# All posts by Mr. Kirk

I have been Mathematics Teacher at Ithaca High School since 2007. In that time, I have taught Algebra 1, Intermediate Algebra, Honors Algebra 2, Honors Pre Calculus, Advanced Placement Statistics, Math AIS, and a one-semester elective course in Fractal Geometry and Chaos Theory.

# Musing: The Largest Vocabularies in Hip Hop

It’s always fun to see what creative things people can do with a little bit of data and some statistical analysis.  Designer and data scientist Matt Daniels analyzed the first 35,000 lyrics in the official works of more than 80 rap and hip hop artists and groups, and sorted them by who has the most extensive vocabularies.  The data analysis may not be perfect — I’m not sure I approve of counting variations of the same word as distinct from each other — but the picture that emerges is quite entertaining.  Check it out here, but be forewarned:  this is an analysis of rap and hip hop music, so there is some strong language on this site.

# Musing: A Rough Guide to Bad Science

This will be of principal interest to my statistics students (young and old!) but this is a nice summary of some of the poor ways science and scientific findings are reported in the news.  Number 4 should look very familiar, as should number 7!

My general recommendation about reading these news articles is, when in doubt, go read the source.  Don’t rely on other people to do your thinking for you.  Go and seek out the information you need and make your own conclusions!

# Musing: The 2048 Game

Here’s an amusing little time waster: http://gabrielecirulli.github.io/2048/

The goal of the game is simple: Get a tile of value 2048.  The controls are also simple: press an arrow key and every tile that can move in that direction, will.  If two tiles of the same value are next to each other, they’ll combine to one tile double that value.  Also, with every move, a two or a four will appear at random in a free spot on the board.  Seems easy, right?

Well keep in mind that to get a 2048 tile, you’ll need to create and combine two 1024 tiles.  To get those two 1024 tiles, you’ll need four 512 tiles, which require eight 256 tiles, which require sixteen 128 tiles.  All of these numbers are powers of two, which are the key to the concept of binary numbers, which at the most basic and fundamental level is how computers operate.  A binary number is one of base two, in the same way that a “conventional,” decimal number is base 10.  Consider the number 2048.  You might remember from elementary school that this could be thought of as two 1000’s, zero 100’s, four 10’s, and eight 1’s.  Those numbers – 1000, 100, 10, and 1 – are all powers of 10 (10^3, 10^2, 10^1, and 10^0, respectively).  A number written in binary uses powers of 2, meaning there is a 2^0 = 1’s place, a 2^1 = 2’s place, a 2^2 = 4’s place, and so on.

Moreover, just as any one place value in a decimal number could be occupied by a digit from 0-9, giving you ten options, a place value in a binary number can only be occupied by two digits: 0 and 1.  To write a decimal number like 459 in binary, you first need to figure out how to “assemble” the number using powers of two.  The biggest power of two that fits is 256 (2^8).  That leaves 203 left, meaning 128 (2^7) fits also.  Subtracting that leaves 75, meaning we can take away 64 (2^6).  This leaves only 11 left, from which we can subtract 8 (2^3), then 2 (2^1), then only 1 (2^0) remaining.  So the decimal number 459 can be rewritten as 11100111.  Taken in the other direction, the binary number 110101 would be interpreted as one 1, one 4, one 16, and one 32, giving a decimal number of 53.

What’s important to note is that 53 and 110101 are referring to the same quantity; they are just different ways of representing that quantity.  It’s the same way as how “the cat,” “el gato,” and “l’chat” all refer to the same animal.  Thinking of decimal and binary as different languages for numbers is actually a great analogy, because binary is how computers think of numbers.  The reason why has to do with how computers are made.  The circuits on the motherboard, inside the processor, and all throughout your computer are essentially tiny wires.  At any instant, the wire either has an active electrical charge running through it or it doesn’t.  If the wire is “on,” it is considered a 1.  If it is “off,” it is considered a 0.  The sender on one end of the wire will turn the current on and off extremely quickly in a manner much like morse code, and the receiver on the other end of the wire will interpret the rapid fire of 1’s and 0’s as binary numbers that can be interpreted in any number of ways.

So far, the best I’ve been able to get in the game is a pair of 256 tiles that I wasn’t able to combine before blocking myself off, so my high score is only 3180.  Think you can beat it?

# Musing: AMC 10/AMC 12 Video Solutions

Did you take the AMC 10 or AMC 12 on Tuesday?  If you did, you might be interested in checking out the video solutions to the last few questions of both tests that the YouTube channel Art of Problem Solving just posted, viewable here.

# Monday Mathematical Musing – Are 50/50 Raffles Worth It?

I recently received the following email from my job at TC3…

2013 “POT OF GOLD” 50/50 RAFFLE

Our 17th year of making some lucky winner \$1,000 richer!

\$20 per ticket – only 100 tickets sold.

This got me to thinking: Is this 50/50 raffle worth the ticket?  Are such raffles ever worth it?  Is it even possible for it to be worth it?

# Monday Mathematical Musing – Finding Fractions

In the first musing post, I mentioned that any decimal that ends or repeats — even if it doesn’t repeat immediately, can be expressed as a fraction.  This post will describe how to find these fractions.

# Monday Math Musing 2 – New Largest Prime Number – So What?

First, check out this news article from Time Magzine.  Give it a skim.  Then come back.

## What happened?

A new largest prime number has been discovered through a program run by a mathematician at the University of Central Missouri.  In case you’ve forgotton, a prime number is one whose factors are merely 1 and itself.  The numbers 3, 5, 7, 11, and 13 are all prime, but 6 isn’t (it can factor into 2*3) and 15 isn’t (3*5).  Prime numbers are really the bread-and-butter of many mathematicians, especially those who study a branch of mathematics called number theory.

# 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.

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.

# 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