How many bits are required to represent the decimal numbers in the range from 0 to 999 using
(a) straight binary code?
(b) Using BCD code?
The correct answer and explanation is :
To represent decimal numbers from 0 to 999, we analyze both straight binary and BCD (Binary-Coded Decimal).
✅ (a) Straight Binary Code:
We need the minimum number of bits N such that:
$$
2^N > 999
$$
Let’s calculate:
- $2^9 = 512$
- $2^{10} = 1024$
So, we need 10 bits.
✅ Answer: 10 bits
✅ (b) BCD Code:
BCD represents each decimal digit using 4 bits. Since 999 has 3 digits (hundreds, tens, units), we need:
$$
3 \text{ digits} \times 4 \text{ bits per digit} = 12 \text{ bits}
$$
✅ Answer: 12 bits
✍️ Explanation (Approx. 300 Words):
When representing numbers digitally, the efficiency of representation depends on the coding method. For the decimal range 0 to 999, two common methods are straight binary and BCD.
Straight binary uses a base-2 positional system. Each bit can be 0 or 1, and the value is calculated as powers of 2. To determine how many bits are needed, we find the smallest power of 2 greater than 999. As calculated above, $2^10 = 1024$ is the first power greater than 999, so 10 bits are sufficient to represent all numbers from 0 to 999. This approach is compact and efficient in terms of bit usage.
BCD (Binary-Coded Decimal), on the other hand, encodes each individual decimal digit separately using 4 bits. For example, the number 345 in BCD is represented as:
- 3 → 0011
- 4 → 0100
- 5 → 0101
So, 345 → 0011 0100 0101 (12 bits total)
For numbers up to 999, which have 3 digits, BCD always uses 3 × 4 = 12 bits, regardless of the actual numeric value. BCD is easier for digital displays and human-readable formats but is less efficient than straight binary for storing pure numeric data because it wastes bit patterns that don’t correspond to decimal digits (e.g., 1010 to 1111 are unused in each 4-bit digit).
In conclusion, straight binary requires only 10 bits, whereas BCD requires 12 bits for the same range, trading storage efficiency for simplicity in decimal digit handling.