Bijection between $N$ and $Q $

Question

We know that set of rational numbers is countable. We have bijection betwee $\mathbb{N} $ and $\mathbb{N} \times \mathbb{N }$, whose graphical representation is beow:
Also the bijection between $\mathbb{N} $ and $\mathbb{Z }$ exists. I am unable to find the bijection between $\mathbb{N} $ and $\mathbb{Q} $. Can anyanybody tell me the bijection between $N$ and $Q$ with its graphical representation.

Answer 1

A *relation $$f:Q^+\to N$$ where $f(p/q)=\frac{(p+q-2)(p+q-1)}{2}+q$ can be defined if you collect the positive rationals diagonally. (Here $Q^+$ denotes the set of positive rationals.) Further you may define a relation between negative rationals $Q^-$ and natural numbers in a similar fashion. And this should give a relation between rationals and naturals. $$*NOTE$$This is not a bijective mapping (infact, not even a mapping; for instance: $f({1\over5})\neq f({3\over15}) $ ), it is just a device to indicate the countability of Rationals. Though you may restrict it to a 1-1 into function from rationals to natural numbers if you "skip" the elements you have previously counted. What I mean by this is: if you count $1/2$, you won't pick $2/4, 3/6, 4/8...$ and so on. You may find it in any elementary set theory text book.