乔治·康托有理数集的第 N 项
原文:https://www . geesforgeks . org/n-th-term-of-George-cantor-set-of-rational-numbers/
名叫数学家乔治·康托给出了有理数集是可枚举的一个证明。我们不需要在这里证明，而是需要确定有理数集合中的第 n 项。 例:
Input : N = 8
Output : 2/3
Input : N = 15
Output : 1/5
See image for reference of counting.
有理数的集合如下图所示:
这里第一项是 1/1，第二项是 1/2，第三项是 2/1，第四项是 3/1，第五项是 2/2，第六项是 1/3 以此类推……
1.将此视为一个矩阵(二维数组)，其中行从 1 到 n，列从 1 到 n。 2。行/列会给出有理数。 3。从上图观察图案，可以清楚如何穿过矩阵，即从左到右向上倾斜，然后从右到左向下，直到我们到达第 N 个位置。 4。有 4 个规则的重复模式 5。每次迭代后，移动到下一个数字，继续使用计数器 k 计算其位置，然后与 N 进行比较
C++
// C++ program to find N-th term in
// George Cantor set of rational numbers
#include <bits/stdc++.h>
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 : "<<i<<" / "<<j;
}
// driver code
int main()
{
int n = 15;
georgeCantor(n);
return 0;
}
Java 语言(一种计算机语言，尤用于创建网站)
// 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
Python 3
# 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
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP program to find N-th
// term in George Cantor set
// of rational numbers
function georgeCantor($n)
{
$i = 1; // let i = numerator
$j = 1; // let j = denominator
// to keep the check
// of no. of terms
$k = 1;
// loop till k is
// not equal to n
while ($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++;
}
}
echo "N-th term : ", $i, "/", $j;
}
// Driver Code
$n = 15;
georgeCantor($n);
// This code is contributed by ajit
?>
java 描述语言
<script>
// 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.
</script>
输出:
N-th term : 1/5
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