Binary to Decimal Conversion: A Complete Guide

Converting binary (base-2) to decimal (base-10) is a fundamental skill for anyone working with computers. This guide covers both directions with clear explanations and working code examples.

Understanding Place Values

In a decimal number like 235, the positions represent powers of 10:

Hundreds    Tens    Ones
10²         10¹     10⁰
(100)       (10)    (1)
2           3       5
= 200       + 30    + 5    = 235

Binary works the exact same way, but with 2 as the base:

2³      2²     2¹     2⁰
(8)     (4)    (2)    (1)

Binary to Decimal: Adding Up the Powers

The Method

To convert a binary number to decimal:

1. Write down the binary number
2. Assign powers of 2 to each digit, starting from the right (beginning at 1)
3. Add up the values only where a “1” appears

Example: Convert 1011 to Decimal

Total: 8 + 0 + 2 + 1 = 11

So, binary 1011 is 11 in decimal.

The “Magic Row” Trick for 8-bit Numbers

For any byte (8 bits), memorize this sequence:

128, 64, 32, 16, 8, 4, 2, 1

If you see 00010101, simply look at which “slots” are active:

Position:  128  64  32  16   8   4   2   1
Binary:      0   0   0   1   0   1   0   1
Active:                 ✓       ✓       ✓

Total: 16 + 4 + 1 = 21

Python Implementation

def binary_to_decimal(binary_str):
    decimal_value = 0
    # Reverse the string to start from the right (2^0)
    reversed_binary = binary_str[::-1]

    for i in range(len(reversed_binary)):
        if reversed_binary[i] == '1':
            # Add 2 raised to the power of the current index
            decimal_value += 2**i

    return decimal_value

# Test the function
binary_input = "1011"
print(f"The decimal version of {binary_input} is {binary_to_decimal(binary_input)}")
# Output: The decimal version of 1011 is 11

How it works:

1. Reverse the binary string so we process from right to left (least significant bit first)
2. For each position that contains a ‘1’, add 2 raised to that position’s power
3. Return the total

Decimal to Binary: The Division Method

Going from decimal to binary involves breaking a number down into powers of 2. The most common technique is the “Repeat Division-by-2” method.

The Method

1. Divide the decimal number by 2
2. Write down the remainder (always 0 or 1)
3. Take the quotient and repeat until the quotient is 0
4. Read the remainders from bottom to top

Example: Convert 13 to Binary

Result: Reading from bottom to top → 1101

Verification

1 1 0 1
│ │ │ │
│ │ │ └─ 1 × 1 = 1
│ │ └─── 0 × 2 = 0
│ └───── 1 × 4 = 4
└─────── 1 × 8 = 8

Total: 8 + 4 + 0 + 1 = 13 ✓

Decimal to Binary: The Subtraction Method

If you know your powers of 2, this method can be faster for small numbers.

Convert 21 to Binary

Powers of 2: 128, 64, 32, 16, 8, 4, 2, 1

Result: 10101

Python Implementation

def decimal_to_binary(n):
    if n == 0:
        return "0"

    binary_digits = []
    while n > 0:
        # Get the remainder (0 or 1)
        remainder = n % 2
        binary_digits.append(str(remainder))
        # Update n to the integer quotient
        n = n // 2

    # Reverse the list because we calculated from least to most significant
    binary_digits.reverse()
    return "".join(binary_digits)

# Test the function
decimal_input = 13
print(f"The binary version of {decimal_input} is {decimal_to_binary(decimal_input)}")
# Output: The binary version of 13 is 1101

How it works:

1. Repeatedly divide by 2 and collect the remainders
2. The remainders (0 or 1) are the binary digits
3. Reverse at the end because we collected them least-significant-first

Quick Reference

Powers of 2 to Memorize

Common Byte Values

Practice Examples

Try these yourself:

Summary