I have watched many videos on YouTube about it from people like vsauce, veratasium and others and even my math tutor a few years ago but still don’t understand.
Infinity is just infinity it doesn’t end so how can there be larger than that.
It’s like saying there are 4s greater than 4 which I don’t know what that means. If they both equal and are four how is one four larger.
In: Mathematics
There is a hotel that has an infinite number of rooms. A travel soccer team shows up, they are the team “infinite even” numbers. They take up every room. The next day team infinite odd shows up and needs rooms. The hotel isn’t big enough, all the rooms are filled. The hotels would need a larger infinite number of rooms to house both teams.
What about when team an infinite fractions shows up? Now we need a larger hotel with more rooms…
I think the easiest way to think about this is to think of dimensions. Say you have a line that goes on to infinity. That’s one dimension of infinity. Now add another dimension to that. Thats infinity times infinity, which is larger.
Other folks are using integers (1, 2, 3, etc) and rational numbers (1.00000000000…1, 1.00000000000…2, etc), but my way of thinking about this is similar to dimensions, where the fractions of an integer are another dimension applied to those integers, so you have integers (infinite dimension) and the values between the integers (infinite dimension). Unlike simply adding a single dimension though, in this case we’re taking infinite infinities, because between each decimal is an infinity, and the set of decimals is also infinite.
You can understand that there are infinite number of fractions between 0-1.
You can understand there are an infinite number of intervals between to subsequent cardinals.
So there is an infinite number of fractions an infinite number of times.
If after watching those videos you don’t understand, you’re probably better giving up anyway because they do a pretty darn good job at it, better than me.
The numbers between 0 and 1 are infinite if you include decimals.
The numbers between 0 and infinite are also infinite however as they also include all the decimals between each number this Infinate is larger than the former.
Imagine an infinity that consists only of all the odd numbers : 1, 3, 5, 7, 9 etc.
Imagine an infinity that consists only of al the even numbers : 2, 4, 6, 8, 10 etc
Either of these groups contains only half of all the numbers, but is still infinite.
Now imagine an infinity that consists of ALL numbers : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 etc
That last group consists of twice as many numbers as either of the first two groups, and so is larger. But they are all infinite.
A list of all numbers ending in 4 is infinitely long. 4, 14, 24, 34, 44… A single value for every set of 10 numbers, going on forever.
A list of all numbers ending in 3 should be the same right? 3, 13, 23, 33, 43… A single value for every set of 10.
A list of all numbers ending in 3 OR 6 is infinite as well. 3, 6, 13, 16, 23, 26, 33, 36, 43, 46… But it has two values for every set of 10.
Both lists are infinite, as numbers don’t end at any point. There’s no “biggest number”. But if you imagine an end or compare a set range, by necessity the list of 3&6 should have 2x as many values as the list of 4.
Infinity isn’t a single concept, but rather a way to describe things that go on forever. There can be different “sizes” of infinity, even though our brains struggle to grasp this idea!
This one helps me understand better: there is an infinite amount of numbers between 1 and 2. There is an infinite amount of numbers between 1 and 3 as well but this infinity has to be twice the earlier infinity.
The main idea behind bigger infinity is that, with such big quantities, we rely on comparison and not actual “counting”.
If I can show that no matter what I do for every element of X there are 10 of y, y is definitely larger, right?
Now, go to Geogebra and plot the function y = X and y = x^2. Remembering that if a function “is over” the other then it’s bigger, x^2 is basically always over y=x and the more you go towards right and more the difference become larger. How larger? We can know it with a ratio!
Lim X -> infinity of X^2 / X = X and indeed, at infinity y=x^2 is infinitely larger that y=x.
This is not exactly what we means with bigger infinity but I hope I made sense in what comparing infinities means.
Let’s say we have two bags of marbles, and we want to know if they both have the same number of marbles, BUT we do not know how to count. How can we check if they have the same amount without counting? We can take one marble out from bag A, one marble out from bag B, pair them up. Keep doing this. If every marble in bag A can be paired with a marble in bag B, with no leftovers, then we know both bags had the same number of marbles.
That’s how we can compare sizes of things without counting them, and how we can compare sizes of things that are “infinitely” large.
In math, we call those bags “sets”. Let’s start with two bags of infinite size that ARE the same size- consider a bag of all the positive whole numbers (1,2,3,..) and a bag of all the positive EVEN whole numbers (2,4,6,…). Since these “bags” contain an infinite number of objects, we cannot “count” how many there are in each to compare the size. So, we have to make pairs, like we did with the marbles. In this case, for every number in bag A, we can pair it with a number in bag B that is twice its value. 1 gets paired with 2, 2 gets paired with 4, 3 gets paired with 6, etc. You can see that for EVERY number in bag A, we can pair it with a number in bag B. So the “size”of all positive integers actually = the “size” of all positive EVEN integers.
Now, there are some “bags” of numbers where it is impossible to make these pairings. No matter how you can pair up numbers, there will always be some numbers leftover that can’t be paired up, meaning that one infinity contains more objects than another infinity, making it “larger”.
This is where it can get hard to explain an example, but we’ll give it a try anyways. Let’s look at these two bags of numbers: bag A will be all positive integers again (1, 2, 3,…) and bag B will be all the possible numbers between 0 and 1 (so for example 0.5, 0.51, 0.501, 0.837362773833333, 0.333, 0.33333333, 0.33333333333333333 repeating, you get the idea- basically all decimal numbers between 0 and 1).
For the sake of argument, let’s say we have come up with some pairing of the numbers in the two bags, and I will write out the first few pairings:
1 with 0.53827263727173000000010000…
2 with 0.8173637363839000000040000…
3 with 0.8387262222233474633000000…
4 with 0.3333333333333333333333333…
Imagine this list being infinitely long, exhausting all the marbles in bag A. However, I can ALWAYS come up with a number from bag B (the decimal number less than 1) that is guaranteed to NOT be on this infinitely long list. I will call that my magic number, and I will construct that number following this rule: I will start at row 1 and look at the digit in position 1 after the decimal point, (so 5 in this case) and for ease of illustration, +1 to that digit and append it to my magic number. At row 2, I will look at the digit in position 2 (1 in this case) and +1 to get 2. Continue down the list, and my magic number will start being 0.6294……
Remember that this list is infinitely long, so if we kept doing this, we would get some decimal number. HOWEVER, and this is the cool part, that magic number is GUARANTEED not to be in the original list. How? Well, let’s go down the list and compare it to all the numbers. Is it the same number as the decimal in row 1? Well it can’t be, since I altered the first digit. Is it the same number as the decimal number in row 2? Well it can’t be, since I altered the second digit. Is it the same number as……? You may start to see the point. So what we have shown is that is impossible to pair up positive integers with decimal numbers between 0 and 1, because no matter how you try to list all the decimals out you can always find a NEW decimal that was not on your original list. This means that the size of the bag containing all the decimals between 0 and 1 must be bigger than the size of the bag containing the positive integers, even though they are both infinitely large.
This is just one example of two different sizes of infinity, but there are many other cool examples that illustrate this. These concepts of infinite have always been one of my favorite things in math 🙂
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