Count total number of digits from 1 to N in C++

We are given a number N as input. The goal is to count the total number of digits between numbers 1 to N. 1 to 9 numbers require 1 digit each, 11 to 99 require 2 digits each, 100 to 999 require 3 digits each and so on.

Let us understand with examples

Input − N=11

Output − Count of total number of digits from 1 to N are: 13

Explanation − Numbers 1 to 9 has 1 digit each : 9 digits 10, 11 have 2 digits each. 4 digits. Total digits= 9+4=13.

Input − N=999

Output − Count of the total number of digits from 1 to N are: 2889

Explanation − Numbers 1 to 9 have 1 digit each : 9 digits 10 to 99 have 2 digits each. : 180 digits. 100 to 999 have 3 digits each : 2700 digits Total digits= 2700 + 180 + 9 = 2889 digits

The approach used in the below program is as follows

We will use two approaches. A first naive approach using a recursive function to calculate digits in a number num. Convert passed num into a string. The length of the string is digits in num. Do this for each number bypassing current num-1 recursively.

• Take a number as a positive integer.

• Function total_digits(int num) take the num and returns digit in numbers between 1 to num.

• To calculate digits in num, convert num to string. (to_string(num)).

• The length of the string is digits in num.

• If num is 1 return 1 . Else return length+ total_digits(num-1) for rest of numbers less than num.

• At last we will get total digits as a result.

Efficient Approach

In this approach, we will use the logic that for each number up to N. We will traverse 10, 100, 1000 up to N. For each 10i , the number of digits is (num-i + 1).

• Take a number as a positive integer.

• Function total_digits(int num) take the num and returns digit in numbers between 1 to num.

• Take the total count as 0 initially.

• Traverse i=1 to i<=num, increment i by 10 in each iteration and add num-i+1 to count.

• Return the count at the end of for loop as a result.

Example (naive approach)

 Live Demo

#include <bits/stdc++.h>
using namespace std;
int total_digits(int num){
   string str = to_string(num);
   int length = str.length();
   if (num == 1){
      return 1;
   }
   return length + total_digits(num - 1);
}
int main(){
   int num = 20;
   cout<<"Count of total number of digits from 1 to n are: "<<total_digits(num);
   return 0;
}

Output

If we run the above code it will generate the following output −

Count of total number of digits from 1 to n are: 31

Example (Efficient approach)

 Live Demo

#include <bits/stdc++.h>
using namespace std;
int total_digits(int num){
   int count = 0;
   for(int i = 1; i <= num; i *= 10){
      count = count + (num - i + 1);
   }
   return count;
}
int main(){
   int num = 20;
   cout<<"Count of total number of digits from 1 to n are: "<<total_digits(num);
   return 0;
}

Output

If we run the above code it will generate the following output −

Count of total number of digits from 1 to n are: 31