My Favorite One-Liners: Part 46

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 one-liner is something I’ll use after completing some monumental calculation. For example, if z, w \in \mathbb{C}, the proof of the triangle inequality is no joke, as it requires the following as lemmas:

  • \overline{z + w} = \overline{z} + \overline{w}
  • \overline{zw} = \overline{z} \cdot \overline{w}
  • z + \overline{z} = 2 \hbox{Re}(z)
  • |\hbox{Re}(z)| \le |z|
  • |z|^2 = z \cdot \overline{z}
  • \overline{~\overline{z}~} = z
  • |\overline{z}| = |z|
  • |z \cdot w| = |z| \cdot |w|

With all that as prelude, we have

|z+w|^2 = (z + w) \cdot \overline{z+w}

= (z+w) (\overline{z} + \overline{w})

= z \cdot \overline{z} + z \cdot \overline{w} + \overline{z} \cdot w + w \cdot \overline{w}

= |z|^2 + z \cdot \overline{w} + \overline{z} \cdot w + |w|^2

= |z|^2  + z \cdot \overline{w} + \overline{z} \cdot \overline{~\overline{w}~} + |w|^2

= |z|^2 + z \cdot \overline{w} + \overline{z \cdot \overline{w}} + |w|^2

= |z|^2 + 2 \hbox{Re}(z \cdot \overline{w}) + |w|^2

\le |z|^2 + 2 |z \cdot \overline{w}| + |w|^2

= |z|^2 + 2 |z| \cdot |\overline{w}| + |w|^2

= |z|^2 + 2 |z| \cdot |w| + |w|^2

= (|z| + |w|)^2

In other words,

|z+w|^2 \le (|z| + |w|)^2.

Since |z+w| and |z| + |w| are both positive, we can conclude that

|z+w| \le |z| + |w|.

QED

In my experience, that’s a lot for students to absorb all at once when seeing it for the first time. So I try to celebrate this accomplishment:

Anybody ever watch “Home Improvement”? This is a Binford 6100 “more power” mathematical proof. Grunt with me: RUH-RUH-RUH-RUH!!!

My Favorite One-Liners: Part 45

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 favorite pedagogical techniques is deliberating showing students a wrong way of solving a certain math problem, discussing why it’s the wrong way, and then salvaging the solution to construct the right way of doing the problem. I think this keeps students engaged in the lesson as opposed to learning a new technique by rote memorization.

Earlier in my teaching career, I noticed an unintended side-effect of this pedagogical technique. A student came to me for help in office hours because she couldn’t understand something that she had written in her notes. Lo and behold, she had written down the wrong way of doing the problem and had forgotten that it was the wrong way. Naturally, I clarified this for her.

This got me to thinking: I still would like to use this method of teaching from time to time, but I don’t want to cause misconceptions to arise because somebody was dutifully taking notes but didn’t mark that he/she was writing down an incorrect technique. So I came across the following one-liner that I now use whenever I’m about to start this technique:

Don’t write down what I’m about to say; it’s wrong.

Hopefully this prevents diligent students from taking bad notes as well as tips them off that they need to start paying attention to see where the logic went wrong and thus construct the proper technique.

My Favorite One-Liners: Part 44

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 quip is something that I’ll use to emphasize that the meaning of the word “or” is a little different in mathematics than in ordinary speech. For example, in mathematics, we could solve a quadratic equation for x:

x^2 + 2x - 8 = 0

(x+4)(x-2) = 0

x + 4 = 0 \qquad \hbox{OR} \qquad x - 2 = 0

x = -4 \qquad \hbox{OR} \qquad x = 2

In this example, the word “or” means “one or the other or maybe both.” It could be that both statements are true, as in the next example:

x^2 + 2x +1 = 0

(x+1)(x+1) = 0

x + 1 = 0 \qquad \hbox{OR} \qquad x + 1= 0

x = -1 \qquad \hbox{OR} \qquad x = -1

However, in plain speech, the word “or” typically means “one or the other, but not both.” Here the quip I’ll use to illustrate this:

At the end of “The Bachelor,” the guy has to choose one girl or the other. He can’t choose both.

My Favorite One-Liners: Part 43

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. q Q

Years ago, my first class of students decided to call me “Dr. Q” instead of “Dr. Quintanilla,” and the name has stuck ever since. And I’ll occasionally use this to my advantage when choosing names of variables. For example, here’s a typical proof by induction involving divisibility.

Theorem: If n \ge 1 is a positive integer, then 5^n - 1 is a multiple of 4.

Proof. By induction on n.

n = 1: 5^1 - 1 = 4, which is clearly a multiple of 4.

n: Assume that 5^n - 1 is a multiple of 4.

At this point in the calculation, I ask how I can write this statement as an equation. Eventually, somebody will volunteer that if 5^n-1 is a multiple of 4, then 5^n-1 is equal to 4 times something. At which point, I’ll volunteer:

Yes, so let’s name that something with a variable. Naturally, we should choose something important, something regal, something majestic… so let’s choose the letter q. (Groans and laughter.) It’s good to be the king.

So the proof continues:

n: Assume that 5^n - 1 = 4q, where q is an integer.

n+1. We wish to show that 5^{n+1} - 1 is also a multiple of 4.

At this point, I’ll ask my class how we should write this. Naturally, I give them no choice in the matter:

We wish to show that 5^{n+1} - 1 = 4Q, where Q is some (possibly different) integer.

Then we continue the proof:

5^{n+1} - 1 = 5^n 5^1 - 1

= 5 \times 5^n - 1

= 5 \times (4q + 1) - 1 by the induction hypothesis

= 20q + 5 - 1

= 20q + 4

= 4(5q + 1).

So if we let Q = 5q +1, then 5^{n+1} - 1 = 4Q, where Q is an integer because q is also an integer.

QED

green line

On the flip side of braggadocio, the formula for the binomial distribution is

P(X = k) = \displaystyle {n \choose k} p^k q^{n-k},

where X is the number of successes in n independent and identically distributed trials, where p represents the probability of success on any one trial, and (to my shame) q is the probability of failure.

 

 

Was There a Pi Day on 3/14/1592?

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

Mean Green Math

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.