Long ago, in a land far away, there was a grand hotel where there were infinitely many rooms. This hotel was attended by a brilliant manager.
One night, a guest arrived, but the hotel was full — each room was occupied by one guest. The newly arrived guest asked if a spare room was available. “Of course we have, we are the Infinite Grand Hotel. There is always a room for everyone,” the manager said proudly.
Now since each room was occupied by a guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to Room 4, and so on. Basically, he told every guest in Room n to move to Room n + 1. Since the hotel had infinitely many rooms, there was no problem in moving, there was always a room to move to. This left Room 1 vacant, and therefore, the guest was accommodated. The guest was happy. The manager was happy. » Read more
This video is a lecture on the Beauty and Truth in Mathematics and Science. Professor Robert May of Oxford talks about the beauty of mathematics and its connection to real life problems. I think it is a good watch for all mathematics students and enthusiasts.
Professor May highlighted the importance of Euclidean Geometry, Complex Numbers, Fractals, Relativeity and more. He also cited the works of mathematicians particularly Paul Dirac.
We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.
As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element.
Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment , there is a corresponding point on segment . » Read more