of my favorite ways to prove things, incidentally – I’ve found I use it automatically when I’m in introductory classes.

So thank you, al-Karaji, for making the concrete abstract and the abstract concrete, and Happy Birthday!

Sources:  MacTutor, Wikipedia, and a Convergence article by Janet Beery.

Good morning!  

Every 4 years, the mathematics community gives an award known as the Fields Medal, to up to four mathematicians under the age of 40.  The award is given at the International Congress of the International Mathematical Union, and so we will learn about the 2026 recipients in July this year.  There is not universal agreement about the value of the age limit – it was designed so that the award could be given to people who were still active in mathematical research, as opposed to becoming a lifetime achievement award – with a consequence is that it restricts the pool of possible recipients to those who are able to have significant mathematical achievements earlier in life.  But the Fields Medal is considered to be one of the most prestigious awards in mathematics. 

Earlier this month one of the recipients of the award, Heisuke Hironaka, passed away. He was born in Yamaguchi, Japan, on April 9, 1931.  His dad ran a textie factory, and Heisuke was one of fifteen children in a blended family.  He became interested in math in junior high when a professor gave a lecture at his school, and in college at Kyoto University was part of a seminar that covered current research in mathematics, including the problem of singularties.  Singularities are problem points in math, like the corner in the graph of y=|x|, where the slope changes abruptly.  

In 1960 Hironaka earned his PhD in mathematics at Harvard University and married Wakako Kimoto, a graduate student at Brandeis University, where he became a professor.  Over the next ten years he had two children [and his NYTimes obiturary refers to a third child, possibly born later] and taught at Brandeis, Columbia, and Harvard. In 1970, at age 39, Dr. Hironaka was awarded the Fields Metal for a technique which made it possible to study functions with singularities by smoothing them out in a way that didn’t distort other parts of the functions.

Some of Dr. Hironaka’s other acheivements at this time are described in a New York Times Obituary on March 25, 2026:

In Japan, Dr. Hironaka gave lectures and wrote popular books. An autobiographical work, “The Discovery of Learning,” which discussed his philosophy toward the study of math and science, inspired many students to pursue those subjects. “He would appear on television, on talk shows,” his daughter Dr. Hironaka said. “He was a household name in the ’70s and ’80s.”

In 1980, the elder Dr. Hironaka started a summer seminar program for Japanese high school and college students. The summer seminars, now run by alumni, continue today.

The daughter mentioned above is Dr. Eriko Hironaka, who also became a mathematician.  Dad Dr. Hironaka stayed at Harvard until 1992, with a joint professorship at Kyoto University where he was Director of the Research Institute for Mathematical Sciences, but came out of retirment to become the president of Yamaguchi University and a visiting professor at Seoul National University.  Heisuke Hironaka passed away on March 18, 2026, in Tokyo, Japan.

Sources: “Heisuke Hironaka, Groundbreaking Mathematician, Is Dead at 94” by Kenneth Chang in The New York Times, Wikipedia, and MacTutor.

Good morning! We’re back from Spring Break here, and while it’s not quite feeling like Spring yet, it does feel like the winter is on its way out.

It’s a good day, too, to celebrate Emmy Noether’s 144th birthday!  Emmy Noether was born on March 23, 1882, in Erlangen, Bavaria, now Germany.  Her first name was Amalie, after her mother (Ida Amalie), but she always went by Emmy.

Emmy’s father, Max Noether, was a math professor who studied algebraic geometry, and Emmy’s younger brother Fritz also became a mathematician.  When Emmy was 18 she wanted to study math at the University of Erlangen, where her father worked, but because she was a woman that was forbidden, although she could audit classes. After a few years the rules changed and she was accepted into the doctoral program, earning her degree in 1907.

She wanted to be a math professor like her father, but, again because she was a woman, she wasn’t allowed to be hired by the University of Erlangen.  They did let her work for free for seven years, and in 1915 she was hired by the University of Göttingen at the behest of two other mathematicians, David Hilbert and Felix Klein.  “Hired” is probably the wrong word because she still wasn’t paid, and sometimes couldn’t even advertise that she was the one teaching (some of her lectures had to be advertised as David Hilbert’s)  because of the whole gender thing again, but in the early 1920s she was finally able to draw a salary, long after the importance of her research was recognized.

That period of working and being paid for it lasted only about 10 years; in 1933 she was fired, this time because she was Jewish.  She and her mathematician brother both moved to other countries – Fritz to the Soviet Union (where he was later killed) and Emmy to the United States.  She worked at Bryn Mawr College and at the Institute for Advanced Study in Princeton, but on April 14,1935, only weeks after learning that she had a large tumor, she passed away. 

Emmy Noether is remembered for her fondness for teaching (her students were often referred to as Noether’s boys and Noether’s girls) and for her significant results in mathematics — Noetherian rings are a fundamental part of Abstract Algebra —  and physics.  One of her theorems, known as Noether’s Theorems, is described below:

What the revolutionary theorem says, in cartoon essence, is the following: Wherever you find some sort of symmetry in nature, some predictability or homogeneity of parts, you’ll find lurking in the background a corresponding conservation – of momentum, electric charge, energy or the like. If a bicycle wheel is radially symmetric, if you can spin it on its axis and it still looks the same in all directions, well, then, that symmetric translation must yield a corresponding conservation. By applying the principles and calculations embodied in Noether’s theorem, you’ll see that it is angular momentum, the Newtonian impulse that keeps bicyclists upright and on the move.

(from ”The Mighty Mathematician You’ve Never Heard of” by Natalie Angier, from 26 March 2012.)

Happy birthday Emmy Noether!  

Sources: Wikipedia and Biographies of Women Mathematicians and Britannica

There was no Monday Morning Math this week because I didn’t write one in time, but today is Pi Day (3/14) so I’m posting next week’s MMM a few days early!  And appropriately the theme is Early, in the sense of the earliest approximations of Pi. 

The first is from Mesopotamia from somewhere between the 1800s and 1600s BCE.  According to a footnote on Wikipedia, in 1936 a tablet was excavated from Susa in modern-day Iran which approximates pi by using a hexagon to get an approximation of pi as 3 1/8 (3.125), which is very close to the actual value of pi. 

The second is from Egypt from about 1600 BCE, but with information that was possibly older.  This approximation uses an octagon, and ends up with 3 13/81, which is approximately 3.16.  

What I personally like is that one of these is an over approximation, and the other an under, and these two areas aren’t all that far apart geographically. What if they had met and exhanged their ideas about this? They might have decided to average their results, leading to 3 185/1296≈3.14.

Happy Pi Day!

Following up the meaning of each letter, here are the greek letters that came up, based on a conversation with a recent grad.

• α alpha, β beta, γ gamma – used for angles, plus α is used in stats for significance levels  γ is the m Euler–Mascheroni constant, about 0.577
  (The uppercase Α  and Β aren’t used for special things, but  Γ is used for the gamma function)
• δ deltahas the feel of derivatives, but doesn’t show up there, although it does show up with ε  as a small change. 
  (The uppercase Δ is used as the difference between two values.)
• ε epsilon is used with a small change.  (The uppercase Ε isn’t used much.)
• ζ zeta shows up in the Riemann zeta function (but nothing for uppercase Z).
• η eta (and uppercase Η) don’t get to be a function.
• θ theta is an angle!  (And the uppercase Θ also shows up as an angle.)
• ι iota seems like it should be a tiny amount of something, but it isn’t used that way mathematically.  (The uppercase Ι also doesn’t seem to have a special meaning.)
• κ kappa gets used in curvature, although honestly it’s hard to distinguish it from k.  (Likewise, the uppercase Κ doesn’t seem to have a special meaning)
• λ lambda is used for eigenvalues!  And wavelength. and Lagrange Multipliers!  (I’m not sure about capital Λ), 
• μ mu is used for the mean/average in statistics (and uppercase Μ for median, but that’s probably not specifically the greek M). This letter is also the answer to the question, “What sound do Greek cows make?”
• ν nu is used for degrees of freedom in stats.  (Nothing for uppercase N.) 
• ξ xi is one of my favorite Greek letters and there’s a Riemann Xi Function.  (The uppercase Ξ doesn’t have a title but one of my teachers wrote a fraction with Ξ in the numerator and Ξ-bar — that is, Ξ with a line on top — in the denominator and that is my actual favorite fraction I’ve ever seen.). 
• ο omicron (and uppercase Ο) aren’t used much
• π pi is the number 3.14… !  Uppercase Π also represents the product of many terms.
• ρ rho is used for density.  (Nothing for uppercase Ρ.)
• σ sigma is used for standard deviation in statistics.  There’s also another lowercase ς that doesn’t seem to show up much, but uppercase Σ is used for the sum of many terms.
• τ tau is a constant 2π  (but nothing for uppercase Τ)
• υ upsilon (and uppercase Υ) don’t show up much in math
• φ psi is an angle, and also the golden ratio ((1+√5)/2), and uppercase Φ is often an angle too.
• χ chi is the chromatic number of a graph (but nothing for uppercase Χ)
• ψ psi is the sum of the reciprocals of the Fibonacci sequence (but nothing for uppercase Ψ )
• ω omega is a root of unity — that is, a complex number that is the solution to xn=1  (and uppercase Ω is an ohm)

Or instead of all that, you could just look at this comic from xkcd

(Hebrew letters א‎ ב‎ ג‎ … and fraktur letters  𝔄 𝔅 ℭ … should be next, but other than  א, used for counting the relative sizes of infinite sets, I haven’t used those much so I’ll defer to others for that.)

Good morning!   Today’s Monday Morning Math comes courtesy of the Math Center.  One day a few months ago I walked in, and the board was covered with the alphabet, explaining what each letter was used for.  It turns out that a couple of our majors were creating a list based on a conversation with Batman, and this is what they came up with:

• a,b,c – constants, triangles,
• d – derivative/sometimes delta
• e – the number e
• f, g, h – functions, but h is also height and also in derivative limits
• i, j, k – the unit vectors, but also i is the imaginary number i and also i,j,k are used in the quaternions and also they are used in infinite sums.  Whoa – these are busy letters!
• l – length, line
• m, n – also lines, and also natural numbers or at least integers.  Plus m is slope!
• o – this gets skipped because it looks like 0
• p, q – prime numbers
• q, r – rational numbers (q is double billing!)
• s – side or arc length
• t – time
• u, v, w – vectors again or variables for substitution
• x, y, z – variables, and z is also a complex variable

I’m pretty sure there are more options, but this seemed like a good start – you are welcome to add to it in the comments!  And thanks to Q and TwoPi for helping me to recreate this list!

Good near-morning!  It was cold driving in this morning: below 0 even without the wind, though the sun is certainly shining brightly!

So in honor of the very cold temperatures, it seems like a good idea to talk about temperature!  Here are some fun facts about Fahrenheit and Celsius:

Fahrenheit was named after the Danish physicist Daniel Gabriel Fahrenheit (1686–1736).  He originally planned to have 0 be the freezing temperature of water, salt, and ammonia [so like the ocean, but with extra ammonia?], then 30 be the freezing point of water, then 90 be the temperature of the human body, and then 240 be the boiling point of water.  Physics didn’t quite agree with that, however: you could use two of those to set the scale, but then the others wouldn’t be quite what was wanted, which is why we have freezing at 32 and boiling at 212 and the human body at …well, see below.  But we’ve had a variation for what he proposed for about 300 years, although today only a few countries use the Fahrenheit scale.

Celsius is named after the Swedish astronomer Anders Celsius (1701–1744), which means that it is almost as old as the Fahrenheit scale!  France, for example, began using it as part of the adoption of the metric system right after the French Revolution.  Interestingly, Celsius is the reason that an average human adult temperature is sometimes considered to be 98.6 degrees Fahrenheit!  The person who did the original study – German physician Carl Reinhold August Wunderlich – did a study of thousands of people and published that 37 degrees Celsius was normal.  I couldn’t find the standard deviation, but just looking at units it is only given to a full degree Celsius, and even a variation of 0.1 degrees Celsius would mean a variation close to 0.2 degrees Fahrenheit.  And there is in fact considerable variation.  The more exact sounding 98.6 degrees Fahrenheit just comes from the conversion of 37.

The rule for conversion is the F=9/5C+32, but I prefer the “double and add 30” that I’m pretty sure I learned watching Strange Brew with Bob & Doug McKenzie.  For converting the other way, I guess it would be “subtract 30 and then halve”.

And, finally, if it cools down much it won’t matter what scale you use:  at -40 both Fahrenheit and Celsius are the same.

Sources: Wikipedia and Wikipedia and Live Science and the US Metric Association

Stay warm everyone!