Whole numbers, number lines, rational numbers, and infinity
Once upon a time there were only whole numbers. The idea of fractional numbers, or rational numbers, did not exist. From this time before the existence of fractions, or anyother kinds of numbers for that matter, people did not have a need for fractions let alone such crazy numbers as we believe exist today. That doesn't mean they were any less knowledgable. It just means they didn't really hava a reason to need them as they were too busy surviving to sit around and think of other kinds of numbers that might be or could be created.
That is, until they did. Here is a story about what kinda mighta happened and why we might have all those different kind of numbers anyway.
First, you have probably heard of a number line?
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 it is made from microscopic super dots.
So, I'll still call it a number line, even though now it is really a line of these super dots. One super dots for each whole number 1, 2, 3, ... .
Since they are microscopic, the only way anyone could see them, would lead them to believe it was a line, a glowing line created by super dot at positions 1, 2, 3, 4, ...
Oh yea, forgot to mention super dots glow and give off enough light that even though they are so incredibly small that a human can't see them we can infer they exist by the glow.
So depending on how far apart the whole numbers are the line might be more holes (not whole) than a line.
Now imagine an acrobat wanted travel from dot. If the integers were spaced outtoo close, this wouldn't be much of a challenge so imagine they were far enough apart they would have to jump from one to another like stepping stones. And to make it more death defying, imagine they crossed a body of water with human eating sharks.
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 (integers) could be used. There were no numbers between the stepping stone integers so no solid number line could be made to cross a pond or our imaginary shark infested body of water.
No numbers to represent a value that wasn't a super dot. No values like 0, 1/2, 1 1/2... yes and even negative numbers, but don't fret we won't include them into 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.
Back to the original story. When people needed values between the super dots, they discovered 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 between 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.
Hum ... Imagine a board with three feet between zero and the number one. Where would we put 1/2? If we put a dot there, then a dot at 1/3, and one at 1/4 and ...
Hmm ... maybe we didn't fill as many gaps with this idea as we thought.
So. What about other fractions?
Is it possible to fill up the gaps with super dots so it becomes a solid number line?
What would it look if we wrote all the fractions between 1/100 and 1/2 between the 0 and 1? Would it give us an idea of how fractions fit on a number line?
Wait a minute you say. Aren't there an infinite amount of fractions? So just jam them in there to fill up every gap and you will get a line.
Numbers which are divisible, are whole numbers.
If the numbers aren't divisible, they can be represented as a fraction or rational number.
But, it would take all the whole numbers, if all infinite possible combinations of fractional representations are to be there, to make a line of super dots solid.
They don't stop somewhere. Why? because if every possible combination is there, then they will never repeat or stop no matter how many fractions or decimal places are used to represent them.
An other way of thinking about this is the idea of .999999999999...
What does that mean?
It goes on forever?
How might that be represented.
It means it will never reach one, because every time a nine is added to bring it closer someone can realize that another nine and another nine can be added and there is 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.
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.