Once's in Binary Representation of a Number

Let’s explore how to count the number of 1s (also known as ones) in the binary representation of a number. We'll take the number n = 7 as an example.

1. Initialization: Start with a count of 0.

2. Binary Representation: The binary representation of 7 is 111, which contains 3 ones. Thus, we increment our count to 1.

3. Bit Manipulation: Now, we perform a bitwise operation to remove the rightmost 1:

   • n = 7 (binary 111)
   • When we apply the operation n & (n - 1), we get:
     • 7 - 1 = 6 (binary 110)
     • 7 & 6 = 6 (binary 110)

   After this operation, the count of 1s is now 2 (for the binary 110).

4. Repeat: We continue this process:

   • Now n = 6 (binary 110)
     • 6 - 1 = 5 (binary 101)
     • 6 & 5 = 4 (binary 100)

   The count increases to 3.

5. Final Steps: We continue until n becomes 0:

   • n = 5 (binary 101)
     • 5 - 1 = 4 (binary 100)
     • 5 & 4 = 0 (binary 000)

   At this point, n is 0, and the final count of ones is 3.

Code Implementation

Here is the code that implements this process:

python

def count_ones(n):
    count = 0 

    while n:
        count += 1
        n = n & (n - 1)

    return count

n = int(input("Enter a number: "))
print("The number of ones in", bin(n)[2:], "is", count_ones(n))

Explanation of the Code

• Function Definition: The function count_ones takes an integer n as input.
• Count Variable: We initialize a counter (count) to 0.
• Loop Until Zero: We use a while loop that continues until n becomes 0.
  • In each iteration, we increment the count and update n by clearing the rightmost 1 using n & (n - 1).
• Return Count: Once the loop ends, the function returns the total count of ones.

Conclusion

This method is efficient for counting the number of 1s in the binary representation of a number using bitwise operations. You can test the code by inputting any integer, and it will display the number of ones in its binary form.