Determining the Number of Bits Required for a Binary Code

Question:

Determine the number of bits required for a binary code to represent a) 210 different outputs and b) letters of the alphabet and digits 0 to 9. Compare its efficiency with a decimal system to accomplish the same goal.

Answer:

a) 8 bits are required for a binary code to represent 210 different outputs. b) 6 bits are required for a binary code to represent the letters of the alphabet and digits 0 to 9.

Explaining the Answer in Detail:

a) The number of bits required for a binary code to represent 210 different outputs:

The number of bits required for a binary code to represent 210 different outputs can be determined by calculating the smallest power of two that is greater than or equal to 210.

2^7 = 128, 2^8 = 256.

Since 2^7 is not enough to represent 210 different outputs, 8 bits are needed. Therefore, 8 bits are required for a binary code to represent 210 different outputs.

b) The number of bits required for a binary code to represent the letters of the alphabet and digits 0 to 9:

To represent the letters of the alphabet and digits 0 to 9, we need to determine the total number of symbols to be represented. There are 26 letters in the alphabet and 10 digits, totaling 26 + 10 = 36 symbols.

The formula n = log2(N) is used to determine the number of bits required, where n is the number of bits and N is the number of symbols to be represented.

n = log2(36) = 5.17.

The number of bits required is always rounded up to the nearest whole number. Hence, 6 bits are required for a binary code to represent the letters of the alphabet and digits 0 to 9.

Comparing Efficiency with Decimal System:

The binary system is more efficient in representing data than the decimal system. This is because the binary system is based on powers of two, while the decimal system is based on powers of ten. Therefore, a binary system can represent data using fewer bits than a decimal system.

For example, to represent the number 210 in decimal requires 3 digits, while in binary it only requires 8 bits (equivalent to 3 decimal digits). Therefore, the binary system is more efficient than the decimal system in representing data.