# Algorithm to generate positive rational numbers

A rational number is of the form p/q where p and q are integers. The problem statement is to generate rational number such that any particular number is generated in a finite time. For a given n, we generate all rational numbers where 1 <= p <= n and 1 <= q <= n

Examples:

```Input : 5
Output : 1, 1/2, 2, 1/3, 2/3, 3/2, 3, 1/4,
3/4, 4/3, 4, 1/5, 2/5, 3/5, 4/5,
5/4, 5/3, 5/2, 5

Input : 7
Output :1, 1/2, 2, 1/3, 2/3, 3/2, 3, 1/4, 3/4,
4/3, 4, 1/5, 2/5, 3/5, 4/5, 5/4, 5/3,
5/2, 5, 1/6, 5/6, 6/5, 6, 1/7, 2/7, 3/7,
4/7, 5/7, 6/7, 7/6, 7/5, 7/4, 7/3, 7/2, 7
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

In mathematical terms a set is countably infinite if its elements can be mapped on a one to one basis with the set of natural numbers.

The problem statement here is to generate combinations of p/q where both p and q are integers and any particular combination of p and q will be reached in a finite no. of steps. If p is incremented 1, 2, 3… etc keeping q constant or vice versa all combinations cannot be reached in finite time. The way to handle this is to imagine the natural numbers arranged as a row, col of a matrix

(1, 1) (1, 2) (1, 3) (1, 4)
(2, 1) (2, 2) (2, 3) (2, 4)
(3, 1) (3, 2) (3, 3) (3, 4)
(4, 1) (4, 2) (4, 3) (4, 4)

These elements are traversed in an inverted L shape in each iteration

(1, 1)
(1, 2), (2, 2) (2, 1)
(1, 3), (2, 3), (3, 3), (3, 2), (3, 1)

yielding

1/1
1/2, 2/2, 2/1
1/3, 2/3, 3/3, 3/2, 3/1

Obviously this will yield duplicates as 2/1 and 4/2 etc, but these can be weeded out by using the Greatest common divisor constraint.

 `// Java program  ` `import` `java.util.ArrayList; ` `import` `java.util.List; ` ` `  `class` `Rational { ` ` `  `    ``private` `static` `class` `RationalNumber { ` ` `  `        ``private` `int` `numerator; ` `        ``private` `int` `denominator; ` ` `  `        ``public` `RationalNumber(``int` `numerator, ``int` `denominator) ` `        ``{ ` `            ``this``.numerator = numerator; ` `            ``this``.denominator = denominator; ` `        ``} ` ` `  `        ``@Override` `        ``public` `String toString() ` `        ``{ ` `            ``if` `(denominator == ``1``) { ` `                ``return` `Integer.toString(numerator); ` `            ``} ` `            ``else` `{ ` `                ``return` `Integer.toString(numerator) + ``'/'` `+  ` `                       ``Integer.toString(denominator); ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``/** ` `     ``* Greatest common divisor ` `     ``* @param num1 ` `     ``* @param num2 ` `     ``* @return ` `     ``*/` `    ``private` `static` `int` `gcd(``int` `num1, ``int` `num2) ` `    ``{ ` `        ``int` `n1 = num1; ` `        ``int` `n2 = num2; ` ` `  `        ``while` `(n1 != n2) { ` `            ``if` `(n1 > n2) ` `                ``n1 -= n2; ` `            ``else` `                ``n2 -= n1; ` `        ``} ` `        ``return` `n1; ` `    ``} ` ` `  `    ``private` `static` `List generate(``int` `n) ` `    ``{ ` ` `  `        ``List list = ``new` `ArrayList<>(); ` ` `  `        ``if` `(n > ``1``) { ` `            ``RationalNumber rational = ``new` `RationalNumber(``1``, ``1``); ` `            ``list.add(rational); ` `        ``} ` ` `  `        ``for` `(``int` `loop = ``1``; loop <= n; loop++) { ` ` `  `            ``int` `jump = ``1``; ` ` `  `            ``// Handle even case ` `            ``if` `(loop % ``2` `== ``0``) ` `                ``jump = ``2``; ` `            ``else` `                ``jump = ``1``; ` ` `  `            ``for` `(``int` `row = ``1``; row <= loop - ``1``; row += jump) { ` ` `  `                ``// Add only if there are no common divisors other than 1 ` `                ``if` `(gcd(row, loop) == ``1``) { ` `                    ``RationalNumber rational = ``new` `RationalNumber(row, loop); ` `                    ``list.add(rational); ` `                ``} ` `            ``} ` ` `  `            ``for` `(``int` `col = loop - ``1``; col >= ``1``; col -= jump) { ` ` `  `                ``// Add only if there are no common divisors other than 1 ` `                ``if` `(gcd(col, loop) == ``1``) { ` `                    ``RationalNumber rational = ``new` `RationalNumber(loop, col); ` `                    ``list.add(rational); ` `                ``} ` `            ``} ` `        ``} ` ` `  `        ``return` `list; ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``List rationals = generate(``7``); ` `        ``System.out.println(rationals.stream(). ` `                  ``map(RationalNumber::toString). ` `                  ``reduce((x, y) -> x + ``", "` `+ y).get()); ` `    ``} ` `} `

Output:

```1, 1/2, 2, 1/3, 2/3, 3/2, 3, 1/4, 3/4, 4/3, 4, 1/5, 2/5, 3/5, 4/5, 5/4, 5/3, 5/2, 5, 1/6, 5/6, 6/5, 6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 7/6, 7/5, 7/4, 7/3, 7/2, 7
```

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