Proofs of Distributing Exponents and Radicals

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?


Quaternions Anniversary

Today in 1843 William Rowan Hamilton invented quaternions (a way of using 4-dimensional numbers to concisely encode 3-dimensional positions) as he walked across Brougham Bridge in Dublin, carving them into the stone there to make sure he didn’t forget later. Begorah, that’s 132 years ago!



Almost Homeless

One of the top study tips many of us try to impart to our students is how mathematics (to a degree greater than any other discipline) builds on itself, with every day being an absolute requirement for what comes next. Much like a metal chain (I will say), if you break any single link, then the whole structure falls apart.

Several weeks ago, I met a visitor at an open-house for my girlfriend's art studio. We get to chatting, and I say that I teach college math; it's a good place to work, my boss treats me great, and there's an enormous need for help on the part of community-college students trying to pass remedial courses. He agrees, saying he was one of those students, and fortunately he did get the help he needed. I say: “For any of us, including myself, the limit on our careers and our aspirations is almost always how much math we were able to master in school.” He says: “I think possibly, maybe three or four days in elementary school, I either zoned out or something wasn't explained clearly... and directly because of that, twenty years later, I almost became homeless.”

Sometimes I use that anecdote on the first day of my remedial classes now, and it does make quite an impact.


Hope for Open Textbooks?

One of my primary arguments against MOOC's being a revolutionary force is by comparing them to books. In truth, while my attitude toward MOOCs is fairly negative, I would be prone to having a distinct hope for free, open-sourced, digital textbooks. The advantages seem multitudinous: (1) effectively free of cost, (2) a force-multiplier for the live classroom environment (as far as both cost and burden of carrying them), (3) ability to actually own them on a mobile device and not be dependent on an outside streaming service, (4) ability to read them without any internet connection whatsoever, (5) ability to share and re-host the work freely, (6) ease of editing for fixes, and tailoring for individual courses and local requirements.

Compared to a suite of video lectures, this would seem fairly easy to do – and yet as far as I can tell, even this relatively simple project has failed to succeed to date. I've spent some time surveying open-source introductory algebra texts, for example, and found them all to be surprisingly deficient (rather reminiscent of some video lectures, in fact – frequently unplanned, technologically difficult to access, or with confusing and unprofessional jokes and puns in the text, etc.) I plan to spend some time writing up particular reviews in the future.

An argument: If making information widely available eliminates the need for live in-person instruction, then why didn't the printing press “tsunami” destroy live colleges (when in fact it did the opposite)? If free MOOCs current low quality is something easily fixed, then why aren't the even simpler open-source textbooks yet representing high quality offerings?

So that said, a few news items regarding open-source text developments that do give me cause for hope:

1. California has passed and signed a law to fund open-source textbook development in 50 core subject areas. While there was a similar attempt under the Schwarzenegger administration, that prior try had ambiguous definitions, weak standards, and no funding. This new law sounds like a much stronger attempt that does give me hope.

2. Finnish researchers and teachers engaged in a 3-day “hackathon” in which they completed an entire open-source textbook. While I would be highly skeptical of the quality of such an offering, it at least signals that there is some amount of buzz and excitement for the idea, which perhaps bodes good things to come.


Bungled Election Probability

Here's a common malformed math problem that really irks me -- The idea that in a voting situation, the ratio of voters indicates the probability of one party winning. For example: a problem might say that out of a group of 50 people, 30 people favor candidate A, and 20 people candidate B, so candidate A has a 60% chance of winning an election. Obviously, this can only be the case if the election is decided by one single voter being chosen by random method, which is not remotely how any elections actually work.

I've seen this pop in one or more publisher-provided testbanks that I use. And yes, it currently also appears in the Udacity Statistics 101 course (Unit 24.4, et. al.; hopefully fixed soon?). For god's sake people, please don't do that.


MOOCs in the News

MIT's Technology Review has one of the best survey/reviews I've seen of current MOOC programs, and also pointedly asks if they might be a temporary fad. The article opens with a fascinating comparison to the correspondence-course craze of about a hundred years ago, which made similar promises of widespread and personalized educational opportunities, and saw millions of prospective students enroll – culminating in very low outcomes and success rates, and ultimately the collapse of those programs.

American Educator magazine has a powerful “Notebook” column assessing Khan Academy, and pointing out the relatively poor quality of the lessons made available there. Quoting a profile from Time magazine, “Sal Khan... explains how he prepares each of his video lessons. He doesn't use a script. In fact, he admits, 'I don't know what I'm going to say half the time'... 'two minutes of research on Google'... is how Khan describes his own pre-lesson routine.” Note that this observation is identical to my #1 criticism of the Udacity Introduction to Statistics course, here.