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Are These Containers Full or Empty?
An Experimental Investigation of Infinity
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First Container:
Imagine this container being large enough to hold an infinite number of balls.  You are going to put in numbered balls in groups of ten.  The first group will be numbered 1-10, the second group will be numbered 11-20, the third group will be 21-30, and so on.  Every time you put in a group of ten numbered balls, you are going to take out the ball labeled with the multiple of 10.  This process will continue infinitely, in other words, forever.




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Second Container:
Imagine this container being large enough to hold an infinite number of balls.  You are going to put in numbered balls in groups of ten.  The first group will be numbered 1-10, the second group will be numbered 11-20, the third group will be 21-30, and so on. When you put in the first 10 balls, you will take out the ball labeled with the number 1. The second set of 10 balls you will take out the number 2, and so on.   This process will continue infinitely, in other words, forever.





Now tell me what you think.
Is the first container full or empty?  What about the second container?

Not sure what to think?
Make your best guess.

Now here is the best part.
Let's do it.  First, cut about 100 small squares of paper and makes two piles of 50
pieces.  Label each set 1-50.  Next find some container that you can use to put the pieces of paper in.  Now start the process.  From your first set of 50 numbers, put in the pieces labeled 1-10 and draw back out the number 1.  Continue with the process until all 50 pieces are in and the numbers 1-5 have been drawn as stated above in the directions to the first container.  Next you will do the process again with your second set of 50 pieces of paper in the manner described about for the second container.

Now what do you think?
Take a look at your guesses from before.  Do you still think you are right?  Here is a hint:  In the course of the entire process of putting in and taking out the numbers of both containers, can you name a single number that will be left when you are finished?


Conclusion:
This is a link to the final answer.  Please do not click on the link until you have finished the above activities.


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