I have just come across the concept that there are different types of infinite numbers (whole integers, irrational numbers etc.) and the concept that for example, the amount of irrational numbers between 0 and 1 is higher than the amount of whole numbers from 1 to infinity.
I guess I just don’t understand how an infinite amount of something can be bigger/smaller than an infinite amount of something else…
Please un-f**k my brain 😀
In: 4
The way I try to explain it to younger learners is that “infinity” is really a placeholder for when we want to say “a lot, so much that it might as well be too much to think about”.
However, there are situations that we want to compare two or more different kinds of “too much to think about”, and that’s why some infinities are “bigger” than others.
Oversimplified, but not sure how advanced the five year old is.
The way I try to explain it to younger learners is that “infinity” is really a placeholder for when we want to say “a lot, so much that it might as well be too much to think about”.
However, there are situations that we want to compare two or more different kinds of “too much to think about”, and that’s why some infinities are “bigger” than others.
Oversimplified, but not sure how advanced the five year old is.
The way I try to explain it to younger learners is that “infinity” is really a placeholder for when we want to say “a lot, so much that it might as well be too much to think about”.
However, there are situations that we want to compare two or more different kinds of “too much to think about”, and that’s why some infinities are “bigger” than others.
Oversimplified, but not sure how advanced the five year old is.
To begin with, infinity is not a number. It’s a concept that allows a lot of complex math to work. Similar to zero. You can’t have zero of something, and you can’t have infinity of something.
The different inifinites are for calculating different types of math. An infinite negative number has to be different than an infinite positive number, but neither of those “exist”.
To begin with, infinity is not a number. It’s a concept that allows a lot of complex math to work. Similar to zero. You can’t have zero of something, and you can’t have infinity of something.
The different inifinites are for calculating different types of math. An infinite negative number has to be different than an infinite positive number, but neither of those “exist”.
Let’s consider the series that happens when you add one to the preceding number: 1, 2, 3, 4, 5, 6, 7, etc. etc. etc.
We know this series will go on to infinity. We also know that we can calculate it’s discrete value at any point along the way.
So far so good?
Now let’s compare that to the series where we double the preceding value: 1, 2, 4, 8, 16, 32, 64, 128, etc. etc. etc.
We know that this series will *also* go on to infinity, and we can calculate a discrete value at any point along the way.
Which of these two series is “bigger”? Again, we know that both will continue to infinity, but one of these definitely approaches infinity at a *much faster rate* than the other.
Let’s consider the series that happens when you add one to the preceding number: 1, 2, 3, 4, 5, 6, 7, etc. etc. etc.
We know this series will go on to infinity. We also know that we can calculate it’s discrete value at any point along the way.
So far so good?
Now let’s compare that to the series where we double the preceding value: 1, 2, 4, 8, 16, 32, 64, 128, etc. etc. etc.
We know that this series will *also* go on to infinity, and we can calculate a discrete value at any point along the way.
Which of these two series is “bigger”? Again, we know that both will continue to infinity, but one of these definitely approaches infinity at a *much faster rate* than the other.
Infinity is a concept, not a number… and cannot be directly compared to them. There is no number “infinity”, it’s about how you got there.
Most commonly infinity comes up when you’re looking at long term trends and patterns. However, even if two different variables will get “infinitely large” given an unlimited amount of time, often one is still always bigger than the other. Hence, an “infinity” divided by an “infinity” could be anything, from 0 to 1 to 7 to another infinity.
If you write an equation like x^2 / 4x, then as x becomes infinitely large, you have an infinitely large top and bottom of the fraction. However you can also very clearly see that beyond x=4, the top is always bigger than the bottom and the difference just keeps growing. So here we have 2 different infinities which are also infinitely apart from each other. They’re both “infinity”, but one is very much bigger than the other.
In your “irrational numbers” example, the trick is to show that you can pair off each integer with a unique irrational number, and then showing that some irrational numbers will still be missed. Therefore, even though the number of integers is infinite, the number of irrational numbers is “more infinite” since we missed some.
Spoiler because this feels like a homework question.
>!For the sake of example, for each integer, I’m going to map it to the irrational number that’s written as “0.” followed by that integer, then that integer+1, then +2, and so on indefinitely. So irrational number 17 is 0.1718192021222324…. Clearly this will be unique for each possible integer, and the pattern should be easily understood for any possible integer, so I’ve fulfilled my obligations about building this pairing. Yet also clearly I’m missing a lot of irrational numbers, like any that would begin with `0.00`. Ergo, there are more irrational numbers between 0 and 1 than there are integers between 1 and infinity.!<
Infinity is a concept, not a number… and cannot be directly compared to them. There is no number “infinity”, it’s about how you got there.
Most commonly infinity comes up when you’re looking at long term trends and patterns. However, even if two different variables will get “infinitely large” given an unlimited amount of time, often one is still always bigger than the other. Hence, an “infinity” divided by an “infinity” could be anything, from 0 to 1 to 7 to another infinity.
If you write an equation like x^2 / 4x, then as x becomes infinitely large, you have an infinitely large top and bottom of the fraction. However you can also very clearly see that beyond x=4, the top is always bigger than the bottom and the difference just keeps growing. So here we have 2 different infinities which are also infinitely apart from each other. They’re both “infinity”, but one is very much bigger than the other.
In your “irrational numbers” example, the trick is to show that you can pair off each integer with a unique irrational number, and then showing that some irrational numbers will still be missed. Therefore, even though the number of integers is infinite, the number of irrational numbers is “more infinite” since we missed some.
Spoiler because this feels like a homework question.
>!For the sake of example, for each integer, I’m going to map it to the irrational number that’s written as “0.” followed by that integer, then that integer+1, then +2, and so on indefinitely. So irrational number 17 is 0.1718192021222324…. Clearly this will be unique for each possible integer, and the pattern should be easily understood for any possible integer, so I’ve fulfilled my obligations about building this pairing. Yet also clearly I’m missing a lot of irrational numbers, like any that would begin with `0.00`. Ergo, there are more irrational numbers between 0 and 1 than there are integers between 1 and infinity.!<
I think it has to do with countability. If you are able to define a set within two arbitrary points, it is considered countable. Uncountable sets are, by definition, larger than countable sets. Therefore, an infinite Uncountable set is larger than an infinite countable set.
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