Why is the binary number system fundamental to digital devices, and how does it support data representation?
Convert the decimal number -45 to binary using two’s complement representation for an 8-bit system.
Explain the process of converting the hexadecimal number 3A7 to its binary equivalent.
What is the significance of using octal numbers in computing, and how would you convert the binary number 10110110 to octal?
Compare signed magnitude and two’s complement representations for the decimal number -19 in a 6-bit system.
Answers and Descriptions
Answer: The binary number system is fundamental because digital devices operate using two states (0 and 1), represented by voltage levels. Binary supports data representation by encoding instructions and data as 0s and 1s.
Description: Binary’s simplicity aligns with electronic circuits, enabling data storage and processing. A byte (8 bits) can represent 256 unique values, sufficient for characters or numbers.Answer: -45: Positive 45 = 00101101; One’s complement = 11010010; Two’s complement = 11010011.
Description: Two’s complement simplifies arithmetic by inverting bits and adding 1. The result represents -45 in 8 bits.Answer: 3A7 = 0011 (3), 1010 (A), 0111 (7) → 001110100111 (binary).
Description: Each hex digit maps to 4 binary bits, efficient for memory addresses.Answer: Octal is compact. 10110110 → 10 110 110 = 266 (octal).
Description: Octal was used in UNIX. Group binary in threes and convert.Answer: Signed magnitude = 110011; Two’s complement = 101101 (for -19).
Description: Two’s complement avoids multiple zeros, preferred in modern systems.