Intuitive way to explain |Q| = aleph zero?

In summary, the standard proofs for the rationals and perfect squares have the same inuitive argument.
  • #1
EnumaElish
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What is an intuitive, calculus-grade way to explain that rationals have the same cardinality as N? (Same question for perfect squares.)
 
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  • #2
I'm not trying to be difficult, but aren't the standard proofs pretty inuitive? Especially for the perfect squares. To every element of the set of perfect squares you can correspond exactly on element of the set of naturals: its square root. The same goes for the other way around. So you have a one to one correspondance between the squares and naturals, so they have the same cardinality. The proof is nearly identical for the rationals, except that the bijection is somewhat more complicated
 
  • #3
Look at http://home.att.net/~numericana/answer/sets.htm [Broken] in the section called "The countability of rational numbers".
 
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  • #4
Heres a pictorial way to show it for the rationals. Start with the positive rationals and start listing them like so
1/1 1/2 1/3 1/4 1/5...
2/1 2/2 2/3 2/4 2/5 ...
3/1 3/2 3/3 3/4 3/5 ...

Now, to count, follow this sequence: 1/2, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3,... (I hope you can see what I'm doing). Now, it should be clear that this list is countable. Now, insert the negative rationals just before their positive counterparts, i.e. -1/1, 1/1, -1/2,..., and just put zero at the top left. Now, using the same arguement, the full set of Q is countable.

I'm not sure whether this was something like what you were after, but it seems quite intuitive!
 
  • #5
Ah, diagonal method - possibly because a proof that Q is countable is called diagonal, and that R isn't yet has the same adjective, people have wrong impressions.One way is a standard method using prime decomposition.

Send a/b to 2^a*3^b (for a and b >0) or 2^a*3^b*5 for negative rationals. This is a bijection from Q onto a subset of N. This proof shows that any finite product of countable sets is countable, and shows an important difference between union and product (in the categorical sense, perhaps).
 
  • #6
I thought both the "perfect squares" and the "rationals" demonstration can use the cross-tabulation square where row and column headings are the natural numbers. For the perfect squares, each cell holds the product "row # * column #"; all perfect squares are on the diagonal, which you can count linearly.

For the rationals, each cell is row #/column # (or the inverse), but I wasn't sure how to count them. Now I know. Thanks.
 

1. What is |Q| = aleph zero?

|Q| = aleph zero is a mathematical concept that refers to the cardinality, or size, of the set of rational numbers. It is the same size as the set of natural numbers (denoted as |N|) and is equal to the infinite number aleph zero.

2. How can the intuitive explanation of |Q| = aleph zero be understood?

The intuitive explanation of |Q| = aleph zero can be understood by imagining that the set of rational numbers can be paired up with the set of natural numbers. This shows that the two sets have the same size, and therefore, |Q| = |N| = aleph zero.

3. What is the importance of understanding |Q| = aleph zero?

Understanding |Q| = aleph zero is important in the field of mathematics as it provides a deeper understanding of the concept of infinity and the different sizes of infinite sets. It also has practical applications in areas such as computer science and cryptography.

4. What are some examples of rational numbers?

Some examples of rational numbers include 1/2, 3/4, -2/5, 0.25, and 2.5. These are numbers that can be expressed as a ratio of two integers, with the denominator not equal to 0.

5. How does |Q| = aleph zero compare to other infinite sets?

|Q| = aleph zero is smaller than the set of real numbers (denoted as |R|), which is considered a larger infinite set. It is also larger than the set of integers (denoted as |Z|), which is considered a smaller infinite set.

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