In my introductory algebra classes these days, I've switched to providing actual proofs for major principles after about the halfway point in the class. As usual, the point of this is (a) to prepare students for what real math classes are like, and (b) provide insight into why things work as they do.
What I was distressed to find yesterday is that you actually can't find any proofs for the pretty rudimentary notions of the fundamental rule (as I call it) or distribution of exponents or radicals (over multiplication or division), even for just integer powers. Not anywhere online on a Google search. Not in any math text in any of my bookcases at home.
So this in turn caused me to have weird dreams and wake up at 5am all pissed off over that fact, scribbling stuff on note paper to make up the gap, in classic MadMath style. Just in case anyone else is ever searching for the same thing, I'll present the distribution proofs below. We assume the usual properties of commuting, associating, and distributing multiplication and addition (which for integers can be proven from the Peano axioms).
Definition of Exponents: For positive integers n, a^n = a*a*...a [n times]. For negative integers, a^(-n) = 1/a^n. For the zero power, a^0 = 1.
Theorem: Exponents distribute over multiplication. That is: (ab)^n = a^n * b^n for any integer n.
Proof: For positive powers, (ab)^n = (ab)(ab)...(ab) [n times] = (a*a*...a)(b*b*...b) [by commuting & associating] = a^n * b^n [definition of exponents]. For negative powers, (ab)^(-n) = 1/(ab)^n [definition of negative exponent] = 1/(a^n * b^n) [positive exponents distribute] = 1/a^n * 1/b^n [definition of multiplying fractions] = a^(-n) * b(-n). For the zero power, (ab)^0 = 1 = 1*1 = a^0 * b^0 [definition of zero power].
Definition of Radicals: Square root √a means a positive number x such that x^2 = a.
Theorem: Radicals distribute over multiplication. That is: √(ab) = √a√b for any positive a, b.
Proof: Let x = √a, which means x^2 = a [definition of square root]. Let y = √b, which means y^2 = b [definition of square root]. Note that √(x^2 * y^2) = xy because (xy)^2 = x^2 * y^2 [distribution of exponents], satisfying the definition of square root. So √(ab) = √(x^2 * y^2) = xy = √a√b [substituting in each step].
[Note: Obviously I've skipped cube roots and Nth roots in the foregoing, but they're easily expanded from the above.]
Questions: Did I get those right (e.g., would it be more complete to use induction for the exponents proof)? Do you think they're illuminating for basic algebra students? And more importantly -- Can you find any place online that I missed or in any textbook that presents proofs for the preceding?
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I can't see why induction would be any better. (Subscribing to comments in case someone else sees it differently.)
ReplyDeleteI don't think this would be illuminating for basic algebra students. They take these things for granted. Comparing the steps of your proof to the steps one might take trying to prove something that isn't true (let's try to prove that sqrt(a^2+b^2) = a+b) might be illuminating.
You know, I presented this in 3 different sections last week, and it actually went better than I would have guessed. Some anticipated benefits -- (1) it got us to review the definition of what radicals really mean, (2) it generated a discussion about how we can turn radical problems into exponent problems (so they really must follow all the same rules), and (3) it challenged and intrigued the more advanced, often bored, students (by asking them to complete the end of each line, a good idea from S. Thrun).
DeleteFor me, the protective work around stuff like sqrt(a^2+b^2) (exp/rad not distributing over add) is best done by (a) frequent verbalizing/quizzing on the issue, in words, and (b) emphasizing that we can only use rules, in the specific format established by a proof. Obviously, the real mathematical demonstration comes from a counterexample.
The ellipses in your proof mean that you are using induction; it's up to you to decide whether that implicit usage is sufficiently formal or whether you want to formalize it via induction.
ReplyDeleteWell put, that adds some clarity. Granted that I want to use this in the basic algebra class, I think I should avoid a dependence on explaining induction.
DeleteAs a follow-up, I've deleted this presentation from my remedial Basic Algebra class (partly because there was a recent reorganization and we don't even test students on distributing exponents anymore). But I am doing it in my College Algebra class, where I'm personally committed to showing proofs for everything.
ReplyDeleteWorks pretty well and was a nice emphasis for students to tell me how to write the "because" statement for radicals in the middle, and thus really understand the meaning of radicals.
4 years too late, but.....I teach these intro proofs in discrete math. The book is Susanna Epp, Discrete Mathematics with Applications edition 4. Your proof on radicals is problem 59 in 4.1. The Epp book still does not provide a comprehensive foundation in doing these types of proofs, but it gets you started down the right path. In particular it actually presents direct proofs. Direct proofs seem to be non-existent in the general math education of american students.
ReplyDeleteThanks for the information, that's good to know! We've been having conversations about our department's Discrete Math class lately (or rather: how to prepare students better for it). Good to know it's in some book somewhere.
DeleteAnd I've heard this before about the lack of direct proofs in U.S. math; in fact, that's specifically the weak point for our community-college math majors in discrete math.
I saw a question on StackExchange a while back to wit, "Why do students always want to use proof by contradiction for everything?".
In my opinion, this is good for a basic algebra class. For more advanced courses that have been introduced to complex numbers though, you may want to touch on how this changes when moving out of R and into C.
ReplyDeleteFor example, sqrt(-9)*sqrt(-9) = -9, but NOT positive 9 because of order of operations. So distributing the square root over negative numbers doesn't work.
I like it, good point!
Deleteit is a very simple but thoughtful explanation. I love it so much!
ReplyDeleteThanks for saying that, I'm glad if it helped!
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