Whole numbers and number lines and rational and irrational numbers and Infinity

One explanation of rational and irrational numbers from the time before the existence of different groups of numbers. In a way they existed long before anyone ever had a need for any numbers let alone such crazy numbers as we know exist now, but humans didn't know they did and really had no reason to use them and were too busy surviving to sit around and think of the kinds of numbers that might be or had been created.

So here is a story about why what kinda happened and partly why there is a need for all those different kind of numbers anyway.

First, you have probably heard of a number line? Right? Well you probably have always thought of it as a line. A rope, steel cable, very long stick, something solid from one end to another. Like a road that a person or vehicle could meander down.
Well, it doesn't have to be thought of that way. Imagine that it is made from microscopic super dots. So if a number line, I'll still call it a number line, but it is really a line of super dots., is made of real number super dots. The only place that anyone could see anything, that would lead them to believe that it was a line, would be a glowing super dot at positions 1, 2, 3, 4, ... Oh yea, forgot to tell you that these super dots. glow and give off enough light that even thought they are so incredibly small that a human can't see them we can infer that they exist by the glow. Even though this number line would be more hole (not whole) than line. If the integers were spaced out far enough a person would have to jump from one to another like stepping stones, if they were trying to use this number line to cross a body of water.
Okay, Okay you say I am getting carried away. Well just hang with me for awhile. Remember one of my favorite sayings is - you have to learn more to remember less. Well this is one of those times.
Okay on with the story. Somewhere back in time when there were only stepping stones and no solid number lines to cross ponds or make numbers someone discovered a need to represent a value that wasn't on the super dot number line. Values like 0, 1/2, 1 1/2... yes and even negative numbers, but don't fret we won't include them in this story we will stay on the positive side of the super dot number line. Later if you want to go to the negative side you are welcome to go there on your own.
Back to the original story. When people needed values between the super dots., they discovered that it was pretty easy to create them by thinking of what they wanted to divide the whole (not hole) by. For example if they wanted a value between 0 and 1 and the value they wanted would divide the one into two pieces, they could just write it as 1/2 (read one divided by two). Back then some things were simpler. Somewhere along the line of time, (not to be confused with time line) some one got the kinda smart kinda dumb idea that one divided by two ought to have its own name and not a name that included two whole numbers and an operation, so one divided by two became know as one-half.
Well, you can imagine the excitement when people didn't have to stretch so far when all of a sudden super dots were discovered to be in the middle of all the whole numbers. It literally cut them in half. Wow! What a marvelous idea and I bet you have figured out that it wasn't too long until someone said, Hey? Why aren't there other super dot stepping stones. I bet we could make one divided by three, and one divided by four, and one divided by five... And you know if we do that we should be able to fill in all the spaces between the whole super dots. and make it solid.

Here I would write the number zero on the chalk board with three feet between it and where I would write the number 1. Then I would ask where to put 1/2 and put a dot there, then 1/3, put a dot there, and ask them to predict where the next one would go.... Then ask if it will fill the gap between 0 and ? would it fill the gap between 1/2 and 0? 1/4 and zero? any number and zero?

Hmm. maybe we didn't find as many values with this idea as we thought at first.

By then someone might mention. What about the other fractions and I would add them. I would also start to convert them to decimal numbers and make a chart with the fractional number in one column and the decimal equivalent in the other.

By then I usually have them hooked into the question, Is it possible to fill up a number line with super dots. so that it becomes solid?

The quest for numbers begins and a look at the patterns that are created by whole numbers and rational numbers over days or weeks or months (depending on the students - I did this with sixth graders and there was no need to rush. It was just one of things that I did to try to convince them that anyone could be a mathematician if they wanted and who wouldn't want to be? Oh yeah an architect, scientist, doctor, engineer, but wait a minute they would understand this too.
Anyway the thoughts usually lead to some additional conclusions. Eventually the idea of infinite comes in and with some additional questioning by me they can speculate that every whole number and every possible way that a decimal number could be represented with every possible combination of digits would need to happen if the number line were made solid.

So, let's review what we have now. If the numbers were divisible, they were whole numbers. If the numbers weren't divisible, but could could be represented in a certain way (I will let you fill in the blank here as this is becoming longer than I first anticipated hopefully not infinite), then voile they are rational numbers. But, there should also be numbers that, if all infinite possible combinations of decimal representations are to be there, to make the line of super dots. solid. that don't repeat forever or divide and stop somewhere. Why? because if every possible combination is there then a randomly arranged pattern they will never repeat no matter how many decimal places are used to represent them, has to be there. And if they are, then those are irrational, because the thought is insane?
The other benefit of this is the idea of .999999999999... and what does that mean? Students can say that it goes on forever, but they still struggle with what that means and how it might be represented. Well, maybe not everyone, but I have known crowds that could get excited about it. And that brings me back to the line idea. Eventually someone suggests that the line can never be solid because every time a nine is added to bring it closer someone might realize that another nine and another nine can be added and there will always be a hole. AH ha! that's what infinity means it goes on forever, so if it goes on forever it has reached one and that is why when .99999999999999 approaches 1.00000 it is one. WOW... What? The only way to make a solid number line out of super dots. (numbers) is not if there are an infinite number of super dots. side by side. But the number of super dots. has to always be approaching infinity, because if it wasn't, then a hole would be created.

Simple? Yes? Well maybe? No? Hope not.
See why one of my favorite sayings is you have to learn more to remember less? or I just had a radical rewrite shoot through my brains. You have to learn more to understand less. Understand less is okay. Understand nothing may be a problem.

Dr. Robert Sweetland's Notes ©