INFINITY
Infinity is without a doubt one of the hardest mathematical concepts to comprehend. Infinity is without doubt one of the hardest mathematical concepts to understand. Many ideas that works with normal numbers don’t work anymore, and instead there are countless paradoxes. Is there a largest number? Is there anything bigger than infinity? What is infinity plus one? What is infinity plus infinity? Today, we will be discussing 2 of infinity’s paradoxes: Hilbert’s Infinite Hotel, and Grandi’s Series.
HILBERT’S INFINITE HOTEL
In 1924, German mathematician, David Hilbert tried to explain some of the properties of infinity using a hotel with infinitely many rooms just to show us how hard it is to wrap your minds around the concept of infinity. …show more content…
Very surprisingly (and luckily), this is not the case for Hilbert’s Infinite Hotel. Once you arrive and request for a room, the hotel night manager at the reception makes a loudspeaker announcement, asking all the guests to shift along, meaning that the guest in room 1 moves to room 2, and the guest in room 2 moves to room 3 and so on. The guest in any room moves to . Since there is no last room in this hotel, every guest will have a room. Of course this will not work with a finite hotel since the guest in the last room will be left without a room. The Hilbert Infinite Hotel doesn’t have a last room, therefore This idea can be expanded: If 19 guests arrive, the night manager will simply ask all the guests to move up 19 rooms. If 999 999 guests arrive, all current guests will have to move up 999 999 rooms – and similarly for any other number. But one day, an infinitely large bus, with infinite number of guests arrive. Now the previous method won’t work anymore because you can’t ask guests to move up infinitely many rooms, they would never arrive at their new …show more content…
Weirdly, there is a third answer as well. We’re going to try to find out what S (sum) equals to. Let’s do
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What we ended up with, is what we started with, the alternating plus and minus 1, meaning that if we take the S to the other side, we’ve got . Now that’s really weird, the sum of adding plus and minus 1 forever, gives you . Well it might be 1, it might be 0, but it also might be .
So what is the answer? Many people, including mathematician believe that the best answer to this sum is . So today, I will try to proof to you why most people believe to be the best answer.
If we use the partial sums to calculate Grandi’s series, it doesn’t work. Partial sums are sums that are part of a sequence which gets closer and closer to a certain value. The first one would be 1, when you add the first 2 together, it would be 0, you add the first 3 together, and you get 1 again, and so on. It simply will alternate between 1s and 0s, meaning it will not be getting closer to a certain value. Therefore, the partial sums don’t work with Grandi’s series.
So we can use another method to calculate the sum by looking at the averages of the partial sums. Here we
Infinity is without a doubt one of the hardest mathematical concepts to comprehend. Infinity is without doubt one of the hardest mathematical concepts to understand. Many ideas that works with normal numbers don’t work anymore, and instead there are countless paradoxes. Is there a largest number? Is there anything bigger than infinity? What is infinity plus one? What is infinity plus infinity? Today, we will be discussing 2 of infinity’s paradoxes: Hilbert’s Infinite Hotel, and Grandi’s Series.
HILBERT’S INFINITE HOTEL
In 1924, German mathematician, David Hilbert tried to explain some of the properties of infinity using a hotel with infinitely many rooms just to show us how hard it is to wrap your minds around the concept of infinity. …show more content…
Very surprisingly (and luckily), this is not the case for Hilbert’s Infinite Hotel. Once you arrive and request for a room, the hotel night manager at the reception makes a loudspeaker announcement, asking all the guests to shift along, meaning that the guest in room 1 moves to room 2, and the guest in room 2 moves to room 3 and so on. The guest in any room moves to . Since there is no last room in this hotel, every guest will have a room. Of course this will not work with a finite hotel since the guest in the last room will be left without a room. The Hilbert Infinite Hotel doesn’t have a last room, therefore This idea can be expanded: If 19 guests arrive, the night manager will simply ask all the guests to move up 19 rooms. If 999 999 guests arrive, all current guests will have to move up 999 999 rooms – and similarly for any other number. But one day, an infinitely large bus, with infinite number of guests arrive. Now the previous method won’t work anymore because you can’t ask guests to move up infinitely many rooms, they would never arrive at their new …show more content…
Weirdly, there is a third answer as well. We’re going to try to find out what S (sum) equals to. Let’s do
)
What we ended up with, is what we started with, the alternating plus and minus 1, meaning that if we take the S to the other side, we’ve got . Now that’s really weird, the sum of adding plus and minus 1 forever, gives you . Well it might be 1, it might be 0, but it also might be .
So what is the answer? Many people, including mathematician believe that the best answer to this sum is . So today, I will try to proof to you why most people believe to be the best answer.
If we use the partial sums to calculate Grandi’s series, it doesn’t work. Partial sums are sums that are part of a sequence which gets closer and closer to a certain value. The first one would be 1, when you add the first 2 together, it would be 0, you add the first 3 together, and you get 1 again, and so on. It simply will alternate between 1s and 0s, meaning it will not be getting closer to a certain value. Therefore, the partial sums don’t work with Grandi’s series.
So we can use another method to calculate the sum by looking at the averages of the partial sums. Here we