How to Convert Binary to Decimal - Computing Fundamentals

To convert binary to decimal, multiply each binary digit by 2 raised to the power of its position index from right to left starting at 0, then add all the results together, like decimal = sum(bit * 2**position).

Examples

Input101

Output5

Input1101

Output13

Input0

Output0

How to Think About It

Think of each binary digit as a switch that is either off (0) or on (1). Each position from right to left represents a power of 2 starting at 0. To find the decimal number, add up the values of all the switches that are on.

Algorithm

1

Get the binary number as a string.

2

Start from the rightmost digit and move left.

3

For each digit, convert it to a number (0 or 1).

4

Multiply this number by 2 raised to the power of the digit's position index.

5

Add this value to a running total.

6

Return the total as the decimal number.

Code

computing_fundamentals

binary = input('Enter binary number: ')
decimal = 0
for i, bit in enumerate(reversed(binary)):
    decimal += int(bit) * (2 ** i)
print('Decimal:', decimal)

Output

Enter binary number: 1101
Decimal: 13

Dry Run

Let's trace binary '101' through the code

1

Start with decimal = 0

decimal = 0

2

Process rightmost bit '1' at position 0

decimal += 1 * 2**0 = 1

3

Process middle bit '0' at position 1

decimal += 0 * 2**1 = 0 (no change)

4

Process leftmost bit '1' at position 2

decimal += 1 * 2**2 = 4

Step	Bit	Position	Calculation	Decimal Total
1	1	0	1 * 2**0 = 1	1
2	0	1	0 * 2**1 = 0	1
3	1	2	1 * 2**2 = 4	5

Why This Works

Step 1: Binary digits represent powers of 2

Each digit in a binary number stands for 2 raised to the power of its position, starting from 0 on the right.

Step 2: Multiply each bit by its power of 2

Multiply the bit (0 or 1) by 2 to the power of its position to find its decimal value.

Step 3: Sum all values to get decimal

Add all these values together to get the final decimal number.

Alternative Approaches

Using built-in conversion function

computing_fundamentals

binary = input('Enter binary number: ')
decimal = int(binary, 2)
print('Decimal:', decimal)

This method is simpler and faster but depends on language built-ins.

Recursive conversion

computing_fundamentals

def bin_to_dec(b):
    if b == '':
        return 0
    return bin_to_dec(b[:-1]) * 2 + int(b[-1])
binary = input('Enter binary number: ')
print('Decimal:', bin_to_dec(binary))

Uses recursion to process bits from left to right; good for understanding but less efficient.

Complexity: O(n) time, O(1) space

Time Complexity

The algorithm processes each binary digit once, so time grows linearly with the number of digits.

Space Complexity

Only a few variables are used regardless of input size, so space usage is constant.

Which Approach is Fastest?

Using built-in functions is fastest and simplest; manual loops give more control and understanding.

Approach	Time	Space	Best For
Manual loop	O(n)	O(1)	Learning and control
Built-in function	O(n)	O(1)	Quick and practical use
Recursive method	O(n)	O(n)	Educational, less efficient