1592?

In honor of Pi Day, here’s a bonus edition of Mean Green Math from two years ago.

March 14, 2015 has been labeled the Pi Day of the Century because of the way this day is abbreviated, at least in America: 3/14/15.

I was recently asked an interesting question: did any of our ancestors observe Pi Day about 400 years ago on 3/14/1592? The answer is, I highly doubt it.

My first thought was that $latex pi$ may not have been known to that many decimal places in 1592. However, a quick check on Wikipedia (see also here), as well as the book “$latex pi$ Unleashed,” verifies that my initial thought was wrong. In China, 7 places of accuracy were obtained by the 5th century. By the 14th century, $latex pi$ was known to 13 decimal places in India. In the 15th century, $latex pi$ was calculated to 16 decimal places in Persia.

It’s highly doubtful that the mathematicians in these ancient cultures actually talked to…

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My Favorite One-Liners: Part 42

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.

The function $f(x) = a^x$ typically exhibits exponential growth (if $a > 1$ ) or exponential decay (if $a < 1$ ). The one exception is if $a = 1$ , when the function is merely a constant. Which often leads to my favorite blooper from Star Trek. The crew is trying to find a stowaway, and they get the bright idea of turning off all the sound on the ship and then turning up the sound so that the stowaway’s heartbeat can be heard. After all, Captain Kirk boasts, the Enterprise has the ability to amplify sound by 1 to the fourth power.

My Favorite One-Liners: Part 41

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.

Every once in a while, my brain, my lips, and my hands get out of sync while I’m teaching, so that I’ll write down what I really mean but I’ll say something that’s different. (I don’t think that this affliction is terribly unique to me, which is why I err on the side of grace whenever a politician or other public figure makes an obvious mistake in a speech.) Of course, such mistakes still have to be corrected, and often students will point out that I said something that was completely opposite of what I meant to say. When this happens, I jocularly wave my fingers at my class with the following playful admonition:

Do what I mean, not what I say.

My Favorite One-Liners: Part 40

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.

In some classes, the Greek letter $\phi$ or $\Phi$ naturally appears. Sometimes, it’s an angle in a triangle or a displacement when graphing a sinusoidal function. Other times, it represents the cumulative distribution function of a standard normal distribution.

Which begs the question, how should a student pronounce this symbol?

I tell my students that this is the Greek letter “phi,” pronounced “fee”. However, other mathematicians may pronounce it as “fie,” rhyming with “high”. Continuing,

Other mathematicians pronounce it as “foe.” Others, as “fum.”

My Favorite One-Liners: Part 39

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.

One of my great pet peeves while I’m teaching is the perennial question “Is this going to be on the test?”, usually after I’ve proven a theorem. Over the years, I’ve come up with the perfect response:

Put this on the test… boy, that’s a great idea.

Then I’ll get some paper, write a “note” to myself to place said theorem on the test, and place it in my pocket. All the while, the rest of the students are grumbling things like “Way to go,” “Thanks for giving him the idea,” and the like.

Since pulling this little song and dance routine, nobody has ever asked me a second time if something’s going to be on the test.

My Favorite One-Liners: Part 38

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.

When I was a student, I heard the story (probably apocryphal) about the mathematician who wrote up a mathematical paper that was hundreds of pages long and gave it to the departmental administrative assistant to type. (This story took place many years ago before the advent of office computers, and so typewriters were the standard for professional communication.) The mathematician had written “iff” as the standard abbreviation for “if and only if” since typewriters did not have a button for the $\Leftrightarrow$ symbol.

Well, so the story goes, the administrative assistant saw all of these “iff”s, muttered to herself about how mathematicians don’t know how to spell, and replaced every “iff” in the paper with “if”.

And so the mathematician had to carefully pore through this huge paper, carefully checking if the word “if” should be “if” or “iff”.

I have no idea if this story is true or not, but it makes a great story to tell students.

My Favorite One-Liners: Part 37

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.

Sometimes, I’ll deliberately show something wrong to my students, and their job is to figure out how it went wrong. For example, I might show my students the classic “proof” that $1= 2$ :

$x =y$

$x^2 = xy$

$x^2 - y^2 = xy - y^2$

$(x+y)(x-y) = y(x-y)$

$x + y = y$

$y + y = y$

$2y = y$

$2 = 1$

After coming to the conclusion, as my students are staring at this very unanticipated result, I’ll smile with my best used-car salesman smile and say “Trust me,” just like in the old Joe Isuzu commercials.

Of course, the joke is that my students shouldn’t trust me, and they should figure out exactly what happened.

My Favorite One-Liners: Part 36

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.

Not everything in mathematics works out the way we’d prefer it to. For example, in statistics, a Type I error, whose probability is denoted by $\alpha$ , is rejecting the null hypothesis even though the null hypothesis is true. Conversely, a Type II error, whose probability is denoted by $\beta$ , is retaining the null hypothesis even though the null hypothesis is false.

Ideally, we’d like $\alpha = 0$ and $\beta = 0$ , so there’s no chance of making a mistake. I’ll tell my students:

There are actually two places in the country where this can happen. One’s in California, and the other is in Florida. And that place is called Fantasyland.

My Favorite One-Liners: Part 35

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.

Every once in a while, I’ll discuss something in class which is at least tangentially related to an unsolved problems in mathematics. For example, when discussing infinite series, I’ll ask my students to debate whether or not this series converges:

$1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \dots$

Of course, this one converges since it’s an infinite geometric series. Then we’ll move on to an infinite series that is not geometric:

$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots$ ,

where the denominators are all perfect squares. I’ll have my students guess about whether or not this one converges. It turns out that it does, and the answer is exactly what my students should expect the answer to be, $\pi^2/6$ .

Then I tell my students, that was a joke (usually to relieved laughter).

Next, I’ll put up the series

$1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \dots$ ,

where the denominators are all perfect cubes. I’ll have my students guess about whether or not this one converges. Usually someone will see that this one has to converge since the previous one converged and the terms of this one are pairwise smaller than the previous series — an intuitive use of the Dominated Convergence Test. Then, I’ll ask, what does this converge to?

The answer is, nobody knows. It can be calculated to very high precision with modern computers, of course, but it’s unknown whether there’s a simple expression for this sum.

So, concluding the story whenever I present an unsolved problem, I’ll tell my students,

If you figure out the answer, call me, and call me collect.

My Favorite One-Liners: Part 34

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.

Suppose that my students need to prove a theorem like “Let $n$ be an integer. Then $n$ is odd if and only if $n^2$ is odd.” I’ll ask my students, “What is the structure of this proof?”

The key is the phrase “if and only if”. So this theorem requires two proofs:

Assume that $n$ is odd, and show that $n^2$ is odd.
Assume that $n^2$ is odd, and show that $n$ is odd.

I call this a blue-light special: Two for the price of one. Then we get down to the business of proving both directions of the theorem.

I’ll also use the phrase “blue-light special” to refer to the conclusion of the conjugate root theorem: if a polynomial $f$ with real coefficients has a complex root $z$ , then $\overline{z}$ is also a root. It’s a blue-light special: two for the price of one.