How is the set of rational numbers countably infinite?

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Because you can construct a mapping of the rationals into the integers.

One easy way is to map the positive rational $\dfrac{a}{b}$ to the integer $2^a 3^b$.

This shows that there are at least as many integers as there are positive rationals.

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DilllyBar
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Updated on October 28, 2022

Comments

  • DilllyBar
    DilllyBar about 1 year

    How is $\mathbb{Q}$ countably infinite?

    The definition says all elements of the set must have a one-to-one relation to the natural numbers. I do not understand this.

    How do the elements in $\mathbb{Q}$ have a one-to-one relation with natural numbers?

    • copper.hat
      copper.hat almost 7 years
      Do you see that $\mathbb{N}$ , $\mathbb{Z}$ and $\mathbb{Z}^2$ all have the same cardinality?
    • Daniel Xiang
      Daniel Xiang almost 7 years
      you've already asked this question and I've posted a solution. Also I'm sure that if you googled "rational numbers countable proof" it would be the first link to appear.
    • Masacroso
      Masacroso almost 7 years
      Just for the record see that.