My Favorite College Algebra Lecture

Well, it’s not an entire lecture, not even quite half a lecture. But my favorite part of a lecture in college algebra is the properties of exponentiation. Sounds strange, right? I basically list off some algebraic identities involving exponents, and I’m done. The value does not lie in the content, but in the presentation of the material, and how it comes the closest to what I enjoy about math.

We start with b\geq 0 and m,n natural numbers. We notice that when looking at b^{n}\cdot b^{m}, this is first multiplying b a total of n times, and then multiplying b another m times. So this is the same as multiplying b a total of m+n times, or b^{m+n} Here I might take a specific m,n like m=4 and n=3 and write it out. So we get the equation b^{m+n}=b^{m}\cdot b^{n}.

Next if we take a look at (b^m)^n, we can expand it out as multiplying b^m a total of n times. So b^m\cdot b^m\dotsb b^m, then being a little suggestive, I expand the b^m out vertically, creating an array of b‘s. Then it becomes clear by appealing to the area of a rectangle that this quantity is b^{mn} the result of multiplying b a total of mn times.

So here’s the part I love. So far, we have appealed to the nature of exponents as repeated multiplication to deduce some properties about it. I then say that these are nice algebraic properties that will make our lives easier when we have to deal with exponentials, wouldn’t it be nice if they were still true even after we consider more general m,n? So let’s pretend they do still hold and see what happens.

We first take a look at m=0, n=1 and by looking at the addition property we get that b^{1+0}=b^{1}\cdot b^{0}, and since 1+0=1 and b^1=b, we see that b=b\cdot b^0, and if we divide both sides by b, we see that b^0=1. So we get a very specific value for b^0=1. It wasn’t something handed down from the heavens, it wasn’t some convoluted aspect of 0, or strange combinatorics. Just we want this addition property to be true, and for it to be so, b^0 must be 1.

Then we take a look at negative numbers. So we let m be any positive whole number and use the addition property and notice that b^{m}\cdot b^{-m}=b^{m+(-m)}=b^{0}=1. So erasing the middlemen, we have b^{m}\cdot b^{-m}=1, or by dividing both sides by b^{m}, we have b^{-m}=1/b^{m}. Now we have that negative exponents correspond to the reciprocals. Again something that pops out, just because we wanted to keep that addition property.

Finally we look at rational numbers. We first we deal with 1/n. Finally the other property comes into play: (b^{1/n})^n=b^{1/n\cdot n}=b^{1}=b. Cutting out the middlemen again, we see that (b^{1/n})^n=b. But there is a special name for a number whose nth power is equal to b (when b>0, and that is the nth root of b, written \sqrt[n]{b}. As for the arbitrary case, we see that since m/n=(1/n)\cdot m, and so b^{m/n}=(b^{m})^{1/n}=\sqrt[n]{b^m}.

Now, I may not have persuaded you that this is a good lecture, let alone one deserving of being called a favorite. The first thing to notice is that there are few hard quantities involved; it’s mostly algebra. When these classes are taught, the first thing a (pure) mathematician notices is that it’s deprived of any interesting abstract mathematical ideas. The teacher is forbidden (for good reason!) from straying too far into his own domain of general theory, and focus on the particular, lest the students become confused and bewildered. The more headstrong mathematician-teacher charges ahead anyway, and is met with sighing, head scratching and yawning. Proving the quadratic formula by completing the square of an arbitrary quadratic functions sounds like a great idea, but it’s a quick way to lose a room full of freshmen. This has a wonderful balance; easy enough that the algebra clarifies rather than obfuscate, yet complicated enough, that it says something actually interesting and not at first obvious.

Also take note that the progression of our ideas follows the same construction of the rationals from the natural numbers. We starting with a counting device; consider repeated counting, also called addition and then repeated addition, which we call multiplication. We consider 0 the neutral element in addition. Then we follow-up with negative numbers, or in more algebraic terms, the additive inverses we get from extending our monoid of natural numbers to the group of integers, which turns out to be a ring. We then form the ring of fractions, which give us multiplicative inverses. We follow the same chain of constructions without mentioning the fact that we are doing so, so we implicitly form a picture of the way numbers work without having to explicitly talk about a great deal of abstract algebra.

But what seems like the most important thing I love about this is the premise of the lecture. We begin with an observation that exponents work in a certain way, when we consider natural numbers. In particular, it’s very natural, almost completely obvious. Now rather than making rules for the other numbers and noticing that we get the same properties, making them more complicated by having so many cases, we say “wouldn’t it be nice if this still held?” And the point isn’t “This is how things are,” but “we want this, and we accept the consequences of our choice.” Too often, math is considered a bunch of strict rules, like a code of law. “You are allowed this action, but not this one.” And I admit I encourage this view, especially when dealing with things like dividing fractions, or adding fractions or… pretty much anything involving fractions. But this isn’t how math is, not the kind I love so much. It allows for so much creativity and invention. The only truly mathematical law is “Be consistent,” which is admittedly more restrictive than at first appears. But we allow every twist and turn of every possible mathematical rule or system and allow a complete freedom provided one thing: we must accept the logical consequences of our choices. “Everything not forbidden is permitted.”


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