Understanding BCD Code Representation

BCD stands for Binary-Coded Decimal. It is a way to represent decimal numbers using binary digits (bits). In BCD, each decimal digit from 0 to 9 is represented by its equivalent 4-bit binary code. This is different from pure binary representation, where the entire decimal number is converted into a single binary number.

Representing (1000)₁₀ in BCD

To represent a decimal number in BCD, we take each digit of the decimal number and represent it using its 4-bit binary equivalent. The number given is (1000)₁₀.

Let's break down the decimal number (1000)₁₀ into its individual digits:

• The first digit is 1.
• The second digit is 0.
• The third digit is 0.
• The fourth digit is 0.

Converting Each Digit to 4-bit BCD

Now, we convert each decimal digit into its 4-bit BCD equivalent:

• Decimal digit 1: The 4-bit BCD code for 1 is 0001.
• Decimal digit 0: The 4-bit BCD code for 0 is 0000.
• Decimal digit 0: The 4-bit BCD code for 0 is 0000.
• Decimal digit 0: The 4-bit BCD code for 0 is 0000.

We can summarize this in a table:

Calculating Total Bits Required for BCD

To find the total number of bits required to represent (1000)₁₀ in BCD, we concatenate the 4-bit BCD codes for each digit in order:

BCD representation of (1000)₁₀ = (BCD of 1) (BCD of 0) (BCD of 0) (BCD of 0)

BCD representation of (1000)₁₀ = 0001 0000 0000 0000

Let's count the total number of bits used:

• Digit 1 uses 4 bits.
• Digit 0 uses 4 bits.
• Digit 0 uses 4 bits.
• Digit 0 uses 4 bits.

Total number of bits = 4 + 4 + 4 + 4 = 16 bits.

Therefore, 16 bits are required to represent the decimal number (1000)₁₀ in BCD code.

Comparison with Pure Binary Representation

It is important to note the difference between BCD and pure binary. The decimal number (1000)₁₀ in pure binary is calculated by finding the powers of 2 that sum up to 1000. \(1000_{10} = 512 + 256 + 128 + 64 + 32 + 8 = 2^9 + 2^8 + 2^7 + 2^6 + 2^5 + 2^3\). This would be represented as \(1111101000_2\). Counting these bits, we get 10 bits for pure binary representation of 1000. However, the question asks for BCD representation, which requires 16 bits.

Revision Table: BCD Concepts

Additional Information: BCD Encoding

BCD encoding is used in applications where decimal arithmetic is preferred or where direct interface with decimal displays (like older calculators or digital clocks) is necessary. While BCD uses more bits than pure binary for the same number (as seen with 1000 requiring 16 bits in BCD vs 10 bits in binary), it simplifies the hardware required for decimal calculations and input/output.

The 4-bit BCD codes for decimal digits 0 through 9 are:

• 0 → 0000
• 1 → 0001
• 2 → 0010
• 3 → 0011
• 4 → 0100
• 5 → 0101
• 6 → 0110
• 7 → 0111
• 8 → 1000
• 9 → 1001

Codes from 1010 to 1111 are invalid BCD codes for single digits.