Euler and 1,000,009

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

1,000,009 = 1000^2 + 3^2

but that he had since discovered a second partition, namely

1,000,009 = 235^2 + 972^2,

which revealed the composite character of the number.

See Wikipedia and/or Mathworld for the details of how this allowed Euler to factor 1,000,009.

Factoring Mersenne “primes”

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.”

The Map of Mathematics

This video made the rounds earlier this year:

My Favorite One-Liners: Part 52

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

y = \overline{y} + r \displaystyle \frac{s_y}{s_x} (x - \overline{x}),

we obtain the equation

y = 69 + 0.5(x-69) = 0.5x + 34.5,

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.

My Favorite One-Liners: Part 49

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 post is certainly not a one-liner but instead is my pseudohistory for how the roots of polynomials were found.

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, a \ne 0:

ax + b = 0 \qquad and \qquad ax^2 + bx + c = 0

These are pretty easy to solve, with solutions well known to students:

x = -\displaystyle \frac{b}{a} \qquad and \qquad x = \displaystyle \frac{-b \pm \sqrt{b^2-4ac}}{2a}

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 x 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:

ax^3 + bx^2 + cx + d = 0

Is there some formula that we can just plug a, b, c, and d 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 a, b, c, and d, 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?

ax^4 + bx^3 + cx^2 + dx + e = 0

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 a, b, c, d, and e, 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.)

ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0

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 x^5 = 0. 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.

green lineReal references:

http://mathworld.wolfram.com/QuadraticEquation.html

http://mathworld.wolfram.com/CubicFormula.html

http://mathworld.wolfram.com/QuarticEquation.html

http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html

http://mathworld.wolfram.com/QuinticEquation.html

http://library.wolfram.com/examples/quintic/

http://library.wolfram.com/examples/quintic/timeline.html

My Favorite One-Liners: Part 19

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. This is a quip that I’ll use when a theoretical calculation can be easily confirmed with a calculator. Today’s post is less of a one-liner than a story.

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, a \ne 0:

ax + b = 0 \qquad and \qquad ax^2 + bx + c = 0

These are pretty easy to solve, with solutions well known to students:

x = -\displaystyle \frac{b}{a} \qquad and \qquad x = \displaystyle \frac{-b \pm \sqrt{b^2-4ac}}{2a}

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 x 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:

ax^3 + bx^2 + cx + d = 0

Is there some formula that we can just plug a, b, c, and d 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 a, b, c, and d, 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?

ax^4 + bx^3 + cx^2 + dx + e = 0

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 a, b, c, d, and e, 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.)

ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0

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 x^5 = 0. 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.

Which bring me to the conclusion of this story: we have 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.

green lineReal references:

http://mathworld.wolfram.com/QuadraticEquation.html

http://mathworld.wolfram.com/CubicFormula.html

http://mathworld.wolfram.com/QuarticEquation.html

http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html

http://mathworld.wolfram.com/QuinticEquation.html

http://library.wolfram.com/examples/quintic/

http://library.wolfram.com/examples/quintic/timeline.html

Tribute to William Rowan Hamilton

This was right up my alley: a mash-up of mathematical physics and musical theater to pay tribute to William Rowan Hamilton, developer of quaterions and one of the founders of quantum physics.

Linus Pauling and the Golden Rule

Source: http://www.azquotes.com/quote/378980

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 9

When teaching students mathematical induction, the following series (well, at least the first two or three) are used as typical examples:

1 + 2 + 3 + \dots + n = \displaystyle \frac{n(n+1)}{2}

1^2 + 2^2 + 3^2 + \dots + n^2 = \displaystyle \frac{n(n+1)(2n+1)}{6}

1^3 + 2^3 + 3^3 + \dots + n^3 = \displaystyle \frac{n^2(n+1)^2}{4}

1^4 + 2^4 + 3^4 + \dots + n^4 = \displaystyle \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}

What I didn’t know (Gamma, page 81) is that Johann Faulhaber published the following cute result in 1631 (see also Wikipedia): If k is odd, then

1^k + 2^k + 3^k + \dots + n^k = f_k(n(n+1)),

where f_k is a polynomial. For example, to match the above examples, f_1(x) = x/2 and f_3(x) = x^2/4. Furthermore, if k is even, then

1^k + 2^k + 3^k + \dots + n^k = (2n+1) f_k(n(n+1)),

where again f_k is a polynomial. For example, to match the above examples, f_2(x) = x/6 and f_3(x) = x(3x-1)/30.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 6

As noted in Gamma (page 65), mathematician meant astrologer in context. Still, what a terrific quote.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.