**Complete 4 pages APA formatted article:**

** Math paradoxes. **

The key concept here is that there are an infinite number of rooms, so that our logic – which would terminate in the ‘real world’- can go on forever. This is called ‘Hilbert’s infinite hotel paradox’ and the famed hotel is often jokingly referred to a “Hilbert’s” analogously to “Hiltons”! Infinity is a very hard concept to understand and possess the most absurd properties of any mathematically definable object. Cantor was the first mathematician to study the properties of infinite sets in greater detail. Suppose you group together all the even numbers (2, 4, 6, 8, 10…) and all the perfect squares (1, 4, 9, 16…) separately into two groups. Which group has more members? If selection was from a small set, say from the first 100 numbers, then the answer is fairly obvious.

There are 50 even numbers in the list from 1 to 100 while there are only 10 perfect squares. As the set grows larger, we expect the ratio to remain the same. However, if the grouping is from the entire set of integers, then lo and behold, we find the rather unusual result that both the groups have exactly the same number of members! This is because, for every even number from the first set we can find a perfect square in the other set. Thus, since for every element in the first set there is a corresponding element in the next set, we have to conclude that no set has more members than the other. as if this were to be so, some even number would have no perfect squares to relate to. Series’ show the remarkable properties of “Convergence and “Divergence”.

These properties happen to be very well studied as they find applications in most branches of engineering. Take an apple pie and cut it in half. Cut one of these halves in half again and repeat the process. Initially you have 1 object (in this case a pie). It then becomes . The third iteration reduces it to . It is easy to see where we are going.