Monday Mathematical Musing 1 – Cardinality

With the new semester, I want to try to actually live up to the second half of the purpose of this website.  Yes, the primary purpose is to provide a location for you to find homework assignments that you missed, project deadlines that you’ve forgotten, and upcoming test dates that you do not yet know, but the name of my website is “Assignments and Mathematical Musings.”  And while there has been a copious amount of the former, there has been none of the latter.  I hope to, as regularly as I can, fix that.  These posts will contain interesting mathematical tidbits that I will try to write so that all of my students could enjoy.  If I encounter an interesting or significant news article, I might write about it here.  If I come across a fun puzzle or nifty proof of an easy-to-understand idea, I’ll try to share it.  If you find something that you think I might think interesting, please send it along and I’ll give you the proper credit.

**Please note: For those math super fans who may be reading these posts, remember that their intended audience is high school math students in 9th-12th grade.  I will try to be mathematically accurate in all of my posts, but I may “fudge” some things here and there for the sake of clarity.  If I make an egregious error, please call me on it, but otherwise permit me a bit of poetic license, as it were.

For my first post, I want to discuss the significance of the picture below, an example of a proof without words.

Cardinality of an Open Interval

No, that isn’t a gong being suspended from the floor with weird ray beams coming out of it.  I’ll explain what it actually is shortly, but first I want to talk about infinity.

Continue reading Monday Mathematical Musing 1 – Cardinality

A question of many Benders

A recent episode of Futurama featured the lovable alcoholic robot Bender creating duplicates of himself that are 60% his size.  Later in the episode the duplicates continue to replicate at 60% size, until the sheer number of sub-atomic Benders start to overwhelm the world and eat away at it.  This is an actual end-of-world scenario called, as it is in this episode, the “grey goo,” but the episode also makes a point of talking about the generations of Benders and how their population increases without bound.

The question that popped into my mind is what generation of Bender would be so small as to influence matter at the subatomic level, as they do in the episode.  In particular, the diminutive Benders are shown altering the molecular structure of water, and so we shall use that as a frame of reference.  The size of a water molecule is 0.942 angstrom, about 94.2 picometers (1/10th a nanometer or 1.0 x 10^-10 meters).  The Benders pulling apart the molecules appear to be roughly three times that size, so we will assume they are roughly 300 picometers in height.

A full sized Bender seems to be the same height as a typical human, which we will assume to be 1.73 meters.  From this, we can derive the formula h(g) = 1.73(.6)^g where g is the number of generations and h(g) is the height of the g-th generation.  If we want to know at what generation the height will reach 300 x 10^-10 meters, we can easily substitute it in and solve for g (I’ll leave that for you to calculate).  According to my math, I get very near 35 generations, which we will use as our number.

So there were 35 generations of Benders out there to get to the sub-atomic sized Benders we saw in the show.  How many Benders does this mean?  We see that each time a Bender replicates himself, two copies are created.  If we assume a Bender only copies himself one time, that means there are 2^35 = 34.36 billion Benders in the 35th generation!  With all the Benders in all the other generations, this works out to be 2^0 + 2^1 + 2^2 + … + 2^35 = 68.72 billion Benders on the entire planet!  And that’s if a Bender only copies itself one time!  If each Bender copies itself twice, we would wind up with 4.7 x 10^21 (4.7 quintillion) Benders!  It seems that Bender’s call for each of his descendants to perform “1 quntillionth” of a task was not too far off!

Mathematical Musing – The Myth of Anonymity in America

I recently read an article discussing how just 10 digits would be enough to end privacy as we know it.  The article is a bit alarmist, but makes some interesting points that I’d like to discuss here.

Firstly, the article claims that a 10-digit code is sufficient to uniquely identify every person alive on earth.  Where do they get that figure?  It has to do with a tool in mathematics called a permutation, which is essentially an ordering of some sequence of numbers or objects.  Consider sports jerseys.

A sports jersey has room for two digits, both of which can be any number from 0 to 9.  There are, therefore, 10 options for both places, giving us a total of 100 possible jersey numbers – from 00 to 99.  What we have just used here is something called the Fundamental Counting Principle (also known as the rule of product).  Essentially, given a number of slots to fill and a number of choices for each slot, the total number of outcomes is equal to the products of all the numbers of choices for each slot.  Since a jersey has two slots with ten choices each, the total number of outcomes is 10*10 = 10^2 =100.

If we instead have a string of ten digits, each place having ten options, the total number of outcomes is 10*10*10*10*10*10*10*10*10*10 = 10^10 = 10,000,000,000, or ten billion.  Considering the world’s population is still less than 7 billion (though is predicted to reach that mark in 2011), a quantity of 10 billion identification numbers would be more than enough to assign one to every living human being. The idea of having your entire persona and identity reduced to a string of numbers is a frightening thing to many people.

Of course, we in the United States already have a system that pretty much does this. Continue reading Mathematical Musing – The Myth of Anonymity in America

Assignments and Mathematical Musings from Mr. Kirk