Math Forum Discussions

Ladnor Geissinger

Posts: 310
From: University of North Carolina at Chapel Hill
Registered: 12/4/04

writing shortcuts lead to misunderstanding of sqrt and abs
Posted: Sep 5, 2005 4:29 PM

Plain Text

I have noticed that shortcuts in writing (even though they may be common mathspeak) seem to cause misunderstanding related to abs and sqrt, and that erupts in arguments on several math lists.

The square root function sqrt (often denoted by the radical sign) was essentially known and used probably even before the users were thoroughly comfortable with using negative numbers. That is, it was known that for any positive number p there exists a unique positive number q such that q^2 = p, and they had ways to find good approximations for q. (The number q might well have been called "THE square root of p".) As used now and imbedded in calculators, sqrt is a function with source and target both being the non-negative real numbers so that sqrt(p) is q, and with the addition of 0 so that sqrt(0) is 0. In fact, for convenience of the user, to warn when something is inappropriate, many calculators also respond when a negative number is input into sqrt by outputting ERROR.

Well, people grew confortable with using negative numbers and started to look for all the solutions to equations -- often the equation was f(t)=0 where f(t) was a polynomial. The solutions are called the "roots of f". If the equation is as simple as t^2=p, then everyone noticed first that that there are no real roots if p is negative, and second that if p is positive then q=sqrt(p) is the only positive root and -q is the only negative root.
This latter sentence is what is almost always shortened to the briefer version: "the square roots of p are sqrt(p) and -sqrt(p)".
And when this occurs in the context of "solving the equation t^2=p" then it is often abbreviated in the form: "If t^2=p then t=sqrt(p) or t=-sqrt(p)."
Or it is rendered in the even more cryptic form (but intended to mean the same thing):
"If t^2=p then t=+-sqrt(p)".
There is nothing really wrong with this except that as is common in mathspeak, a universal quantifier is missing and understood. We should really begin the sentence with "For all real numbers t, if t^2=p then ...."

A similar thing occurs with the much simpler abs function which has domain all real numbers and produces only nonnegative output. Suppose we have an equation like abs(t)=p and we wish to "solve for t, or for some variable that occurs in the expression t". First we should note that there cannot exist a numerical expression t such that abs(t)=p if p is negative - hence no solutions. And when p is nonnegative, we assume that t is some numerical expression such that abs(t)=p and then ask what we may properly deduce from that. We may correctly conclude that "t=p or that t=-p", from the usual definition of the abs function. Then likely we will break this into two cases and see what further can be said in each case.

In both cases we get a conclusion that is of the form "t=u or t=-u" which is correct and only seems to cause problems when it is shortened to "t=+-u" and then the "or" is neglected or misunderstood.