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The WEIRDNESS of Infinity… Or how to melt your brain!. J.P. McCarthy, UCC & CIT. Weird Fact 1. Weird Fact 1:. This is impossible right?! Surely the more numbers I add up the bigger the sum?. How about if I use fractions? How about this sum:.
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The WEIRDNESS of Infinity… Or how to melt your brain! J.P. McCarthy, UCC & CIT.
Weird Fact 1: This is impossible right?! Surely the more numbers I add up the bigger the sum? How about if I use fractions? How about this sum: The dots mean that the pattern goes on for ever and ever and ever…
Weird Fact 1: Meet this flea… we will call him… “Michael from the Red Hot Chilli Peppers.“ We will start him on the endline of a basket ball court.
Weird Fact 1: There he is Michael from the RHCP:
Weird Fact 1: He jumped half the length of the court: He has half the court left… what is a half of a half?
Weird Fact 1 That is right. He now jumps quarter the length: He has a quarter of the court left and will jump half-way to the end again…
Weird Fact 1: He’s getting there! An eighth of the way to go!
Weird Fact 1: Will he ever get there, he is slowing up!
Weird Fact 1: Five jumps and he is close but not there!
Weird Fact 1: …but he is getting closer all the time! Let us say the length of the court is one
Weird Fact 1: How far has the flea jumped after one jump… How far after two jumps… How far after three jumps… What are these numbers getting closer to? What are they never bigger than?
Weird Fact 1: This shows that the following sum gets closer and closer to one and we say the infinite sum is equal to one!
Weird Fact 2: This is has to be impossible?! Let us list the counting numbers and the square numbers: 1,2,3,4,5,6,7,8,9,10,… 1,4,9,16,25,36,49,… There are ten counting numbers between 1 and 10 and only three squares!
Weird Fact 2: O.K. I have to be in trouble here right… Let us get things straight. What does it mean to say there four penguins in this picture:
Weird Fact 2: What we do is as follows… to have four penguins what we mean is that we can count four… but what we do is say this one is one, this one is two, this one is three and this one is four:
Weird Fact 2: These red lines that go from one collection of things to another are called a map…
Weird Fact 2: We could have put arrows but we actually don’t want that… we want the arrows to go both ways. Let us call arrows that go between two collections like this a perfect matching… every penguin has his own unique number and every number has his own unique penguin.
Weird Fact 2: We say that two collections have the same size when there is a perfect matching between them… for example:
Weird Fact 2: Now how can there be as many square numbers as counting numbers if there are more counting numbers… But are there really more counting numbers… How many counting numbers are there? How many square numbers are there?
Weird Fact 2: Seemingly the same… but there must be a perfect matching… can you find one? How did you find a perfect matching for the penguins…
Weird Fact 2: Or more visually, what about this perfect matching?
Weird Fact 3: How can a hotel take in more guests even when it is full? SURELY, this is impossible!! If there are no rooms… where are the guests Going to stay?
Weird Fact 3: Well it works like this… when a guest comes up to the reception he says this… “I know you got no free rooms but you can still make space for me… just look down the corridor!”
Weird Fact 3: So the receptionist looks down the corridor…
Weird Fact 3: And Kung Fu Panda says… move the guest in room 1 to room 2…
Weird Fact 3: …move the guest in room 2 to room 3…
Weird Fact 3: …move the guest in room 3 to room 4…
Weird Fact 3: …move the guest in room 4 to room 5… and repeat forever…
Weird Fact 3: …now the first room is empty!
Weird Fact 3: Suppose an infinite bus with an infinite number of passengers drives up to the hotel:
Weird Fact 3: Can we fit in all of these passengers? We move all the guests to the room with double their current room number… then the infinity of the odd numbered rooms are free!
Weird Fact 4: We have already seen that there are as many squares as counting numbers…
Weird Fact 4: We still think there are more counting numbers… but between any two squares there aren’t too many counting numbers…
Weird Fact 4: Do we think there are more fractions than counting numbers? What is a fraction though, really? Fractions have a top …and a bottom… and they are both… BUT… the bottom can’t be zero!
Weird Fact 4: Now there are many, many fractions between each counting number… how many do you think? For example, between 1 and 2 we have, only for example… There are as many as you want… an infinite number!
Weird Fact 4: How could there be as many fractions as counting numbers?! Especially since between any two counting numbers there is an infinite number of fractions! Recall before that to do a perfect matching all we have to do is count! Can we count ALL the fractions?!
Weird Fact 4: This is how we do it… we set up a grid like this:
Weird Fact 4: Each point is given by a pair of coordinates…
Weird Fact 4: …and we can count fractions… let the first number be top… …second be bottom!
Weird Fact 4: Let us start…
Weird Fact 4: Just remember not to double count!
Weird Fact 4: Just remember not to double count!
Weird Fact 4: Just remember not to double count!
Weird Fact 4: Just remember not to double count!
Weird Fact 4: Just remember not to double count!
Weird Fact 4: Just remember not to double count!
