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.

To Infinity and Beyond!

You all know that infinity isn’t actually a thing; it’s an idea.  It’s the idea that there is no end to some lists of numbers.  It may not surprise you that there are different types of infinity — different degrees of its size.  What probably will surprise you is that these sizes of infinity don’t quite work in the ways they seem like they ought to.

The size of a set of numbers (i.e., the number of numbers there are) is referred to as the set’s Cardinality.  The set {2, 3, 4, 5} has a cardinality of 4, since there are four numbers.  We can easily compare the cardinality of finite sets of numbers by simply counting how many numbers are in each set and comparing them.  If two sets have the same cardinality, we could create a direct connection — called a mapping — from each element of one set to each element of the other set, like in the picture below.

Such a mapping is called a Bijection, which is a term and idea that you will learn more about in Calculus, though you have been using for most of your high school career.

The most fundamental infinite set of numbers is the set of Natural Numbers.  You may know them as the counting numbers, as they’re the numbers we use to count things — 1, 2, 3, 4, and so on.  There is no end to this list of numbers.  Tell me a number n that you say is the largest natural number, and I’ll tell you that n + 1 is bigger.  The cardinality of this set is, naturally, infinity.  Mathematicians call this level of infinity ℵ₀, which is read “aleph-null” or “aleph-zero” as it uses the first letter of the Hebrew alphabet.  This is the “smallest” degree of infinity.

Now consider the Integers, which is the set of all positive and negative whole numbers as well as zero.  Intuitively, it should seem like the infinity that describes the set of integers should be larger than the infinity that describes the natural numbers.  After all, the natural numbers only increase in one direction while the integers go to infinity in both the negative and positive direction.  This intuition is incorrect, however.  Their cardinalities are the same.  We can show this is true by describing a bijection that will map the set of natural numbers to the set of integers.  In other words, if we can describe a way to write the integers in a specific order, then we can count them using the natural numbers.

It works like this:  We start with zero, then 1, then -1.  After that comes 2 and -2, then 3 and -3, 4 and -4, 5 and -5 and so on.  The zero is the first integer, 1 is the second, -1 the third, on to infinity.

Therefore, the set of integers has the same number of numbers as the integers.

Be Rational for Once!

What about Rational Numbers then?  Surely there’s way more fractions than there are whole numbers.  After all, there’s an infinite number of fractions even between 0 and 1 (1/2, 1/3, 1/4, 1/5, 1/6, …).  Well it turns out that no, there is actually the same number of rationals as there are naturals.  It works like this.

Consider the standard coordinate plane.  We will use the x– and y-axis to create rational numbers of the form x/y.  We just need a place to start, and a convenient starting location is the origin.  This gives us the fraction 0/0, which is not defined.  So we move one space to the right, at (1,0).  This gives us 1/0, which is still not defined.  Move up one to (1,1) and we get 1/1, which equals 1.  This will be our first rational number.  Move to the left to (0,1) and we get 0/1 = 0, our second rational number.  Moving to (-1,1) gives us -1/1 = -1, our third rational number.  Continuing around in this spiral pattern, ignoring any undefined numbers and skipping anything that repeats — (-1,-1) will produce -1/-1 = 1 again, for example — and we can get a (somewhat inconvenient) way of reaching every possible rational number.

A special note/reminder needs to be made here: a rational number is any number that can be written as a fraction.  This includes any decimal that ever stops (0.12 = 12/100 for example) as well as any number that repeats (0.33333… = 1/3).  In this post, we discuss how you can find the fraction that produces whatever such decimal/fraction combination you’d like (for example, 0.1234545454545… = 679/5500).

Oh Get Real…

So what about the Real Numbers, then?  Remember that these include all the rational numbers as well as all the decimals that don’t stop or repeat — aka the irrational numbers.  Surely that set, which is represented with a simple number line, has an infinity greater than the naturals.

It turns out that yes, it does.  The cardinality of the real numbers is greater than the cardinality of the naturals.  Fundamentally, Natural Numbers are countable whereas Real Numbers are uncountable.  It is impossible to create a bijection that maps the real numbers to the naturals (for one explanation on why, read about Cantor’s Diagonal Argument).  The infinity that represents the number of real numbers is greater than the infinity that represents the number of naturals.  Mathematicians refer to this level of infinity as ℵ1 – aleph-one.  There are further degrees of infinity.  The set of all subsets of real numbers, for example, or the set of all possible functions that map the set of real numbers to itself both have a cardinality of 2^ℵ1.  

So what’s the deal, then, with the proof-without-words that we started with?

The line along the bottom is the number line, representing all real numbers.  The segment along the top represents an open interval — i.e., an interval like 2<x<5 or (2,5), where 2 and 5 are not included in the interval but all real numbers between them are.  What this picture shows is that the cardinality of any-sized open interval — from (2,5) to (2,2.000000000001) — is the same as that of the set of real numbers.  No matter where you are in that segment, as long as you’re not on the actual endpoints, you can find a point along the edge of the top part of the circle and follow the blue line through the circle’s center to a unique real number on the number line.  What this means is that there is an uncountable degree of infinity in even the tiniest, most minuscule of intervals, and there is the same number of numbers between 1 and 2 as there are between 1 and a googolplex.

Infinity is weird.