Tuesday, January 10, 2012


Thanks to the book Group Theory in the Bedroom, by Brian Hayes, I finally found someone to blame for my geometry students' daily growing hatred of mathematical language.  His name is Robert Recorde.  For those of you who, like pre-this-week me, have never heard of Robert Recorde, don't worry: you've seen his handiwork.  Recorde was a 16th-century Welsh doctor, mathematician, and author of Whetstone of Witte (1557), the book in which the modern equals sign first appears.

Hayes only mentions Recorde's contribution in passing, as part of an extremely interesting essay on the mathematical and philosophical nature of identity, but includes a curious quotation from Whetstone that explains why Recorde chose the particular symbol he did.  Here is a reproduction of the original text:

And here is the content of the second sentence, in somewhat more recognizable English:
"And to avoid the tedious repetition of these words, 'is equal to,' I will set, as I do often in work use, a pair of parallels...lines of one length, thus: ====, because no two things can be more equal."
On one hand, what a beautiful example of simple, useful notation arising organically from one man's dissatisfaction with the clunky and cumbersome status quo.  Here's a guy who was really tired of writing "is equal to" over and over again, and so he invented some shorthand to make things more comprehensible in his daily efforts.  (If you want an example of just how awkward mathematical ideas can be when posed in natural language instead of symbols, here is an example, drawn more or less at random from Euclid's Elements: "If from a point without a circle, straight lines are drawn to the circumference; of those falling upon the concave circumference the greatest is that which passes through the center, and the line which is nearer the greatest is greater than that which is more remote."  Sheesh.)

On the other hand, there is something about Recorde's explanation of his notational choice that strikes the modern mathematical reader as odd, particularly if that modern mathematical reader is a teacher of high school geometry.  Namely, parallel lines are not even close to equal.

To be fair to Mr. Recorde, these matters were not so nicely settled in his day, but there is a non-negligible amount of irony in the fact that the "modern" equals sign was chosen for a reason completely at odds with the "modern" concept of equality.  How so?

To answer that, let's set aside equality for a moment and consider sameness.  Specifically, let's consider that there are (at least) two perfectly sensible interpretations of "the same" in everyday speech.  If I pull into the parking lot at school tomorrow, and in the spot next to me is someone who also drives a blue 2002 Subaru Impreza WRX, I might say something to the other driver like, "Hey, we have the same car!"  That is, on some fundamental level, our cars share enough defining characteristics to be considered identical, even though we both agree that the two vehicles are composed of distinct sets of molecules.  Compare this situation to one in which I say to my sister, "Hey, we have the same father!"  This time there is only one referent, one object, one guy.

In our everyday lives, the type of sameness in question is easily discerned by the context.  I am very rarely confused about whose car I'm driving, or about how two people can share a parent.  In math though, these two concepts of sameness have distinct technical denotations.  And, since most students think of "equal" and "same" as being, well, the same, there arises a creeping discomfort from having to make a seemingly picayune distinction where it doesn't ordinarily exist.  And geometry class is the first place this confusion manifests in earnest.

Geometric objects are considered "equal" if and only if they are the same as sets.  They contain exactly the same points.  To say, for instance, that one quadrilateral is equal to another is like saying that my father is the same as my sister's father.  There is really only one object in question.

If, however, we have geometric objects whose fundamental characteristics are identical, we replace the word "equal" with "congruent."  To say one quadrilateral is congruent to another is to say that their corresponding sides have the same length, and that their corresponding angles have the same measure, because side lengths and angle measures are the defining characteristics of quadrilaterals.  This is the Subaru scenario.

The question of equality v. congruence is already confusing to students, but it gets much, much worse.  For one thing, the ideas are not strictly distinct.  Equality in geometry is actually a subset of congruence: if two things are equal (selfsame), then they are necessarily congruent (characteristically identical), but the converse need not be true.  For another thing, objects that are not equal, but are congruent, have measures that are not congruent, but equal.  How can that be, since I just said that equality implies congruence?  Because the measures are numbers, and numbers aren't geometric objects, so they don't play by the rules of congruence.  (I apologize to the mathematically fastidious reader who is already typing nasty things in my comments box about modular arithmetic.  Yes, numbers can sometimes be congruent, but not really in the same sense.)

As an example, imagine I have two unique line segments with endpoints A&B and C&D, respectively, and both segments measure 4 units in length.  Then I might say, "Segment AB is congruent to segment CD because the measure of segment AB is equal to the measure of segment CD;  furthermore, segment AB is both congruent to and equal to segment AB, but is only congruent to segment CD, which also happens to be both congruent to and equal to itself."  That sentence, in a nutshell, is why my students hate me.

More to the original point, now we see why Recorde's choice of parallel line segments to represent equality is so strange.  Segments are equal if and only if they define precisely the same set of points, and parallel segments don't even have a single point in common; in fact, that's roughly the definition of "parallel."  In a real sense, no two things can be less equal!