乔治·坎托的第N个有理数集

乔治·康托尔(George Cantor)，一位名叫数学家的数学家，证明了有理数集是可数的。我们不必在这里进行证明，而必须确定有理数集中的第N个项。
例子 ：

Input : N = 8
Output : 2/3

Input : N = 15
Output : 1/5
See image for reference of counting.

有理数集如下图所示：

这里第一项为1/1，第二项为1/2，第三项为2/1，第四项为3/1，第五项为2/2，第六项为1/3，依此类推……

1. View this as a matrix(2-D array) with rows from 1 to n and columns from 1 to n. 
2. rows/columns will give the rational numbers. 
3. Observe the pattern from the above image it will be clear how to traverse through the matrix i.e diagonally from left to right upwards and then from right to left downwards till we reach the Nth position. 
4. There are 4 regular repeating pattern 
5. After each iteration move to next number to keep counting its position using counter k, then compare it with N

C++

// C++ program to find N-th term in
// George Cantor set of rational numbers
#include 
using namespace std;
 
void georgeCantor(int n)
{    
    int i = 1; // let i = numerator
    int j = 1; // let j = denominator
    int k = 1; // to keep the check of no. of terms
         
    // loop till k is not equal to n
    while (k < n)
    {
        j++ , k++;
         
        // check if k is already equal to N
        // then the first term is the required
        // rational number
        if (k == n)
            break;
         
        // loop for traversing from right to left
        // downwards diagonally
        while (j > 1 && k < n) {
            i++, j--, k++;
        }
         
        if (k == n)
           break;
         
        i++, k++;
         
        if (k == n)
           break;
         
        // loop for traversing from left
        // to right upwards diagonally
        while (i > 1 && k < n) {
            i--, j++, k++;
        }       
    }   
    cout << "N-th term : "<

---

Java

// Java program to find N-th term in
// George Cantor set of rational number
import java.io.*;
 
class GFG
{
    static void georgeCantor(int n)
    {
        // let i = numerator
        int i = 1;
         
        // let j = denominator
        int j = 1;
         
        // to keep the check of no. of terms
        int k = 1;
             
        // loop till k is not equal to n
        while (k < n)
        {
            j++ ;
            k++;
             
            // check if k is already equal to N
            // then the first term is the required
            // rational number
            if (k == n)
                break;
             
            // loop for traversing from right to left
            // downwards diagonally
            while (j > 1 && k < n) {
                i++;
                j--;
                k++;
            }
             
            if (k == n)
            break;
             
            i++;
            k++;
             
            if (k == n)
            break;
             
            // loop for traversing from left
            // to right upwards diagonally
            while (i > 1 && k < n) {
                i--;
                j++;
                k++;
            }
        }
        System.out.println("N-th term : "+i +"/" +j);
    }
     
    // Driver code
    public static void main (String[] args)
    {
        int n = 15;
        georgeCantor(n);
         
    }
}
 
// This code is contributed by vt_m

Python3

# Python3 program to find N-th term in
# George Cantor set of rational numbers
 
def georgeCantor(n):
     
    # let i = numerator
    i = 1
     
    # let j = denominator
    j = 1 
     
    # to keep the check of no. of terms
    k = 1 
         
    # loop till k is not equal to n
    while k < n:
        j += 1
        k += 1
         
        # check if k is already equal to N
        # then the first term is the
        # required rational number
        if k == n:
            break
         
        # loop for traversing from right
        # to left downwards diagonally
        while j > 1 and k < n:
            i += 1
            j -= 1
            k += 1
         
        if k == n:
            break
         
        i += 1
        k += 1
         
        if k == n:
            break
         
        # loop for traversing from left
        # to right upwards diagonally
        while i > 1 and k < n:
            i -= 1
            j += 1
            k += 1
    print ("N-th term : %d/%d"%(i, j))
 
# Driver code
n = 15
georgeCantor(n)
 
# This code is contributed
# by Shreyanshi Arun

C#

// C# program to find N-th term in
// George Cantor set of rational number
using System;
 
class GFG
{
    static void georgeCantor(int n)
    {
        // let i = numerator
        int i = 1;
         
        // let j = denominator
        int j = 1;
         
        // to keep the check of no. of terms
        int k = 1;
             
        // loop till k is not equal to n
        while (k < n)
        {
            j++ ;
            k++;
             
            // check if k is already equal to N
            // then the first term is the required
            // rational number
            if (k == n)
                break;
             
            // loop for traversing from right to left
            // downwards diagonally
            while (j > 1 && k < n)
            {
                i++;
                j--;
                k++;
            }
             
            if (k == n)
            break;
             
            i++;
            k++;
             
            if (k == n)
            break;
             
            // loop for traversing from left
            // to right upwards diagonally
            while (i > 1 && k < n)
            {
                i--;
                j++;
                k++;
            }
        }
        Console.WriteLine("N-th term : "+i +"/" +j);
    }
     
    // Driver code
    public static void Main ()
    {
        int n = 15;
        georgeCantor(n);
         
    }
}
 
// This code is contributed by vt_m

PHP

 1 && $k < $n)
        {
            $i++; $j--; $k++;
        }
         
        if ($k == $n)
        break;
         
        $i++; $k++;
         
        if ($k == $n)
        break;
         
        // loop for traversing from left
        // to right upwards diagonally
        while ($i > 1 && $k < $n)
        {
            $i--; $j++; $k++;
        }
    }
    echo "N-th term : ", $i, "/", $j;
}
 
// Driver Code
$n = 15;
georgeCantor($n);
 
// This code is contributed by ajit
?>

Javascript

// JavaScript program to find N-th term in
// George Cantor set of rational number
 
    function georgeCantor(n)
    {
        // let i = numerator
        let i = 1;
           
        // let j = denominator
        let j = 1;
           
        // to keep the check of no. of terms
        let k = 1;
               
        // loop till k is not equal to n
        while (k < n)
        {
            j++ ;
            k++;
               
            // check if k is already equal to N
            // then the first term is the required
            // rational number
            if (k == n)
                break;
               
            // loop for traversing from right to left
            // downwards diagonally
            while (j > 1 && k < n) {
                i++;
                j--;
                k++;
            }
               
            if (k == n)
            break;
               
            i++;
            k++;
               
            if (k == n)
            break;
               
            // loop for traversing from left
            // to right upwards diagonally
            while (i > 1 && k < n) {
                i--;
                j++;
                k++;
            }
        }
        document.write("N-th term : "+i +"/" +j);
    }
   
// Driver Code
 
        let n = 15;
        georgeCantor(n);
 
// This code is contributed by susmitakundugoaldanga.

输出 ：

N-th term : 1/5