Entrance exam at MIT

Here’s a story that I’ll tell my students when, for the first time in a semester, I’m about to use a lemma to make a major step in proving a theorem. (I think I was 13 when I first heard this one, and obviously it’s stuck with me over the years.)

At MIT, there’s a two-part entrance exam to determine who will be the engineers and who will be the mathematicians. For the first part of the exam, students are led one at a time into a kitchen. There’s an empty pot on the floor, a sink, and a stove. The assignment is to boil water. Everyone does exactly the same thing: they fill the pot with water, place it on the stove, and then turn the stove on. Everyone passes.

For the second part of the exam, students are led one at a time again into the kitchen. This time, there’s a pot full of water sitting on the stove. The assignment, once again, is to boil water. Nearly everyone simply turns on the stove. These students are led off to become engineers. The mathematicians are ones who take the pot off the stove, dump the water into the sink, and place the empty pot on the floor… thereby reducing to the original problem, which had already been solved.

Laws of Logarithms

One of the most common student mistakes with logarithms is thinking that

$\log_b(x+y) = \log_b x + \log_b y$ .

When I first started my career, I referred to this as the Third Classic Blunder. The first classic blunder, of course, is getting into a major land war in Asia. The second classic blunder is getting into a battle of wits with a Sicilian when death is on the line. And the third classic blunder is thinking that $\log_b(x+y)$ somehow simplfies as $\log_b x + \log_b y$ .

Sadly, as the years pass, fewer and fewer students immediately get the cultural reference. On the bright side, it’s also an opportunity to introduce a new generation to one of the great cinematic masterpieces of all time.

One of my colleagues calls this mistake the Universal Distributive Law, where the $\log_b$ distributes just as if $x+y$ was being multiplied by a constant. Other mistakes in this vein include $\sqrt{x+y} = \sqrt{x} + \sqrt{y}$ and $(x+y)^2 = x^2 + y^2$ .

Along the same lines, other classic blunders are thinking that

$\left(\log_b x\right)^n$ simplifies as $\log_b \left(x^n \right)$

and that

$\displaystyle \frac{\log_b x}{\log_b y}$ simplifies as $\log_b \left( \frac{x}{y} \right)$ .

I’m continually amazed at the number of good students who intellectually know that the above equations are false but panic and use them when solving a problem.

Interdisciplinary Studies (Part 1)

An eccentric investor hired a biologist, a mathematician, and a physicist to design and train the perfect racehorse. After studying the problem for a couple of weeks, they returned to the investor to present their results.

The biologist said, “I’ve come up with a plan to breed the perfect racehorse. It’ll just take a thousand or two generations of breeding.”

The mathematician said, “I haven’t been able to solve this problem yet, but I’ve made some preliminary findings. So far, I’ve been able to show that for each horse race there will exist a winner, and furthermore that winner will be unique.”

So the investor’s hopes were pinned on the physicist, who began, “I think I’ve solved this race horse problem, but I had to make a few simplifying assumptions. First, let’s assume that each horse is a perfect frictionless rolling sphere…”

Newton’s Three Laws

The following is a true story that I’ll use to engage my students as we’re about to begin a section requiring the use of Newton’s Second Law.

One night, just before I was about to fall asleep, my wife asked if I could double-check that the front door was locked. Here was the ensuing conversation.

Me: “I’m tired… Newton’s First Law.”

Her: “Newton’s Second Law.”

Me: “Newton’s THIRD Law!!!”

In case you need a translation into plain English:

Me: “A body at rest tends to remain at rest. I don’t want to get up.”

Her: “Force equals mass times acceleration. Then I will push you out of bed.”

Me: “For every action there is an equal and opposite reaction. Then you’ll get pushed out of bed too!”