Tag Archives: math

C’mon, Verizon, You Can Do Better

I generally dislike Verizon’s current commercials, but the commercial about infinity has to be my least favorite, largely because it implies things that are mathematically wrong.

In the commercial, the man asks the kids, “what’s the largest number you can think of?” The commercial is of course a bit tongue-in-cheek, with one child answering “10”, but the part that I can’t get over is when one kid says “infinity”, then another says “infinity and one”, and then finally one says “infinity times infinity”, after which the man implies that his mind is blown.

Ignoring the fact that infinity isn’t strictly a number (but it’s often treated like a number, and I’ll likely confound this in my post; I’m also going to assume that infinity, unless otherwise specified, refers to aleph naught, or the cardinality of the set of natural numbers), this commercial teaches people bad math: infinity, infinity + 1, and infinity * infinity are all actually the same size. None is “larger” than any other, so you can’t one-up someone else by saying that infinity + 1 is larger than infinity or that infinity * infinity is larger than infinity + 1.

The part that should really blow your mind is not infinity * infinity by itself, but rather that infinity * infinity is the same size as infinity. An example: you can find a bijection between the rational numbers and the natural numbers, which means that the sets of the natural numbers and the rational numbers are the same size. And in general, assuming that you believe in the axiom of choice, any infinite cardinal number is equal to its square, so using “times” will not get you larger cardinalities.

But of course, there is something larger than infinity, and that’s the power set of infinity. Thanks to Cantor’s theorem, we know that for any set, the set of all subsets of that set has a strictly greater cardinality, which means that you can always find a greater infinity. Crazy, right? In fact, not only are there infinitely many infinities, but the “cardinality” of the collection of infinities is larger than any of the infinities that it contains.

I don’t expect a 30-second TV commercial to get into all of the intricacies of infinity, but I would hope that the makers of the commercial would at least not propagate misinformation about math.

How to Date Efficiently Part 2

…or why you shouldn’t settle down until you’re at least 27.

Another of my favorite math problems is the secretary problem. Let’s say that you’re trying to hire a secretary. You have n applicants for the job, and you know a priori that you have a strict ordering of the candidates once you’ve seen them (i.e. if you’ve seen m candidates, you can rank them in order), but you’ll see them one by one in a random order, and for each applicant, you have to decide to hire him/her or else reject him/her forever. What’s the strategy to choose the best candidate?

It turns out, the optimal solution is to automatically reject the first n/e candidates (where e is the base of the natural logarithm), and then to accept the first candidate who is better than everyone you’ve already seen. In essence, you recognize that you need to have a training set of a certain size to learn what’s out there, and then you hope that you can find someone who’s better than everyone in your training set.

This means that you shouldn’t settle down with your first boyfriend/girlfriend since he/she is probably not the best person out there for you, even if he/she seems wonderful at the time. You don’t have anything to compare to, so you don’t know if your first is the best match for you. This seems to be supported by the fact that the younger you marry, the more likely you are to divorce.

Applied to real life, let’s say that you start seriously dating at age 20 and you have 20 years of prime dating years (okay, this maybe isn’t practical for woman). But 20/e ~ 7, so you should date until you’re 27, and then marry the next person that you find who’s better than everyone else you’ve dated so far.

Of course, there are caveats to this: this strategy maximizes the probability that you choose the best candidate instead of optimizing the expected value of your mate (you wind up with the last person you see the 37% of the time that the best person was in the first n/e that you automatically rejected); in real life, once you say no to someone, you don’t necessarily say no to him/her forever (see the reasonably enjoyable romcom What’s Your Number?); you can’t necessarily provide a strict ordering of your mates, etc. You can also learn about relationships from observing others, so you don’t necessarily have to date someone to know if he/she’s good for you, and you can potentially get your training set vicariously, so maybe you can know whether or not the first person that you date is better or worse than the average relationship that you’ve observed second-hand.

Anyway, I know this strategy is likely to be much more controversial than my first tenet of dating efficiently, but personally, I think it means that I won’t be completely comfortable settling down until I’m at least a little bit older. What are your thoughts about the need to wait until you’re older before settling down permanently?

How to Date Efficiently

…or why you should always ask people out.

One of my favorite math problems is the stable marriage problem. Let’s say that you have n heterosexual men and n heterosexual women where each man has ranked each woman in order of mating preference, and each woman has ranked each man the same way. Can we find a matching such that all marriages are stable (i.e. two people won’t leave their current partners because they’d be happier with each other)?

The answer, perhaps surprisingly, is yes, we can always find such a matching. And one straightforward way to do this is to use the Gale-Shipley algorithm. Essentially, each man goes down his list of women in order of preference, starting with his most desired mate, and proposes to her. Each woman looks amongst her suitors, chooses the one that she prefers most, and rejects the rest, and then the rejected men propose to their next most desired mates on their lists. This process repeats until each man is paired with a woman (for a more thorough explanation, see the Wikipedia article). There are two interesting results: 1) this algorithm provides the most optimal solution to the proposers (i.e. each man ends up with the best possible mate that he could end up with in any stable matching) and 2) this algorithm provides the least optimal solution to the proposees (i.e. each woman ends up with the worst possible mate that she could end up with in any stable matching).

The reason why I love this problem is because it has a real life lesson embedded within: if you ask people out, you’re going to end up with a more optimal mate than if you wait to be asked out. Think about it: if you take the initiative, you can start by asking out your dream date. If he/she says no, who cares? Just move on to the next best person on your list. Eventually, you’ll end up with the best person you could have because you’ve already asked out (and been rejected by) anyone who could be better. By taking control, you give yourself the opportunity to maximize your mate preference.

On the other hand, if you never ask anyone out, you only get to select from the people who ask you out, which is a subset of all people you could date, so your choices are inherently more limited than they could be (or at least no better than they could be). Thus, your choices are non-optimal and you could likely do better.

Taken another way, let’s say that you’re in the market for a new blender. You have two strategies: go online and search for the best blender that you can find, or just buy a blender from the traveling salespeople who knock on your door. Do you think you’ll get a better blender if you take the initiative and search for it yourself, or do you think you’ll fare better if you wait for someone to try to sell you one?

Granted, there are complications to this theorem when trying to apply it to real life. We can’t rank people strictly, we don’t always know our preferences until after we actually start dating people, marriages aren’t just one-sided affairs where only the happiness of one person matters, not everyone gets married at the same time, there are societal norms that say that women shouldn’t ask men out, etc. But I think the message remains the same: take the initiative.

I know, it’s harder than I’m making it out to be, particularly for women, but a common theme in being efficient is taking initiative and being okay with rejection. I’ll write more regarding these themes later. Until then, I’d love to hear: how has taking the initiative and asking people out worked for you?