The Square Root Wars

The Square Root Wars; Tales of a Radical

Rod Rodrigues

This is a tale about what has happened to square roots. Square roots are easy, but they are made difficult by those who perpetrate a careless language. After many years of this careless treatment in the textbooks, the radicals are getting worse and worse treatment, to the rightful confusion of the students. Why shouldn’t our students be confused, when their textbooks are confused, their teachers are confused. I have been concerned about this for some time, but my position was bolstered with the article, When Equalities are not Equal: Missing Mathematical Precision in Teaching Texts, and Technology by Michael J. Bossé amd N. R. Nandakumar in the November issue, 2003 of The College Mathematics Journal. They say , “Algebra texts, teachers, and calculators all say some things are equal when they are not. They really shouldn’t.” The authors address more than simply the radical. I will focus here on the radical.

Let’s start with the definitions.

The square root(s) of a number, a, is a number whose square is a. Put into symbols, a square root, x, of a number a, has the property that x² = a. Given the rules of signs, if follows that only non-negative numbers can HAVE Real Number square roots. Furthermore, all positive numbers have 2 square roots. For example,

The square roots of 4 are 2 and -2 because 2² = 4 and (-2)² = 4.
Another example is the square roots of 25. They are 5 and -5 because 5² = 25 and (-5)² = 25.

There is a problem if you speak about “taking the square root of something” especially if it contains a variable expression. Since the variable could be either positive OR negative, you simply don’t know what to say. Is the square root of x² equal to +/- x? After all, if the “square root of 16 is +/- 4”, then the square root of x² is +/- x. Furthermore, if the square root can have two values, then there is no such thing as THE square root, and there is no such thing as a square root function, is there? A function must be single-valued, by definition. Isn’t this poor math language? .If mathematics is to be a precise language, this is a real violation. And most teachers of calculus know the harm that is imbued with early misuse of the language of square roots at this level.

Theoretically, we solve this double-value problem by introducing the radical. The radical clarifies everything for us and allows us to speak clearly and unambiguously. The radical is defined as the principal, or positive square root. The radical can give only one (positive) answer. So and . In particular the VALUE of a radical must be positive, by definition. This allows us also to graph it, since it is single-valued.

When variables are introduced inside of the radical, there are TWO things to worry about. The item inside of the radical still must be a positive quantity AND the answer can only be positive. Put another way the implicit domain for the expression is , and the range of the function defined by the rule is the nonnegative reals. Any other interpretation is inconsistent and sets the student up for confusion.

What about ? The domain for this one is the entire set of reals. But the range is still the positive reals, since the radical must of necessity be positive. Therefore, we cannot say without qualifying; although the domain is the set of reals, the identity is not true for all numbers in the domain. An unqualified equation like that is incorrect. In the texts, how far away is the qualifying condition? Do you think that students read the qualifying conditions? If we are lucky, the students will read the qualifying conditions. If the condition is too far away from the statement, it is most likely that the condition will be overlooked.

Here is where things get really sticky! If x < 0, then and for we have . It happens that we have a perfectly good function lying around with the same property – the absolute value function. So we can say without a problem. This guarantees a positive, single-valued unambiguous result every time.

The ingrained method for solving a simple quadratic equation like creates a real hassle. The ingrained method is the sloppy language that tells the student to take the square root of both sides. Repeat, there is no such thing as THE square root; we do have THE radical, but not THE square root. Since the radical is a single-valued unary operator, we can apply it to both sides of the equation to get . This gives us the two roots, 11 and -11.

Note that I said apply the single-valued UNARY operator, the radical. I did not say the square root, since the “square root” is not single-valued, not a function. Here is where textbooks are causing problems, and software is no better.

Textbooks

Textbooks usually define it properly. Sometimes you have to really dig, however. I will use Mark Dugopolski, Elementary and Intermediate Algebra, as an example (this book is not unique) for a more detailed look at what happens.

In the index it appears as “square root function” (p. 585)– and there is no such thing

On page 585, The square root function is defined as the function with rule . This is not mathematically correct. This is a violation of the language of mathematics.

“Square root” is defined on page 468

“We use the idea of roots to reverse powers. Because 3² and (-3)² = 9, both 3 and -3 are square roots of 9.”
If a = bⁿ for a positive integer n, then b is an n^th root of a. If a = b², then b is a square root of a…”
If n is a positive even integer and a is positive, then there are two real n^th roots of a. We call these roots even roots. The positive even root of a positive number is called the principal root.
If n is a positive even integer and a is positive, then denotes the principal n^th root of a.
“We read as “the n^th root of a.”

Math is supposed to be a precise language. (2) is a correct usage for the term square root; it refers to a square root, and not THE square root. (3) is correct, but it is already setting the student up for confusion of terms. This avoids the use of radical and sets up the expression principal root – separating the issue from the term radical. (4) is correct

On page 482, using fractional exponents, he goes back to doing it right using the wrong words. To quote:

Square Root of x² (italics mine)

for any real number. Note that can also be written

In solving the quadratic equation x² = 4 Dugopolski avoids the muddy language, at least. The book opts for a discussion leading to the direct solution without saying “Take the square root of both sides.” Instead, it cites “the even root property.” The even root property skips from the equation to the solution without use of absolute values or factoring. It is simply something to be memorized. The end result is to encourage the sloppy language.

Thought: When a book states that , how far away is the conditional that goes with it? Where do you find the condition “provided that x is positive? Students will often stop reading beyond the line where you find this.

Other Books

Meserve, Sobel, Dossey, Introduction to Mathematics, 6^th edition

p. 114: …. d² = 2, that is, d = , the square root of 2 (the italics is theirs, the bold is mine), the positive number whose square is 2.

In a single sentence they have undone any distinction between square root and radical, and they ignore negative square roots. I could not find a definition of square root prior to this in the book.

Goodman and Hirsch, Precalculus; Understanding Functions

Introduces it correctly on page 37. For example, they say since 3² = 9. Note that even though (-3)2 = 9, we are interested only in the principal or nonnegative square root.

They introduce The Square root Theorem (to bypass the derivation all the time). If x² = d, then . This avoids saying “Take the square root of both sides.” Instead, you say, “Apply the Square Root Theorem. I still don’t like using the words Square roots, though; this does not entirely sidestep or resolve the issue.

Larson and Hosteller, Calculus, 6^th edition

p. 23. The radical function is called “The Square Root Function.”

I could not find the definition of square root – same for radical. It is assumed in this text, though misapplied.

Yashiwara and Yashiwara, Introductory Algebra,

p. 438. They ease into the radical correctly. They define the radical as the principal or positive square root.

p. 439 They undo it when they describe how to find the roots of a quadratic equation.:

thus undoing what they set up properly on the previous page. They continue the same verbiage in other examples, thus riveting in the incorrect notion that there are two radicals.

Sullivan, Michael and Sullivan, Michael III, College Algebra, 3^rd edition.

Concept introduced on page 18 – correctly.

p. 76 – they avoid the incorrect terminology. When working the initial examples, they use the factoring method as a justification for the two roots. When they summarize, they call it The Square Root Method. Later, when solving an equation, they do NOT say “Take the square root of both sides.” Instead, they say “use The Square Root Method.” This still contributes to the problem, since it promotes the notion that square rooting is single-valued, despite the result you get.

Cleaves, Cheryl and Hobbes, Margie, Basic Math, Algebra, and Geometry with Applications, 2004

p. 484. This is probably the worst I have seen. In the green instructions box, they tell you to take the square roots of both sides, thus drawing your attention by highlighting the incorrect usage. They continue to use the verbiage through the examples. Their third item is the most disturbing. They explicitly state that , a total violation of the accepted definition of the radical.

Computer Languages

In most computer languages, they have chosen to use SQR or SQRT for the positive square root. Even Excel uses SQRT for the radical. We are getting used to this. It contributes to the problem, however. Using “square root” without including the sense of the radical is a mistake and fosters error.

Web Sites

University of Idaho Polya Institute

This is the worst of any I have seen. The idea is good – provide videos. But the instructor actually states that and “justifies” it. That is how far this problem has come. Here, the error is carried out to its logical end – absolutely incorrect instruction at the University level.

www.purplemath.com is generally an excellent site, but it falls down on this one. For example, on the page http://www.purplemath.com/modules/radicals.htm they say

Do not confuse “simplifying” with “solving.” if you have “x² = 4”, and you take the square root on either side, you will get , because you could square either 2 or -2 to get +4. But because is specifically the positive square root.

The language gets even worse as they continue, contributing to the confusion, rather than clarifying.

What happened to “Keep it simple?” Recommended course of action.

Simply define the radical as the principal root, and stick to a consistent pattern of language. A clear definition and unambiguous, consistent application can go a long way to avoiding confusion.

Define the radical right away as the principal square root, calling it the square root radical (in words)
In applying this to the expression , observe that, from the definition, the only correct way to handle this is to use the fact that . This is clear and unambiguous. This establishes the radical as a unary operator returning unique, positive results.
Using this, solving the equation is a snap. Avoiding the pitfalls, it goes this way:
Taking this approach, one does not reinforce the false notion that the radical of 25 gives two results. It is consistent. It is unambiguous. And it sets the student up for success with radicals.

If you would like to avoid the middle step, I would recommend calling The Root Extraction Method rather than the misleading Square Root Method.
Note that some books avoid the issue entirely by resorting primarily to the quadratic formula. This is an option, but for the simpler type of equation that we see in the previous note, it is overkill. One of my professors used to call that “killing a fly with a cannonball.” It will work, but is it necessary?