BCD to Binary Converter

What is BCD (Binary Coded Decimal)?

Binary Coded Decimal (BCD) is a binary-encoded representation of decimal numbers where each decimal digit is encoded as a fixed-length group of 4 binary bits. Unlike pure binary which converts the entire number to base-2, BCD encodes each digit individually, making it easier to interface with decimal displays and financial systems.

For example, the decimal number 25 is represented as 0010 0101 in BCD (2 = 0010, 5 = 0101), while pure binary represents it as 11001. Converting BCD to pure binary involves first decoding the BCD groups to their decimal digits, then converting the resulting decimal number to standard binary.

The 8421 BCD Encoding System

The most widely used BCD system is the 8421 weighted encoding, where each bit position in a 4-bit group carries a specific weight:

• Most significant bit (MSB): Weight of 8
• Second bit: Weight of 4
• Third bit: Weight of 2
• Least significant bit (LSB): Weight of 1

The decimal value of a BCD nibble \(b_3 b_2 b_1 b_0\) is computed as: $$D = (b_3 \times 8) + (b_2 \times 4) + (b_1 \times 2) + (b_0 \times 1)$$ Only values 0 through 9 are valid; nibbles 1010 through 1111 (decimal 10-15) are invalid in standard BCD.

How to Convert BCD to Binary

The conversion from BCD to pure binary follows a two-step process:

1. Decode BCD to Decimal: Split the BCD string into 4-bit groups. Decode each group using 8421 weights to get individual decimal digits. Combine the digits to form the complete decimal number.
2. Convert Decimal to Binary: Repeatedly divide the decimal number by 2, recording the remainders. Read the remainders from bottom to top to get the pure binary representation.

Example: Converting 0010 0101 (BCD) to Binary

Let's walk through converting the BCD value 0010 0101 to pure binary:

Step 1 - BCD to Decimal:

• First nibble 0010: (0x8)+(0x4)+(1x2)+(0x1) = 2
• Second nibble 0101: (0x8)+(1x4)+(0x2)+(1x1) = 5
• Combined decimal: 25

Step 2 - Decimal to Binary:

• 25 / 2 = 12 remainder 1
• 12 / 2 = 6 remainder 0
• 6 / 2 = 3 remainder 0
• 3 / 2 = 1 remainder 1
• 1 / 2 = 0 remainder 1
• Reading remainders upward: 11001

Therefore, BCD 0010 0101 = Decimal 25 = Binary 11001.

Related Tools

Check out these related conversion tools: BCD to Decimal Converter, Binary Converter, and Decimal to BCD Converter.