Different sizes of infinities
I know the following piece is a bit long, but I REALLY hope you will take the time to read and try to understand the thoughts (if you don't know them already). This is one of the areas where YOU can show your students something about mathematics that is FASCINATING, AMAZING, awesome, mind-boggling!!! ... show them that mathematics is NOT just mundane, mind-numbing manipulation of numbers in calculations. This topic is not extra-curricular; it IS taught in pre-algebra / algebra 1 (real numbers), BUT you can explain some parts of it even to 4-6th graders.
This sounds really strange. How could there be different kinds or sizes of infinities?
I first encountered this idea in my university studies, and I thought it was absolutely FASCINATING!
The video below goes through the basics and explains them well, though really fast. I recommend you pause the video at those points where it goes too fast for you, and "rewind" or rewatch those parts. I will explain some of the points FROM the video below.
Did you know... that in ONE sense, there are AS many even numbers AS there are whole numbers.
I know... in one sense, there aren't... but in another, there are. That is because to COMPARE the sizes of two sets (even infinite sets), we check if there is a one-to-one match between the numbers in the sets. And it turns out that, YES...
...it IS possible to make a one-to-one match between the set of even numbers and the set of all counting numbers:
This one-to-one correspondence (bijection) SHOWS or proves that the set of counting numbers and the set of positive even numbers are the "same-size". Both are infinite sets... but the "size of their infinity", so to speak, is the same.
Listing ALL the fractions?Next, consider the set of all the fractions.
Could you easily come up with a way to LIST all the possible fractions?
Think about that... if you can make such a list, then you can put counting numbers and fractions into ONE-TO-ONE correspondence! (Because just by the fact it is a list, the items in that list are in order... and you can start COUNTING them with the counting numbers.)
The image below shows how to make the list. We put all fractions into a two-dimensional GRID, and then start going diagonally through the grid, starting with 1/1, which is 1. (Follow the arrows.) You will encounter equivalent fractions, which will be skipped over.
Again, think about this: ALL possible fractions are in the list. For example 145/8793 will be in the grid at the intersection of the 145th row and 8793rd column, and will eventually get listed in your list.
Fractions (rationals) are a COUNTABLE setThis shows that the rational numbers (fractions) are a COUNTABLE set - in other words, you can "count them" in the sense that you can make a LIST of them. While the set of rational numbers is infinite, the infinity of rational numbers is "of the same size" as the infinity of the counting numbers.
A BIGGER infinity?Now... this is where it gets interesting! Did you know there exists a BIGGER infinity than the infinity of the set of counting numbers? It is kind of astounding to think about.
The set of all real numbers (the set of all decimals, both rational and irrational numbers) IS such a set!
The PROOF is on the video, starting at 2:58. Again, it goes fast, BUT I feel it is well explained, so just pause the video and run the explanation several times till you get it.
We call the set of all real numbers UNCOUNTABLE, because you cannot "count" or "list" all of them. And... the uncountableness comes from the non-rational (irrational) numbers. There are SO many of them!
I had not heard of the analogy of the night sky before (as in the video)... that rational numbers (fractions) are like the visible stars, and the irrational numbers are the rest, the blackness. That is kind of the way I've thought about it, though... that rationals are like individual dots on the number line, and irrationals are the "fill ins" that totally cover the number line.
Infinities of different sizesSo, there exist INFINITIES of different sizes. Some infinities are "bigger" than others!!!!! Mind-boggling, isn't it? (But I feel that it's GOOD that our human minds get boggled sometimes : )
Even the concept of infinity in itself is hard to grasp. For example, my daughter has a difficult time whenever she thinks about eternal time or space... something unending. I had always thought that seeing how the number line is unending would be HELPFUL in that respect — but it seemingly hasn't helped her (yet anyway).
(Near the end, the video shows a little "crack" in the word "MATH". Well... I am not sure. I think that it's NOT mathematics itself that would have a deficit or a crack, but that there is some kind of a limitation somewhere... maybe in our HUMAN thinking.)