Radical Notations

February 19, 2008 in Notations

The Square Root Function (let’s call it “Squirt”) is—of course— a certain bijection on the non-negative reals: symbolically, $\sqrt{}: [0,\infty) \rightarrow [0,\infty)$ (“squirt maps the interval zero-to-infinity to itself”). Specifically, if you want to get all technical about it, $\sqrt{\null} := \{(x,y) | x, y \in [0,\infty); y^2 = x\}$ . Conventionally, one writes $\sqrt{x}$ in place of $\sqrt{\null}(x)$ ; the argument ( $x$ ) is called the “radicand” and is said to be “under the radical”. We’ll observe this convention throughout the rest of the discussion. Squirt is a continuous function and enjoys the property that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ whenever both sides of the equation are defined (functions with this property are said to be “multiplicative”). All of this ought to be completely uncontroversial.

Now, it’s perfectly possible—of course—to extend squirt to a certain discontinuous nonmultiplicative nonbijection ( $f$ , say) on the Complex Number Field. But, and I only wish that this were uncontroversial, we sure as sunrise shouldn’t call $f$ “The Square Root Function” (or denote it by $\sqrt{\null}$ )—continuity, bijectivity, and multiplicativity are all very useful properties and shouldn’t be given up at the careless stroke of a keypad.

Yesterday, for my sins, I was made to write out somesuch ghastly nonsense as $\sqrt{-5}\sqrt{-7} = -\sqrt{35}$ right in front of my 104 students. Obviously, I couldn’t bring myself to do it without complaining about it—these people mostly seem to trust me and I’d like to try to deserve it. But whenever I’m made to differ with the text it not only undermines their faith in me personally, but also gives support to the all-too-common idea that Mathematics Is Management: that our ways are arbitrary and meaningless and subject to change at some the whim of some unseen authority figure.

I suppose I know why this section of the textbook is there. We’re about to develop QF—the famous “Quadratic Formula” (assume $A\not=0$ ; one then has $Ax^2 + Bx + C = 0$ if and only if $x = {{-B \pm \sqrt{B^2 - 4AC}}\over{2A}}$ ; I’ve spelled it out mostly out of sheer joie de symbolisme but also to take the opportunity to beg other teachers to adopt the convention that Constants Get Capitalized). When the radicand in QF (also known as the “discriminant”, $B^2 - 4AC$ ) is negative, the solutions to $Ax^2 + Bx + C = 0$ are non-real; students of Algebra need to learn about such solutions. The moment has come: the (so-called) Real Numbers are no longer enough for our purposes. This is, not only well and good, but dearer to my heart than I like to admit in public.

But for pity sake, now that we’re letting these struggling beginners in on this earth-shaking idea (that confused great mathematicians for hundreds of years), why make it any harder than it has to be? Why not just admit what every professional knows: that the symbol $``\sqrt{\null}''$ as applied to a negative number is slang and should never appear without the symbol $``\pm''$ ?

Let me be as clear as I know how. I understand why the textbooks (and the furshlugginer graphing calculator) get this wrong: publishers and computer manufacturers are capitalist pigs, not only indifferent to the truth but actively hostile to the truth. What I don’t get is this: where are the mathematicians? How can you go to work every day and allow this kind of thing to go on in your department at your university, in the name of “mathematics”? What the devil do you think tenure is for?

5 Comments

Lsquared

February 24, 2008 at 5:05 am

Yes, I always feel I must give the same rant when taking a root of a negative number (when taking the root of a positive number, the radical symbol denotes the positive or principal square root of the number, but when doing the same with a negative or complex number, there is no consistent way of defining the principal square root, and so the root symbol should really denote both square roots, and so squirt(-1) does not equal i, it equals i and -i). This rant was instilled deeply in me during some complex analysis class or other, and so I always have feel great reluctance when I encounter the statement squirt(-1)=i. I do not, however (sigh), feel that I can reasonably expect my clueless freshmen to keep this straight, however, so I mostly restrict myself to asking problems they can’t muck up…

Reply
vlorbik

February 25, 2008 at 2:38 pm

JD’s write-up at the carny (below)
has the relevant foot-notes
(links to a related discussion
we had in his blog when i first
found out about it …).

Reply

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