Computer Number Systems and Data Representation Principles

Number Systems and Conversions

• Hexadecimal Conversion: The amount (A9F.4)₁₆ is equivalent to decimal 2719.250.
• Memory Addressing: The hexadecimal system is used to number memory locations.
• Binary Conversion: The number (10101.111)₂ is equivalent to decimal 21.875.
• BCD Conversions:
  • The BCD number 0011 0101 1001 0110 is equivalent to decimal 3596.354 (represented as 0011 0101 0100).
  • The decimal number 9832.65 equals the BCD: 1001 1000 0011 0010, 0110 0101.
  • The decimal number 9872.64 equals the BCD: 1001 1000 0111 0010, 0110 0101.
• Scientific Notation: The amount 0.234 x 10^-9 is equivalent to 0.00000000234.
• Additional Conversion: The number 0100 0011 0101 0110.1111 is equivalent to 4356.98.

Coding Systems and Error Detection

• Gray Codes: These are not useful for correcting arithmetical errors. However, Gray codes are useful for unambiguously determining positions and are used in the construction of position detectors to avoid uncertainties.
• Biquinary Code: This code is useful for transmitting data without error and with low power consumption.
• Number Complements: The complement of a number is the module minus the number; it represents what a number lacks to arrive at the module.
• Instruction Controls: The binary system is used to indicate the controls of an instruction.

Arithmetic and Rounding Methods

• Summation: The result of the sum 0.367 x 10⁴ + 0.528 x 10⁵ equals 0.53167 x 10⁵.
• Guard Bits: These are used in the Arithmetic Logic Unit (ALU).
• Rounding to the Next: The result is rounded to the nearest representable number.
• Rounding to +Infinity: The result is rounded to the next higher representable number by truncating the value and moving to the next higher representable increment.
• Rounding to -Infinity: The result is rounded to the lesser amount immediately representable (truncating to the lower representable number).
• Rounding Toward Zero: The rounding is performed toward the nearest representable number in the direction of zero.

Computer Architecture and Standards

• ASCII: American Standard Code for Information Interchange.
• CMG: Computer Measurement Group.
• Amdahl's Law: Improved performance achieved using a faster mode of execution is limited to the fraction of time during which that faster mode can be maintained.
• RM9: The new module is called the remaining residue left after dividing a number by nine.