Recent editions of Monday Math have seen us working pretty hard. So how about we lighten the mood a bit and think about fractions.

Let us start with the obvious. Fractions have tops and bottoms. Got that? Numerators and denominators exist only in elementary and middle school math classes. Their sole purpose is to make mathematics as offputting and unpleasant as possible. If you refer to the top of the fraction and the bottom of the fraction everyone knows what you mean. Say numerator and denominator (which come respectively from Latin words meaning roughly “one who numbers” and “that which is named”) and you may as well be speaking a different language.

When mathematicians look at a fraction like *1/x* we say, “One over *x*” At least, that’s what American mathematicians say. British mathematicians use the far more elegant phrase, “One upon *x*.” They do that because they speak the language much better than we do. At any rate, we certainly do not say, “one *x*th” or some such thing as that.

And why is it that every kid, when learning how to add fractions, is told of the terrible things that occur upon failing to use the least common denominator? (Yes, *FINE*, here I shall allow the term, on account of “common bottom” sounding ridiculous.) What, pray tell, is the BFD? It is bad enough they must struggle with the whole idea of a common denominator, a notion invariably presented as a rule from on high possessing neither rhyme nor reason. But then they are told, having figured out the trick where you just multiply together the given denominators, that such techniques are not acceptable. Why? Because let’s say you are adding one sixth to one fourth. If you naively multiply the denominators you will get twenty-four as the common denominator, when actually it should be twelve. I mean, really, check out the carnage:

\[

\frac{1}{6}+\frac{1}{4}=\frac{4}{24}+\frac{6}{24}=\frac{10}{24}.

\]

Are your eyes bleeding yet? Have you fully appreciated the horror? Your answer is not even reduced! Kids actually get points taken off for this. And then we wonder why they don’t like math.

From where did we ever get the notion that when adding fractions it is terribly important that your sum be pre-reduced? If it bothers you to have a non-reduced fraction as your answer then just reduce it after adding. It’s OK, really.

But while we’re at it, let’s ruminate about this miserable obsession with reducing fractions. *10/24* is a perfectly good fraction. If you have some specific reason for wanting the smallest numbers possible then by all means go ahead and reduce. But don’t do it because you think the reduced fraction is necessarily simpler, or because you think the state of the world is happier when tops and bottoms (!!) have no common factors, or anything along those lines.

Let me tell you what happens when you are morbidly obsessed with reducing fractions. A while back I had to install a new doorknob on a closet door in my house. This meant buying one of those small, hand-held power drills, since new holes were needed for the screws that would hold the small metal dingus to the door. When I prepared to begin drilling I discovered that the five thirty-seconds bit was too fat, but the three thirty-seconds bit was too thin. So I went hunting around for the four thirty-seconds bit. Do you think I found it? Of course not, because whoever stamped the widths onto the bits came from the zombie-like, “must reduce my fractions” school of fraction manipulation, and therefore I had to find the one eighth bit. That’s right! My bits were stamped in halves, fourths, eighths, sixteenths and thirty-seconds, all because of an educational system that thinks reducing fractions is an inherent good, like charity and kindness to others. Of course, as a highly trained math professional I can do these conversions in my head. But I shudder to think of all the missized holes that have been drilled in the nations door jambs over this.

And where this really stops being funny is when you get to calculus. Suddenly you are asked to find the derivative of

\[

f(x)=\frac{x^3+7x-13}{x^4+5x^2}.

\]

You remember the jingle, “If the quotient rule you want to know, it’s low dee high minus high dee low,” and since no one ever has any trouble remembering what goes on the bottom (!!) of one of these goobers we get

\[

f'(x)=\frac{(x^4+5x^2)(3x^2+7)-(x^3+7x-13)(4x^3+10x)}{(x^4+5x^2)^2}.

\]

How many times have I explained to students that they should stop at this point, seeing as how the problem asked them to find the derivative and they have, in fact, found it? That, with probability approaching one, they will mess it up if they set their miserable algebra skills to the task of multiplying everything out? And how many times have they ignored me, multiplying away with the same enthusiasm with which a small child offers to “help” you with some household chore, in the bizarre but sincere belief that they are somehow “simplifying” the fraction? And

*then*mind you, after wasting ten minutes in this futile and totally unnecessary effort, they turn to me and complain they did not have enough time to finish the test!

But you know what makes this *really* not funny? The fact that if they multiply it out *then I have to do likewise*, on the off chance that they actually did it right.

Highly vexing.

And while we’re at it, who came up with mixed numbers? WTF? As far as I’m concerned if you write

\[

2 \ \frac{7}{9}

\]

you’re talking about fourteen ninths. The proper way of expressing two and seven ninths is

\[

\frac{25}{9}.

\]

Improper fraction my ass! That is how fractions

*should*be written. It is how they

*would*be written if the world was the product of a just and loving God, which it is not. Instead of that perfectly clear, impossible to misinterpret fraction, we get the hellish, confusing miasma of mixed numbers. Try breaking a college student of

*that*little habit!

And stop using diagonal lines for your fraction bars, people! Major yuck! A proper faction bar, in nearly all cases, is a horizontal line perfectly parallel to the edge of the paper. (Notice, by the way, that “parallel” has two ells, then one ell. Just mentioning it, even though common math spelling errors is actually a different post.) The only exception is when you are dealing with very complicated fractions with elaborate tops and bottoms (!!), in which case for typographical reasons it is sometimes convenient to use a diagonal line.

And if you insist on using decimals (whose primary pedagogical purpose, so far as I am concerned, is for teaching people that clever little trick for turning a repeating decimal like .347347347347… back into a rational number), then at least make sure your decimal point is thick and dark and easy to read. Don’t leave me wondering whether that smudge on the paper is a decimal point or not!

Oh, I could go on. You *know* I could. But since it is barely conceivable that I am trying your patience, and since the season premiere of *House* is less than ninety minutes away, I shall call it a day. See you next week!