Math Digests August 2021

New mathematical record: what’s the point of calculating pi?

The Guardian, August 17, 2021

Pi is something more than just a number. Pi has infinitely many digits after its decimal point and no observable pattern. It’s a classic example of so-called “transcendental numbers,” which can’t be calculated from any combination of ratios, powers, and roots of whole numbers. You may know $\pi$ as 3.14 or 3.14159, but in a new study, Swiss researchers used a supercomputer to calculate a world record 62.8 trillion digits. The new estimate—yep, it’s still technically an estimate—surpasses the previous record of 50 trillion digits with a calculation more than three times as fast. “It’s an impressive and time-consuming feat that prompts the question: why?” writes Donna Lu in The Guardian. Lu’s article explores the history and motivations of humans’ obsession with $\pi$.

Classroom activities: geometry, pi, circles, accuracy vs precision, transcendental numbers

  • Pi appears in nature as the fundamental ratio between any circle’s circumference and its diameter. But the elusiveness of its exact value has baffled humans for thousands of years. Why do precise estimates of $\pi$ matter?
    • (All ages) Ask students to look up the first 10 digits after the decimal point in $\pi$, then calculate the circumference of a circle with radius 10 meters based on each different estimate: (i.e. 3; then 3.1; then 3.14; and so on.) Discuss how the precision changes with each additional decimal. In what situations does this increased precision matter?
    • (Middle school) Repeat the above activity, but this time calculating the circumference of Earth at the equator based on the different $\pi$ estimates. (The equatorial radius of the earth is 3,963.161 miles.) How many miles longer is the estimated circumference if you use 10 digits in your $\pi$ estimate instead of just one? How many digits of $\pi$ do you need before the difference between consecutive estimates is less than 1 mile? Less than 0.01 miles? Discuss in which situations these differences matter.
    • (High school) Repeat the above activity, now with different levels of precision for both $\pi$ and the equatorial radius. How do the values compare? What does this reveal about the level of precision needed in calculations? (If students are familiar with significant figures, this can also be a chance to practice the concept.)
  • (High school/Algebra/Geometry) Amelia is a pilot. Her small plane holds 50 gallons of fuel. Each gallon can take her 20 miles. If Amelia flies around the earth along the equator, how many times will she need to stop for more fuel? Solve the problem using $\pi$ with two digits after the decimal, and again with 10 digits after the decimal. Does accounting for her altitude above the earth (say, 10,000 feet) significantly change the answer?
  • (High school) Learn about transcendental numbers, and why it’s so fascinating that some numbers (most numbers!) can’t be produced using algebra, by watching this Numberphile video.

—Max Levy

Israeli data: How can efficacy vs. severe disease be strong when 60% of hospitalized are vaccinated?

Covid-19 Data Science, August 17, 2021

In places with high COVID-19 vaccination rates, like Israel, a high proportion of people hospitalized with the disease are vaccinated. Some commentators have cited that fact to incorrectly claim that the COVID-19 vaccines are not effective. The first mistake in this claim is the use of raw counts ($X$ people) rather than rates ($Y$ people per 100,000). The second mistake stems from a statistical phenomenon called Simpson’s paradox: In some data sets, a certain trend is present when the data are put into groups but reverses when the data are combined. In a blog post (and accompanying Twitter thread) drawing upon detailed Israeli data, statistical data scientist Jeffrey Morris of the University of Pennsylvania explains how the data actually show that the vaccines are highly effective in preventing severe COVID-19. (A Washington Post article by Jordan Ellenberg also covers the topic.)

Two graphs with the same data. On the left, the entire data set shows a positive trend. On the right, the data is divided into two sets, each of which shows a negative trend.
A visual example of Simpson’s paradox: the overall trend reverses when data is grouped by some color-represented category. Credit: Towards Data Science

Classroom activities: Simpson’s paradox, statistics

  • (Introductory statistics) Introduce Simpson’s paradox using this MinutePhysics video. For more examples of the phenomenon, see this article from Towards Data Science.
  • (Introductory statistics) Once students understand the basics of Simpson’s paradox, walk them through the reasoning in the COVID-19 article. Present each data table in the article one at a time, asking students to explain what they see. Conclude with a discussion of the caveats that Morris mentions.

—Scott Hershberger

Mathematicians are deploying algorithms to stop gerrymandering

MIT Technology Review, August 12, 2021

In an ideal democracy, everyone’s vote counts equally. But that’s not always the case—especially when politicians can draw electoral districts to their party’s advantage. This practice is called gerrymandering, a reference to the salamander-shaped district drawn by the administration of Massachusetts governor Elbridge Gerry in 1812. But gerrymandering isn’t always that obvious. A map may look perfectly normal, yet still produce election results that don’t match the overall will of the citizens of the region. To deal with this, mathematicians from all over the country have been working on software that compares redistricting maps with randomly generated “samples.” If a map produces wildly different electoral results than the samples, that’s a clue that it may have been drawn to intentionally benefit one party over another. With this software in hand, mathematicians can help the courts and the public identify biased maps, and perhaps help make elections fairer for everyone.

Classroom activities: politics, gerrymandering, probability, law of large numbers, central limit theorem

  • (Middle school) Teach about gerrymandering with this lesson from KQED (PDF).
  • (Introductory statistics) The redistricting approaches described in the article implicitly use probabilistic ideas like the law of large numbers and the central limit theorem. Teach the law of large numbers through a hands-on simulation as a homework or in-class assignment.
    • Give each student a six-sided die (or a similar “random number generator”). They will each roll their die 10-20 times, and the class will pool their data (a total of $N$ rolls) to find \[N(i)=\text{number of times } i \text{ was rolled, } i=1,2,3,4,5,6.\]
    • Before rolling the dice, ask students to predict $N(i)/N$ for each $i$ and write down a confidence interval. This should be the narrowest interval that they think will match the data.
    • Once the data is in, you will probably find that $N(i)/N \approx 1/6$ with some small variation. This is the expected outcome due to the law of large numbers.
    • Have students write a paragraph relating this activity to the redistricting software described in Roberts’ article.

—Leila Sloman

The Delta Variant Isn’t As Contagious As Chickenpox. But It’s Still Highly Contagious

NPR, August 11, 2021

Although the delta variant of SARS-CoV-2 is one of the most contagious respiratory viruses that we know of, it’s still not as contagious as chickenpox. In this story for NPR, Michaeleen Doucleff uses the concept of $R_0$, the average number of people that a sick person will infect when the entire population is vulnerable to the virus, to compare the transmission rate of the delta variant with that of several other viral infections. For the original coronavirus, $R_0$ was between 2 and 3. Recent estimates place the $R_0$ of the delta variant between 6 and 7. That increase makes a huge difference in how fast the virus spreads due to the mathematics of exponential growth.

Classroom activities: exponential growth, exponential decay

  • (Middle school) To bring to life how fast exponential growth is, make a large $x$-$y$ plane on the floor and have students map out an exponential curve with their feet.
  • (High school) The National Council of Teachers of Mathematics offers a set of resources on the math of the pandemic. In particular, use this interactive tool to simulate the spread of a virus.
    • Have students predict how the spread depends on the number of days a person is contagious, the chance of contracting the virus per contagious contact, and the number of contacts per day.
    • How do those three numbers relate to $R_0$?
  • (High school) This lesson from the New York Times introduces exponential growth and exponential decay using data from the coronavirus pandemic. The lesson also includes links to NY Times activities on herd immunity, vaccine efficacy, and vaccine hesitancy.

—Scott Hershberger

Animals Can Count and Use Zero. How Far Does Their Number Sense Go?

Quanta Magazine, August 9, 2021

1 minus 1 seems like the easiest math problem in the world. But the brain processes required to solve it reveal a messy world where the borders between mathematics, biology, and psychology become hazy. Monkeys, baby birds, and even bees can do arithmetic. These animals understand the relative positions of numbers on a number line (e.g. two bananas are greater than one banana)—a concept called numerosity. But can they understand where zero falls on the number line? Zero is special—it’s not a quantity, it’s an absence. “Even humans struggle with zero,” writes Jordana Cepelewicz. Humans only began acknowledging zero in the seventh century. Its complexity led many to think that only humans are in the know. But new research on animal cognition is proving that assumption wrong. Cepelewicz writes about numerosity in animal brains and how crows can, remarkably, understand zero.

Classroom activities: What is zero?, number systems

  • (All levels) Challenge students to explain why 0 falls on a number line before 1 without referencing the fact 1 minus 1 is 0. There are no real “right” answers here; this should just be a way of reinforcing the notion of zero being weird.
  • (All levels) Although other animals have a sense of numerosity, humans are unique in having a “symbolic” understanding of mathematics, using characters like 0 and 1 to express numbers abstractly. This simplifies our communication and allows us to reach a deeper understanding of mathematics.
    • (Middle school) Ask students to invent their own set of symbols to replace our standard digits 0-9. Then have them teach their new systems to each other. Discuss the challenges of learning these symbolic values, and what you think humans gain from this ability.
    • (High school) Ask students to invent their own number system, in base 10 or another base (or another system entirely). Show them the number systems used by the ancient Romans, Mayans, and/or Babylonians for inspiration. Then have the students teach their new systems to each other. Discuss the challenges of learning these symbolic values, and what you think humans gain from this ability.

—Max Levy

Do we still need math?

Big Think, August 4, 2021

In the age of computers, many people believe they never need to think about math. “The supermarket checkout totals the bill, sorts out the special meal deal, adds the sales tax,” writes Ian Stewart in this excerpt from his book What’s the Use?: How Mathematics Shapes Everyday Life. But, he explains, these technologies actually rely heavily on advanced mathematics. Computers are an invaluable tool to mathematicians, with their ability to do millions of calculations in an instant. And the algorithms that make this possible rely on all kinds of mathematical ideas, from linear algebra to topology.

Classroom activities: algorithms, programming, computer science

  • (Middle school) Stewart mentions Google’s PageRank algorithm as a prime example of the reliance of modern life on mathematics. Introduce the concept of algorithms with Teach Engineering’s lesson plan, and explore the PageRank algorithm further by playing the game in the associated Acting Like an Algorithm activity.
  • (Introductory programming) One way to take advantage of computers’ ability to do thousands of calculations per second is by programming loops.
    • Teach for and while loops in Python with this online tutorial. Have students write a loop that calculates $f(x)$ for 10 different choices of $x$ and their choice of the function $f$. Use the timeit.timeit() function to time how long their code takes to run.
    • Time students doing the same calculation by hand. How long does it take?

—Leila Sloman

Some more of this month’s math headlines: