# 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.

# 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 $f_3(x) = x(3x-1)/30$.

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.

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 5

Check out this lovely integral, dubbed the Sophomore’s Dream, found by Johann Bernoulli in 1697 (Gamma, page 44):

$\displaystyle \int_0^1 \frac{dx}{x^x} = \displaystyle \frac{1}{1^1} + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4} + \dots$.

I’ll refer to either Wikipedia or Mathworld for the derivation.

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 4

For $s > 1$, Riemann’s famous zeta function is defined by

$\zeta(s) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$.

This is also called a p-series in calculus.

What I didn’t know (Gamma, page 41) is that, in 1748, Leonhard Euler exactly computed this infinite series for $s = 26$ without a calculator! Here’s the answer:

$\displaystyle 1 + \frac{1}{2^{26}} + \frac{1}{3^{26}} + \frac{1}{4^{26}} + \dots = \frac{1,315,862 \pi^{26}}{11,094,481,976,030,578,125}$.

I knew that Euler was an amazing human calculator, but I didn’t know he was that amazing.

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 3

At the time of this writing, it is unknown if there are infinitely many twin primes, which are prime numbers that differ by 2 (like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc.) However, significant progress has been made in recent years. However, it is known (Gamma, page 30) the sum of the reciprocals of the twin primes converges:

$\displaystyle \left( \frac{1}{3} + \frac{1}{5} \right) + \left( \frac{1}{5} + \frac{1}{7} \right) + \left( \frac{1}{11} + \frac{1}{13} \right) + \left( \frac{1}{17} + \frac{1}{19} \right) = 1.9021605824\dots$.

This constant is known as Brun’s constant (see also Mathworld). In the process of computing this number, the infamous 1994 Pentium bug was found.

Although this sum is finite, it’s still unknown if there are infinitely many twin primes since it’s possible for an infinite sum to converge (like a geometric series).

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.

# Report Cards for Famous Mathematicians

Evariste Galois gets an A+ in math and gets a D in behavior. Evariste is a very creative mathematician with potential for greatness. But he needs to get along better with others. Getting into fights is not the way to succeed.

Gödel is excellent at logic but his work is often incomplete.

I am missing Heisenberg’s report card here. “Werner refuses to give an exact answer and seems to be proud of his uncertainty.”

Benoit Mandelbrot C+, great at drawing designs but keeps going on and on and on about perimeters. Next time he should work on calculating areas as his answers were always zero…

# Katherine Johnson: NASA Pioneer and “Computer”

I recently stumbled on the article “This Black NASA Mathematician Was the Reason Many Astronauts Came Home — Their Life Depended on Her Calculations” featuring Katherine Johnson, one of the first African-Americans to work at NASA. The article features an interview with her; I recommend the whole thing highly.

# Computing e to Any Power (Part 2)

In this series, I’m looking at a wonderful anecdote from Nobel Prize-winning physicist Richard P. Feynman from his book Surely You’re Joking, Mr. Feynman!. This story concerns a time that he computed $e^x$ mentally for a few values of $x$, much to the astonishment of his companions.

Part of this story directly ties to calculus.

One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple.

When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small.

As noted, this refers to the Taylor series expansion of $e^x$, which is can be used to compute $e$ to any power. The terms get very small very quickly because of the factorials in the denominator, thus lending itself to the computation of $e^x$. Indeed, this series is used by modern calculators (with a few tricks to accelerate convergence). In other words, the series from calculus explains how the mysterious “black box” of a graphing calculator actually works.

Continuing the story…

“Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey.

I say, “That’s easy. It’s 27.11.”

Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?”

Another guy says, “You know Feynman, he’s just faking it. It’s not really right.”

They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say.

They find it in the table. “It’s right! But how’d you do it!”

For now, I’m going to ignore how Feynman did this computation in his head and instead discuss “the table.” The setting for this story was approximately 1940, long before the advent of handheld calculators. I’ll often ask my students, “The Brooklyn Bridge got built. So how did people compute $e^x$ before calculators were invented?” The answer is by Taylor series, which were used to produce tables of values of $e^x$. So, if someone wanted to find $e^{3.3}$, they just had a book on the shelf.

For example, the following page comes from the book Marks’ Mechanical Engineers’ Handbook, 6th edition, which was published in 1958 and which I happen to keep on my bookshelf at home.

Look down the fifth and sixth columns of this table, we see that $e^{3.3} \approx 27.11$. Somebody had computed all of these things (and plenty more) using the Taylor series, and they were compiled into a book and sold to mathematicians, scientists, and engineers.

But what if we needed an approximation better more accurate than four significant digits? Back in those days, there were only two options: do the Taylor series yourself, or buy a bigger book with more accurate tables.