Archive for February, 2023

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.


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!