Types of Infinity

Set A in year 8 have recently covered a topic of work on primes, factors and multiples. Whilst considering if the square root of any square number resulted in two answers whilst identifying that the cubed root of any cube number can only have one distinct solution, the concept of infinity arose. It was decided that this might be a good stopping off point for further investigation.

1. Firstly, the class had to identify how large infinity actually is and without putting too fine a point on it, infinity is infinity large. However, it was soon discovered that some versions of infinity were clearly of a different size to others
2. Consider the set of even integers 2, 4, 6, 8, 10,…There are clearly an infinite number of numbers in this group.
3. However, now consider the set of natural numbers 1, 2, 3, 4, 5,…Now, as this includes all of the even numbers and all of the odd numbers, this is twice the size of the set of even integers.
4. Therefore, the set of natural numbers = 2 x (the set of even numbers)

                                                                                   =  2 x infinity

                                                                                   = infinity

We then looked at all the decimals between 0 and 1 to find out there are more numbers in this set than there are natural numbers.

1. We have already agreed that the set of natural numbers (1, 2, 3, 4,…) is infinite.
2. If we now put 0. in front of each number (0.1, 0.2, 0.3,…) this is essentially the same set which is infinitely large.
3. We can also repeat this with 0.0 (0.01, 0.02, 0.03,…) to give another infinitely large set
4. Now carry this on, adding an extra 0 place holder each time (0.001, 0.002,….)
5. You can repeat this an infinite number of times.
6. Therefore, the set of decimals between 0 and 1 = infinity x (the set of natural numbers)                                           
7. = infinity x infinity  = infinity²   = infinity

It was at this point that we decided to move on, before it all got too big to comprehend!

If you are interested in the concept of infinity, then the following video clips may help you get started:

Infinity is bigger than you think – Numberphile      https://www.youtube.com/watch?v=elvOZm0d4H0

The Infinite Hotel Paradox – Jeff Dekofsky               https://www.youtube.com/watch?v=Uj3_KqkI9Zo

Mr S Cheal

Mathematics Department

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