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 is a positive integer, then is a multiple of 4.

Proof. By induction on .

: , which is clearly a multiple of 4.

: Assume that 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 is a multiple of 4, then 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 . (Groans and laughter.) It’s good to be the king.

So the proof continues:

: Assume that , where is an integer.

. We wish to show that 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 , where is some (possibly different) integer.

Then we continue the proof:

by the induction hypothesis

.

So if we let , then , where is an integer because is also an integer.

QED

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

,

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