There is a proof that there are just as many positive numbers as there are integers (both positive and negative numbers). Some call it a paradox, but it makes sense from a mathematical point of view. Before getting into that, though, we need to talk about sets.
A set is simply a collection of objects. There can be a set of marbles in a bag, a set of birds in a flock, or a set of people in a class. Each set has a size or cardinality, which is the number of objects, or elements, in the set. Sometimes, we want to know whether two sets have the same size. For example, a farmer wants to know whether the number of sheep going out to graze is the same as the number going back in, so that he knows if he's lost any sheep.
One way would be to count the number of sheep going in, and then the number of sheep going out. If they're the same, the farmer is good to go. However, there is an easier and clever way, and the farmer doesn't need to know how to count.
First, the farmer watches the sheep go in, and places a pebble on the ground for each sheep. Then, he removes a pebble for each sheep that goes back. If there are none left, then the number of sheep going in is the same as the number going out. In set theory terms, the set of pebbles and the set of sheep have a one-to-one correspondence, i.e. you can match up each element in one set with exactly one element in another set.
Back to the paradox, though. First, we'll lay out the integers, like so:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
And the positive numbers under them.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
These two sets do not seem to have the same number of elements at all. Clearly, the integers have all of the negative numbers, so it must be twice as big!
However, this reasoning is flawed. These sets have a one-to-one correspondence, if they're just arranged like this:
0 1 2 3 4 5 6 7 8
0 1 -1 2 -2 3 -3 4 -4
Every positive integer is followed by its negative counterpart, and every integer appears some finite number of elements down on the list, so there are just as many integers as there are positive integers. Both of these sets are infinite, but they have the same cardinality. The number of elements in each set is the smallest, countable infinity, which is small enough that all of its elements can be systematically listed, without any element appearing infinitely far down the list. In math, this infinite "number" is expressed as aleph-null.
Even the number of rational numbers is countably infinite. Here is an infinite table of the positive rational numbers:
A set is simply a collection of objects. There can be a set of marbles in a bag, a set of birds in a flock, or a set of people in a class. Each set has a size or cardinality, which is the number of objects, or elements, in the set. Sometimes, we want to know whether two sets have the same size. For example, a farmer wants to know whether the number of sheep going out to graze is the same as the number going back in, so that he knows if he's lost any sheep.
One way would be to count the number of sheep going in, and then the number of sheep going out. If they're the same, the farmer is good to go. However, there is an easier and clever way, and the farmer doesn't need to know how to count.
First, the farmer watches the sheep go in, and places a pebble on the ground for each sheep. Then, he removes a pebble for each sheep that goes back. If there are none left, then the number of sheep going in is the same as the number going out. In set theory terms, the set of pebbles and the set of sheep have a one-to-one correspondence, i.e. you can match up each element in one set with exactly one element in another set.
Back to the paradox, though. First, we'll lay out the integers, like so:
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
And the positive numbers under them.
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
These two sets do not seem to have the same number of elements at all. Clearly, the integers have all of the negative numbers, so it must be twice as big!
However, this reasoning is flawed. These sets have a one-to-one correspondence, if they're just arranged like this:
0 1 2 3 4 5 6 7 8
0 1 -1 2 -2 3 -3 4 -4
Every positive integer is followed by its negative counterpart, and every integer appears some finite number of elements down on the list, so there are just as many integers as there are positive integers. Both of these sets are infinite, but they have the same cardinality. The number of elements in each set is the smallest, countable infinity, which is small enough that all of its elements can be systematically listed, without any element appearing infinitely far down the list. In math, this infinite "number" is expressed as aleph-null.
Even the number of rational numbers is countably infinite. Here is an infinite table of the positive rational numbers:
It seems like the number of rational numbers is the square of the number of positive integers, but again, they can be systematically listed by reading the table by diagonals. The list would start:
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1...
However, this list is not yet perfect. For one thing, some of these fractions are not in simplest form. Let's simplify those:
1, 1/2, 2, 1/3, 1, 3, 1/4, 2/3, 3/2, 4, 1/5, 1/2, 1, 2, 5...
There are redundancies, too, so those can be omitted:
1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5...
0 and the negative rational numbers have been ignored at this point. Let's add 0 to the beginning and write every negative rational number after its positive counterpart:
0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 1/4, -1/4, 2/3, -2/3, 3/2, -3/2, 4, -4, 1/5, -1/5, 5, -5...
Now, this list has every rational number exactly once, and no rational number is infinitely far down. Mission accomplished!
Could we go even further and list the real numbers too, giving them a one-to-one correspondence with the natural numbers? The rational numbers are nested infinitely on the number line - between every pair of rationals, there is always another one - and we could list those, so why not the reals? As it turns out, there are other, much larger, uncountable kinds of infinity. There are so many real numbers that they can't even be listed. In fact, just the set of reals between 0 and 1 is uncountable. The most famous proof of this fact is Georg Cantor's diagonal argument, which works as follows:
Let's assume that you have supposedly listed all real numbers between 0 and 1. The beginning of your list could look something like this:
0.15027...
0.58513...
0.23969...
0.84568...
0.64218...
...
Now, I will create a number that is not on your list. Start with a 0.
0
I'll take the first digit of your first number and add 1 to it to make the first digit of my number. If that turns out to be a 10, I'll change it to a 0.
0.2
Now, I'll do the same for the second digit of your second number.
0.29
I repeat this process for all numbers in your list, and get an infinitely long string of digits.
0.29079...
the 1st digit of my number (2) is not the 1st digit of the 1st number on your list (1),
so it can't be equal to the 1st number.
the 2nd digit of my number (9) is not the 2nd digit of the 2nd number on your list (8),
so it can't be equal to the 2nd number.
The number of real numbers is uncountably infinite, and this type of infinity is called aleph-one.
There is an infinite hierarchy of infinities, where every type of infinity is 2 to the power of the last. Aleph-one is two to the power of aleph-null, aleph-two is two to the power of aleph-one, etc. All of these types of infinities are called cardinal numbers. There are even other types of infinities, called ordinals, which describe not just the size of a set, but how its elements are ordered. Where we saw just one countably infinite cardinal aleph-null, there is an uncountable number of countably infinite ordinal numbers with infinitely complex structure. abc
For more information, see Vihart's and VSauce's videos on this.
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1...
However, this list is not yet perfect. For one thing, some of these fractions are not in simplest form. Let's simplify those:
1, 1/2, 2, 1/3, 1, 3, 1/4, 2/3, 3/2, 4, 1/5, 1/2, 1, 2, 5...
There are redundancies, too, so those can be omitted:
1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5...
0 and the negative rational numbers have been ignored at this point. Let's add 0 to the beginning and write every negative rational number after its positive counterpart:
0, 1, -1, 1/2, -1/2, 2, -2, 1/3, -1/3, 3, -3, 1/4, -1/4, 2/3, -2/3, 3/2, -3/2, 4, -4, 1/5, -1/5, 5, -5...
Now, this list has every rational number exactly once, and no rational number is infinitely far down. Mission accomplished!
Could we go even further and list the real numbers too, giving them a one-to-one correspondence with the natural numbers? The rational numbers are nested infinitely on the number line - between every pair of rationals, there is always another one - and we could list those, so why not the reals? As it turns out, there are other, much larger, uncountable kinds of infinity. There are so many real numbers that they can't even be listed. In fact, just the set of reals between 0 and 1 is uncountable. The most famous proof of this fact is Georg Cantor's diagonal argument, which works as follows:
Let's assume that you have supposedly listed all real numbers between 0 and 1. The beginning of your list could look something like this:
0.15027...
0.58513...
0.23969...
0.84568...
0.64218...
...
Now, I will create a number that is not on your list. Start with a 0.
0
I'll take the first digit of your first number and add 1 to it to make the first digit of my number. If that turns out to be a 10, I'll change it to a 0.
0.2
Now, I'll do the same for the second digit of your second number.
0.29
I repeat this process for all numbers in your list, and get an infinitely long string of digits.
0.29079...
the 1st digit of my number (2) is not the 1st digit of the 1st number on your list (1),
so it can't be equal to the 1st number.
the 2nd digit of my number (9) is not the 2nd digit of the 2nd number on your list (8),
so it can't be equal to the 2nd number.
The number of real numbers is uncountably infinite, and this type of infinity is called aleph-one.
There is an infinite hierarchy of infinities, where every type of infinity is 2 to the power of the last. Aleph-one is two to the power of aleph-null, aleph-two is two to the power of aleph-one, etc. All of these types of infinities are called cardinal numbers. There are even other types of infinities, called ordinals, which describe not just the size of a set, but how its elements are ordered. Where we saw just one countably infinite cardinal aleph-null, there is an uncountable number of countably infinite ordinal numbers with infinitely complex structure. abc
For more information, see Vihart's and VSauce's videos on this.