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By Jason Rosenhouse

A recent satirical essay in the Huffington Post reports that congressional Republicans are trying to legislate the value of pi. Fearing that the complexity of modern geometry is hurting America’s performance on international measures of mathematical knowledge, they have decreed that from now on pi shall be equal to three. It is a sad commentary on American culture that you must read slowly and carefully to be certain the essay is just satire.

It has been wisely observed that reality is that which, when you stop believing in it, doesn’t go away. Scientists are especially aware of this, since it is sometimes their sad duty to inform people of truths they would prefer not to accept. Evolution cannot be made to go away by folding you arms and shaking your head, and the planet is warming precipitously regardless of what certain business interests claim to believe. Likewise, the value of pi is what it is, no matter what a legislative body might think.

That value, of course, is found by dividing the circumference of a circle by its diameter. Except that if you take an actual circular object and apply your measuring devices to it you will obtain only a crude approximation to pi. The actual value is an irrational number, meaning that it is a decimal that goes on forever without repeating itself. One of my middle school math teachers once told me that it is just crazy for a number to behave in such a fashion, and that is why it is said to be irrational. Since I rather liked that explanation, you can imagine my disappointment at learning it was not correct.

In this context, the word “irrational” really just means “not a ratio.” More specifically, it is not a ratio of two integers. You see, if you divide one integer by another there are only two things that can happen. Either the process ends or it goes on forever by repeating a pattern. For example, if you divide one by four you get .25, while if you divide one by three you get .3333… . That these are the only possibilities can be proved with some elementary number theory, but I shall spare you the details of how that is done. That aside, our conclusion is that since pi never ends and never repeats, it cannot be written as one integer divided by another.

Which might make you wonder how anyone evaluated pi in the first place. If the number is defined geometrically, but we cannot hope to measure real circles with sufficient accuracy, then why do we constantly hear about computers evaluating its first umpteen million digits? The answer is that we are not forced to define pi in terms of circles. The number arises in other contexts, notably trigonometry. By coupling certain facts about right triangles with techniques drawn from calculus, you can express pi as the sum of a certain infinite series. That is, you can find a never-ending list of numbers that gets smaller and smaller and smaller, with the property that the more of the numbers you sum the better your approximation to pi. Very cool stuff.

Of course, I’m sure we all know that pi is a little bit larger than three. This means that any circle is just over three times larger around than it is across. The failure of most people to be able to visualize this leads to a classic bar bet. Take any tall, thin, drinking glass, the kind with a long stem, and ask the person sitting nearest you if its height is greater than its circumference. When he answers that it is, bet him that he is wrong. Optically, most such glasses appear to be much taller than they are fat, but unless your specimen is very tall and very thin you will win the bet every time. The circumference is more than three times larger than the diameter at the top of the glass. A vessel so proportioned that this length is nonetheless smal

Jason Rosenhouse is Associate Professor of mathematics at James Madison University in Virginia and the author of The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser, which looks at one of the most interesting mathematical brain teasers of recent times.  In the excerpt below Rosenhouse explains what it is like to be a professional mathematician and introduces The Monty Hall Problem.

Like all professional mathematicians, I take it for granted that most people will be bored and intimidated by what I do for a living.  Math, after all, is the sole academic subject about which people brag of their ineptitude.  “Oh,” says the typical well-meaning fellow making idle chitchat at some social gathering, “I was never any good at math.”  Then he smiles sheepishly, secure in the knowledge that his innumeracy in some way reflects well on him.  I have my world-weary stock answers to such statements.  Usually I say, “Well, maybe you just never had the right teacher.”  That defuses the situation nicely.

It is the rare person who fails to see humor in assigning to me the task of dividing up a check at a restaurant.  You know, because I’m a mathematician.  Like the elementary arithmetic used in check division is some sort of novelty act they train you for in graduate school.  I used to reply with “Dividing up a check is applied math.  I’m a pure mathematician,” but this elicits puzzled looks from those who thought mathematics was divided primarily into the cources they were forced to take in order to graduate versus the ones they could mercifully ignore.  I find “Better have someone else do it.  I’m not good with numbers” works pretty well.

I no longer grow vexed by those who ask, with perfect sincerity, how folks continue to do mathematical research when surely everything has been figured out by now.  My patience is boundless for those who assure me that their grade-school nephew is quite the little math prodigy.  When a student, after absorbing a scintillating presentation of, say, the mean-value theorem, asks me with disgust what it is good for, it does not even occur to me to grow annoyed. Instead I launch into a discourse about all of the practical benefits that accrue from an understanding of calculus.  (”You know how when you flip a switch the lights come on? Ever wonder why that is?  It’s because some really smart scientists like James Clerk Maxwell knew lots of calculus and figured out how to apply it to the problem of taming electricity.  Kind of puts your whining into perspective, wouldn’t you say?”)  And upon learning that a mainstream movie has a mathematical character, I feel cheated if that character and his profession are presented with any element of realism.

(Speaking of which, do you remember that 1966 Alfred Hitchcock movie Torn Curtain, the one where physicist Paul Newman goes to Leipzig in an attempt to elicit certain German military secrets?  Remember the scene where Newman starts writing equations on the chalkboard, only to have an impatient East German scientist, disgusted by the primitive state of American physics, cut him off and finish the equations for him?  Well, we don’t do that.  We don’t finish each other’s equations.  And that scene in Good Will Hunting where emotionally troubled math genius Matt Damon and Fields Medalist Stellan Skarsgard high-five each other after successfully performing some feat of elementary algebra?  We don’t do that either.  And don’t even get me started on Jeff Goldblum in Jurassic Pa