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Asked by lwebb to Adam, Geoff, Rob, Sheila, Suzie on 18 Mar 2011 in Categories: General.
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Adam Tuff answered on 17 Mar 2011:
A hard question to answer! There are many theoretical ideas around this…you get the idea that you could count from 1 to infinity…I know we can’t picture infinity, but there we go. Ok, so what about something similar…counting from 0 to 1?
I could count it by going 0, 1. Or I could count it 0, 0.5, 1. Or I could count it 0.01, 0.02…etc. Or I could start at 0.00001…or I could start at 0.00000000000000001….do you get where I’m going? You could take an infinite time just to count to one! If you are interested in things like this – you might be interested in Chaos theory – it’s full of stuff like this!
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Geoff McBride answered on 17 Mar 2011:
I like this one yes there are. The infinity we all know is 1 to infinity. Mathematicians talk of ‘The Continuum’ for example take all the even numbers to infinity and then all the odd numbers to infinity then if we add those two sets of infinity numbers. You could say that it is bigger than 1 to infinity. You can slit the sets as much as you like every tenth number giving ten sets of infinity or every one hundredth number giving one hundred sets of infinity it just goes on. I like this sort of thing I’m actually not that good at maths but I do enjoy it.
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Sheila Kanani answered on 17 Mar 2011:
Hi again lwebb, with another ace question. One I can’t answer, I didn’t realise there were different sizes of infinity 😀
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Suzie Sheehy answered on 18 Mar 2011:
Another good question – there are different types of infinity ‘countable’ and ‘uncountable’ infinities. And different sizes of infinity as well! For example if you counted up all the numbers from 1,2,3 etc… you’d get an infinite number, but if you counted them in steps (0.5, 1, 1.5 etc…) you’d get an infinity that was twice as big!
For most of the time it doesn’t matter, in physics we usually just use infinity as a mathematical concept. I usually think of infinity as 1 divided by 0. -
Robert Simpson answered on 18 Mar 2011:
Here’s a story I always liked that might help you understand the horrible problem of dealing with infinity:
Imagine there is a hotel with an infinite number of rooms: Hotel Infinity. It’s a busy, summer weekend and the hotel is fully booked. Around midday a couple arrive and ask for a room. The manager tells them they are fully booked.
“Well can’t you just ask the person in Room 1 to move to Room 2, and the one in Room 2 to move to Room 3 and so on?” asked the couple. “Then Room 1 would be free for us to stay in.”
“Well the hotel is infinite,” says the manager, “so I guess that will work.”
So everyone staying in the hotel picks up all their things and moves up one room, the people in Room 10 move to Room 11 and the people in Room 4,678,437 move to 4,678,438. Since there are an infinite number of rooms, this is all fine. The couple check in, and things are going well when suddenly a bus load of people arrive, and they all want rooms to. There are fifty of them! The manager tells them they are fully booked and then realised that it doesn’t matter – this Hotel Infinity!
“No problem!”, he says, “we’ll just ask everyone to move up fifty room along from where they are.”
So everyone does just that, the people in Room 11 now move to Room 61 and the people in Room 4,678,438 move to Room 4,678,488. Since there are an infinite number of rooms, this is all fine. However, around mid-afternoon, nother bus arrives and this one is an Infinite Bus.
“Have you got an infinite number of rooms for us?” asks the tour guide leading the bus.
The manager thought long and hard. “Actually, I think I do.” he said. “We’ll just everyone to move to the room with twice the number of their current room.”
So everyone does just that, the people in Room 1 go to Room 2, and Room 2 goes to Room 4, Room 3 goes to Room 6. Suddenly all the odd numbered rooms are free, are there an infinite number of odd numbers, meaning that the infinite bus load of people have a place to stay.
It was going really well! The last problem arose in the evening, they got word from the Infinite Bus Company that there were a whole load of buses all heading for Hotel Infinity and all of them were carrying an infinite number of new hotel guests.
“How many buses are coming?” asked the manager.
“An infinite number of them!” said the Infinite Bus Company. “We’ve got as many buses, as you’ve got hotel rooms. Will you be able to take them all?”
This time, the manager was stumped. He thought long and hard. To buy some time, he did what he had done earlier and asked everyone to move to twice their room number. This freed up the infinite number of odd-numbered rooms. “Aha!” he cried, “I’ve got it”.
The buses arrived and began queueing to get into the infinite car park. Everyone on the first bus was told to look at their seat number. The person in seat 1 went to room 3, the person in seat 2 went to room 3*3 (=9), the person in seat 3 went to Room 3*3*3 (=27) and so on until they were all inside the Hotel.
The people on the second bus did the same but this time using the number 5 (Rooms 5, 25, 125, etc, the third bus used the number 7 (Rooms 7, 49, 343 etc). The hotel manager was sure to use only the prime numbers (there are an infinite number of them, after all) and this made sure that no Room was allocated twice.
The Hotel Infinity made a lot of money that weekend, an infinite amount in fact 🙂 Hopefully this helps illustrate some of the problems and paradoxes of infinity.
[This has to be the longest answer I have given so far!]
Comments
Sheila commented on :
On reading what the other scientists wrote I now understand. Thank you! 🙂
freddie commented on :
In fact, all infinities containing entirely rational numbers are the same size. To determine if an infinite set is the same size as another you need to show that you can pair up every number from the first with every number from the second. For example, the set of all integers to infinity is the same size as the set of all even numbers to infinity. (Even though the latter would appear smaller at first glance.) To show this, you can pair up every integer n with 2n, and therefore the two sets are the same size. There is a bigger infinity than this however. The set of all irrational numbers to infinity is a larger infinity. I’ll leave that one out there for anyone to prove!
Geoff commented on :
I’m not sure I agree by including irrational numbers your simply adding an axis to the number line. However you could argue that that is similar to the Continuum I mentioned earlier.
But could you then have a number line with infinite axis? [I only just thought of that]. Another thought how about you count to infinity using the unary numeral system [base 1] does that not include all possible numbers real or otherwise?
This is the sort of research I’ll look at if I ever retire.
Suzie commented on :
Good point Freddie. I am trying to answer these questions without google or wikipedia to help me so sometimes I’m going to use dodgy examples to make my point! I probably should have used a different one than the one I gave…
lwebb commented on :
Thanks for your answers! 😀
An interesting story about the hotel – a nice way to explain… but surely an infinite number of buses would need an infinite amount of diesel, which is finite! 😉
I asked this after thinking about another question I asked previously (about whether the universe has an edge). If the universe is infinite, but expanding, then wouldn’t the size of the universe be bigger today than it was yesterday (for example)? In which case, the universe would not have been infinite yesterday, and therefore there would have been a boundary or edge between the universe and the “nothing” into which it is expanding?
Is this an example of how infinity can be different sizes?