Archive for November, 2021

Monday Morning Math: Dorothy Vaughn

November 29, 2021

Dorothy Vaughn was born on September 20, 1910 in Kansas City. Missouri. She graduated at the age of 19 from Wilberforce University, a historically black college in Wilberforce, Ohio. Dorothy Vaughn supported her family as a math teacher for 14 years prior to working at NASA as part of the National Advisory Committee for Aeronautics’ (NACA) West Area Computing unit in 1943.

The West Area Computing unit was a group of black women who, as a result of Jim Crow Laws, were segregated at NASA while they performed mathematical calculations on slide rules and graph paper to support the space race and the NASA astronauts’ flight missions to space. Dorothy Vaughn was an expert in the computer programming language FORTRAN and she became NASA’s first black supervisor of the group in 1949 where she taught the women programming to prepare them for the future which she believed would be machine computers.

In addition to her work at NASA, Dorothy Vaughan raised her family of 6 children, one of whom also worked for NASA.

She retired from NASA in 1971 and died on November 10, 2008. She was featured in Margot Lee Shetterly’s book Hidden Figures and portrayed in the film based on the book by Octavia Spencer.


Written by Tracy Lyn Lause

Monday Morning Math: Hippocrates of Chios

November 15, 2021

No, not that one!

Hippocrates of Chios was a Greek mathematician who followed Pythagoras and lived around 470-410 BCE. He was a contemporary of the more famous Hippocrates, the physician Hippocrates of Kos (c460-370 BCE). He is perhaps best known for his work on the classical geometry problems of squaring the circle and doubling the cube.

Astute readers will note that both of the geometry problems above are impossible to solve in general, but Hippocrates’ work led him to discover how to compute the areas of certain lunes, regions bounded by two circular arcs. (See this video for a pleasing application of Hippocrates’ discovery.)

The “Lune of Hippocrates”

The Greek philosopher Proclus credits Hippocrates with writing the first version of Elements of Geometry, much of which Euclid would later incorporate into his own Elements. In this text, Hippocrates introduced the use of letters to represent points, as well as naming objects using the points that defined them (e.g., “triangle ABC”). It is also believed that he had developed a method of “proof by exhaustion” (approximating circles by polygons with an increasing number of sides), later used by Eudoxus and Euclid. Unfortunately, Hippocrates’ text has been lost to history.

While respected as a mathematician, Hippocrates was viewed as “stupid and incompetent in the business of ordinary life” by Aristotle. He lost a large sum of money due to fraud, and had to teach geometry in Athens to make up for it.


Monday Morning Math: Grace Murray Hopper

November 8, 2021

In honor of Veterans Day, our mathematician this week is Grace Murray Hopper.

Grace Murray was born in New York in 1906. She earned her BA in math and physics from Vassar College. Over the next few years she married, earned an MA and PhD in math from Yale University, and became a professor at Vassar.

In 1943, at the age of 37, she joined the Navy in response to World War II and began working with computers. She worked on the Harvard Mark I (which was over 50 feet long, 8 feet tall, and 2 feet deep) and later the Mark II and III. She learned programming and was instrumental in both conceptualizing and creating the first compiler.

At the time of her retirement in 1986 she was at the rank of Rear Admiral and the oldest active military officer. She continued to work even after her retirement and died in 1992. She is buried in Arlington Cemetery.

One of her legacies is the popularization of the term computer bug. She invented the term “debugging” in response to an actual bug (shown in the photo below!)

Hopper found the first computer “bug” a dead moth that had gotten into the Mark I [possibly Mark II] and whose wings were blocking the reading of the holes in the paper tape. The word “bug” had been used to describe a defect since at least 1889 but Hopper is credited with coining the word \debugging” to describe the work to eliminate program faults.

Courtesy of the Naval Surface Warfare Center, Dahlgren, VA., 1988, public domain


Monday Morning Math: Number Systems

November 1, 2021

For today’s Monday Morning Math, we’ll do some counting!

In the Indo Arabic number system we have ten digits: 0123456789 grouped by tens, so the number 23 can be thought of as 2 tens and 3 ones.  But in Mesopotamia (modern day Iraq) 4000 years ago they grouped numbers by 60s.  In that system the number 2 3 would be 2 60s and 3 ones, or what we would think of as 123.  That grouping by 60s, incidentally, has passed down to us in how we count time: it is the reason we have 60 seconds in a minute and 60 minutes in an hour.

For an example closer to home, the Mayans 2000 years ago, in modern-day Mexico, Guatemala, Belize and Honduras, grouped numbers by 20s.  In this number system the number 2 3 would be 2 20s and 3 ones, or what we could think of as 43. Numbers below numbers less than 20 were formed using dots for 1s and lines for 5s, with a shell like oval for the number 0, as in the image below.

Some Mayan numbers in the thousand year old Dresden Codex, named after where it was taken away to.

Several indigenous tribes in North American group numbers by 4s and 8s, particularly the Yuki in northern California, and the Chumash people along the central and southern coast.

(From “Numeral Systems of the Languages of California” by Roland B. Dixon and A. L. Kroeber, published in American Anthropologist in Oct-Nov 1907 (p. 665)

Grouping numbers by 5s and 10s is often attributed to fingers on the hand, but the interesting (to me) thing is that grouping numbers by 4s and 8s is also based on the hands:  the Yuki used the spaces between the fingers to count.  As Dixon and Kroeber wrote above, “It does not follow that because people count by their fingers they count by fives.”

November 1 marks the beginning of National Native American Heritage Month!  Visit the website for Indigenous Mathematicians to learn about more mathematicians!

 I first read about the Yuki in Ethnomathematics: A Multicultural View of Mathematical Ideas by Marcia Ascher.