How Much Farther Is It To Infinity And Beyond?

Aleph - used to symbolize infinite cardinality

Several days ago, as part of my ongoing posts about fractals (specifically That’s Unpossible! The Elusive Dimensions), I made the following statement:

…there are more transcendental numbers than algebraic ones…

And I stand by that statement.  But what pray tell does that even mean?  How can there be more of one type of number than the other?  And how do we know this?  Before we address this statement specifically (and in fact, we’ll address it in another post), we are going to talk about infinity and what that actually means.  Strap yourselves in folks, this is going to get EPIC.

First, let me state that infinity is not a number; it is a concept.  Say what?  No, really it’s true.

Image via http://www.charlespetzold.com/blog/2006/04/101110.html

Generally speaking, we use infinity to describe some thing that is greater than any quantity that we could attempt to assign to that thing.  Let me repeat – it’s something that in some way is so big that we can’t quantify it.  Infinity is a label for the unquantifiable.  To put this into context, think of the largest possible number in your head.  Do you have a really, really big number in mind?  Can you add 1 to it to make it larger?  Of course you can.  Now keep doing that.  Every time you add 1 to whatever number you had, you still get a quantity.   What we need to realize is that there is always, always something larger than whatever quantity we have.  The limit of that realization is this concept we call infinity.  Note that infinity is also used to describe the size of a set, and not just some far off number that we can’t quite label with a value.

Okay, so it’s not a number, it’s a concept.  But what’s the point?  Good question.  Let’s consider, for example, the integers¹.  What if I asked you to tell me how many integers there are?  I’m sure most of us would immediately respond that there are an infinite number of integers.  If you start at any integer, you can add 1 to it and get another.  And another.  And another.  Off to infinity.

But what if I were to ask you – how many rational numbers are there²?  Well, if there are an infinite number of integers, it seems reasonable to believe there would have to be an infinite number of rational numbers, right?  But think about this further.  Let’s consider all of the rationals between the integers 0 and 1.  We have where of course, we have removed fractions that represent the same quantity (i.e., , etc.).  If we just consider those rationals with 1 in the numerator, we have an infinite number of integer denominators.  But we can also use 2 in the numerator and come up with another infinite set of rationals between 0 and 1.  And 3, and 4, and so on, off to infinity.  That is, we have the following sets, one for each of the infinite possible numerators, and each containing an infinite number of rationals:

And these are just those rationals that fall between the integers 0 and 1.  We could repeat the process for all rationals between 1 and 2, 2 and 3, and so on.

So we have an infinite number of sets each containing an infinite number of rationals between each and every pair of subsequent integers (of which there are an infinite number of pairs).  Clearly there must be more rationals than there are integers.

But what if I told you that the number of integers is the same as the number of rationals?  That is, we say they have the same Cardinality.  Surely I must be joking?!?

Image via Wikipedia

To see this, imagine arranging the numbers such that you had pairs on a grid, with each pair representing the ratios (as in the picture to the left).  To see how there are just as many rationals as integers, we simply start counting them as indicated by the arrows, skipping any rational that is equivalent to a rational that has already been counted (i.e., ).  And since we can label each rational as either the first, second, third, and so on, we can say there are just as many of them as there are integers.  To put this another way, since I can pair up 1 unique rational with 1 unique integer, and I can do this by simply following the pattern in the grid, there must be the same number of rationals as there are integers.  Was your mind just blown?

In a case like this, we say that the rationals (and integers) are infinite but countable.  That is, this type of infinity is known as a countable infinity.

Of course, this begs the question: Are there other types of infinity?

And of course dear reader, the answer to that is yes, yes, a thousand times yes.  But we’ll leave that discussion for another day.  Let me end however with a question for you to ponder – how many real numbers are there³?

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¹ The Integers are the set of numbers …, -3, -2, -1, 0, 1, 2, 3, …. and are denoted as .

² The Rational Numbers (denoted as ) are all numbers that can be represented as a fraction where and (such as , or ).

³ The Real numbers (denoted as ) are all numbers that are considered Rational or Irrational.  Irrational numbers (denoted as ) are any numbers that can’t be represented as a Rational (i.e., as , where and (such as or ).

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Image from Dinosaur Comics