I recently came across the following computational trick: to estimate , use
where is the closest perfect square to . For example,
I had not seen this trick before — at least stated in these terms — and I’m definitely not a fan of computational tricks without an explanation. In this case, the approximation is a straightforward consequence of a technique we teach in calculus. If , then , so that . Since , the equation of the tangent line to at is
The key observation is that, for , the graph of will be very close indeed to the graph of . In Calculus I, this is sometimes called the linearization of at . In Calculus II, we observe that these are the first two terms in the Taylor series expansion of about .
For the problem at hand, if , then
if is close to zero. Therefore, if is a perfect square close to so that the relative difference is small, then
One more thought: All of the above might be a bit much to swallow for a talented but young student who has not yet learned calculus. So here’s another heuristic explanation that does not require calculus: if , then the geometric mean will be approximately equal to the arithmetic mean . That is,
I’ve always enjoyed reading about the history of both mathematics and physics, and so I really appreciated this perspective from Physics Today magazine about the importance of this field. One of many insightful paragraphs:
And a more human physics is a good thing. For starters, it makes physics more accessible, particularly for students. Many promising students drop out of the sciences because the material seems disembodied and disconnected from their lives. Science education researchers have found that those lost students “hungered—all of them—for information about how the various methods they were learning had come to be, why physicists and chemists understand nature the way they do, and what were the connections between what they were learning and the larger world.” Students can potentially lose the wonder and curiosity that drew them to science in the first place. Historical narratives naturally raise conceptual, philosophical, political, ethical, or social questions that show the importance of physics for the students’ own lives. A field in which people are acknowledged as people is much more appealing than one in which they are just calculating machines.
I got a kick out of reading this retrospective of Texas high school slide rule competitions… including a 1959 picture of Janis Joplin on her high school slide rule team and a 1980 Dallas Morning News article eulogizing the competition.
The year was 1933. Members of the University’s mathematics department and the Institute for Advanced Study were celebrating the Institute’s opening with a party at the Princeton Inn, which is now Forbes College. “By chance,” an attendee later recalled, he entered just behind the Institute’s most famous faculty member, Albert Einstein. “As we walked across the lobby of the hotel, a Princetonian lady, of the Princetonian variety, strolled toward us. She was fairly tall and almost as wide, beautifully dressed, and she had an air of dignity. She strolled up to Einstein, reached out, put her hand up on Einstein’s head, ruffled his hair all over the place, and said, ‘I have always wanted to do that.’ ”
The source of this marvelous anecdote is Edward McShane, a distinguished mathematician, and the context is an intriguing series of interviews that the University conducted in the 1980s with people who had studied in the mathematics department in the 1930s. These interviews sought to capture the spirit of mathematics at Princeton during a golden age, a time when Einstein, Kurt Gödel, John von Neumann, and other analytical greats crossed paths on campus. In the process, the interviews captured something unexpected: a catalog of weirdness, a palette of colorful and off-kilter adventures that were going on in the background while the big papers were being written.
Here’s a tale of one the great mathematicians of all time that I heard for the first time this year: the great mathematician published a mistake… which, when it occurs today, is highly professionally embarrassing to modern mathematicians. From Mathematics in Ancient Greece:
In a paper published in the year 1774, [Leonhard] Euler listed [1,000,009] as prime. In a subsequent paper Euler corrected his error and gave the prime factors of the integer, adding that one time he had been under the impression that the integer in question admitted of the unique partition
but that he had since discovered a second partition, namely
which revealed the composite character of the number.
I love hearing and telling tales of legendary mathematicians. Today’s tale comes from Frank Nelson Cole and definitely comes from the era before calculators or computers. From Wikipedia:
On October 31, 1903, Cole famously made a presentation to a meeting of the American Mathematical Society where he identified the factors of the Mersenne number 267 − 1, or M67. Édouard Lucas had demonstrated in 1876 that M67 must have factors (i.e., is not prime), but he was unable to determine what those factors were. During Cole’s so-called “lecture”, he approached the chalkboard and in complete silence proceeded to calculate the value of M67, with the result being 147,573,952,589,676,412,927. Cole then moved to the other side of the board and wrote 193,707,721 × 761,838,257,287, and worked through the tedious calculations by hand. Upon completing the multiplication and demonstrating that the result equaled M67, Cole returned to his seat, not having uttered a word during the hour-long presentation. His audience greeted the presentation with a standing ovation. Cole later admitted that finding the factors had taken “three years of Sundays.”
In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s story is a continuation of yesterday’s post.
When I teach regression, I typically use this example to illustrate the regression effect:
Suppose that the heights of fathers and their adult sons both have mean 69 inches and standard deviation 3 inches. Suppose also that the correlation between the heights of the fathers and sons is 0.5. Predict the height of a son whose father is 63 inches tall. Repeat if the father is 78 inches tall.
Using the formula for the regression line
we obtain the equation
so that the predicted height of the son is 66 inches if the father is 63 inches tall. However, the prediction would be 73.5 inches if the father is 76 inches tall. As expected, tall fathers tend to have tall sons, and short fathers tend to have short sons. Then, I’ll tell my class:
However, to the psychological comfort of us short people, tall fathers tend to have sons who are not quite as tall, and short fathers tend to have sons who are not quite as short.
This was first observed by Francis Galton (see the Wikipedia article for more details), a particularly brilliant but aristocratic (read: snobbish) mathematician who had high hopes for breeding a race of super-tall people with the proper use of genetics, only to discover that the laws of statistics naturally prevented this from occurring. Defeated, he called this phenomenon “regression toward the mean,” and so we’re stuck with called fitting data to a straight line “regression” to this day.
When I teach Algebra II or Precalculus (or train my future high school teachers to teach these subjects), we eventually land on the Rational Root Test and Descartes’ Rule of Signs as an aid for finding the roots of cubic equations or higher. Before I get too deep into this subject, however, I like to give a 10-15 minute pseudohistory about the discovery of how polynomial equations can be solved. Historians of mathematics will certain take issue with some of this “history.” However, the main purpose of the story is not complete accuracy but engaging students with the history of mathematics. I think the story I tell engages students while remaining reasonably accurate… and I always refer students to various resources if they want to get the real history.
To begin, I write down the easiest two equations to solve (in all cases, :
These are pretty easy to solve, with solutions well known to students:
In other words, there are formulas that you can just stick in the coefficients and get the answer out without thinking too hard. Sure, there are alternate ways of solving for that could be easier, like factoring, but the worst-case scenario is just plugging into the formula.
These formulas were known to Babylonian mathematicians around 2000 B.C. (When I teach this in class, I write the date, and all other dates and discoverers, next to the equations for dramatic pedagogical effect.) Though not written in these modern terms, basically every ancient culture on the globe that did mathematics had some version of these formulas: for example, the ancient Egyptians, Greeks, Chinese, and Mayans.
Naturally, this leads to a simple question: is there a formula for the cubic:
Is there some formula that we can just plug , , , and to just get the answer? The answer is, Yes, there is a formula. But it’s nasty. The formula was not discovered until 1535 A.D., and it was discovered by a man named Tartaglia. During the 1500s, the study of mathematics was less about the dispassionate pursuit of truth and more about exercising machismo. One mathematician would challenge another: “Here’s my cubic equation; I bet you can’t solve it. Nyah-nyah-nyah-nyah-nyah.” Then the second mathematician would solve it and challenge the first: “Here’s my cubic equation; I bet you can’t solve it. Nyah-nyah-nyah-nyah-nyah.” And so on. Well, Tartaglia came up with a formula that would solve every cubic equation. By plugging in , , , and , you get the answer out.
Tartaglia’s discovery was arguably the first triumph of the European Renaissance. The solution of the cubic was perhaps the first thing known to European mathematicians in the Middle Ages that was unknown to the ancient Greeks.
In 1535, Tartaglia was a relatively unknown mathematician, and so he told a more famous mathematician, Cardano, about his formula. Cardano told Tartaglia, why yes, that is very interesting, and then published the formula under his own name, taking credit without mention of Tartaglia. To this day, the formula is called Cardano’s formula.
So there is a formula. But it would take an entire chalkboard to write down the formula. That’s why we typically don’t make students learn this formula in high school; it’s out there, but it’s simply too complicated to expect students to memorize and use.
This leads to the next natural question: what about quartic equations?
The solution of the quartic was discovered less than five years later by an Italian mathematician named Ferrari. Ferrari found out that there is a formula that you can just plug in , , , , and , turn the crank, and get the answers out. Writing out this formula would take two chalkboards. So there is a formula, but it’s also very, very complicated.
Of course, Ferrari had some famous descendants in the automotive industry.
So now we move onto my favorite equation, the quintic. (If you don’t understand why it’s my favorite, think about my last name.)
After solving the cubic and quartic in rapid succession, surely there should also be a formula for the quintic. So they tried, and they tried, and they tried, and they got nowhere fast. Finally, the problem was solved nearly 300 years later, in 1832 (for the sake telling a good story, I don’t mention Abel) by a French kid named Evariste Galois. Galois showed that there is no formula. That takes some real moxie. There is no formula. No matter how hard you try, you will not find a formula that can work for every quintic. Sure, there are some quintics that can be solved, like . But there is no formula that will work for every single quintic.
Galois made this discovery when he was 19 years old… in other words, approximately the same age as my students. In fact, we know when wrote down his discovery, because it happened the night before he died. You see, he was living in France in 1832. What was going on in France in 1832? I ask my class, have they seen Les Miserables?
France was torn upside-down in 1832 in the aftermath of the French Revolution, and young Galois got into a heated argument with someone over politics; Galois was a republican, while the other guy was a royalist. More importantly, both men were competing for the hand of the same young woman. So they decided to settle their differences like honorable Frenchmen, with a duel. So Galois wrote up his mathematical notes one night, and the next day, he fought the duel, he lost the duel, and he died.
Thus giving complete and total proof that tremendous mathematical genius does not prevent somebody from being a complete idiot.
For the present, there are formulas for cubic and quartic equations, but they’re long and impractical. And for quintic equations and higher, there is no formula. So that’s why we teach these indirect methods like the Rational Root Test and Descartes’ Rule of Signs, as they give tools to use to guess at the roots of higher-order polynomials without using something like the quadratic formula.