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Integer Number Representation

Introduction to 64-bit Binary Numbers:

  • Decimal vs. Binary: Computers understand binary, while humans understand decimal. A conversion mechanism is required to translate between these two systems.
  • Example: A 64-bit binary number is used to represent data in computer systems, such as instructions processed by a CPU.
  • Importance: Understanding binary number representation is crucial for understanding how data is stored, transferred, and manipulated in a system.

Basic Structure of 64-bit Representation:

  • Word Definition: A word in computer systems refers to a fixed-size unit of data, commonly 32 or 64 bits.
  • Least Significant Bit (LSB): The lowest bit in a binary number.
  • Most Significant Bit (MSB): The highest bit in a binary number.
  • Byte: A byte consists of 8 bits. Four bytes make up a word, and two words make up a double word.

Unsigned Binary Representation:

  • Formula: Using n bits, the number of possible patterns is 2^n. For example, using 3 bits, the maximum number is 2^3 = 8 patterns, ranging from 0 to 7.
  • 64-bit Representation: For a 64-bit system, there are 2^64 possible patterns, which results in a huge range of numbers.

Understanding Large Numbers:

  • Representation: Large binary numbers are broken down into their individual bit values and summed to represent the decimal equivalent.
  • Overflow Flag: When performing arithmetic on unsigned numbers in a 64-bit system, an overflow flag will be triggered if the result exceeds the maximum representable number.

Maximum Unsigned Number in 64-bit:

  • The largest number that can be represented in a 64-bit unsigned system is 2^64 - 1.

Signed vs. Unsigned Numbers:

  • Unsigned Numbers: These are always positive and range from 0 to a maximum value (e.g., 2^64 - 1).
  • Signed Numbers: Signed numbers can represent both positive and negative values, and a special mechanism (like two's complement) is used to distinguish between the two. The discussion of signed numbers is introduced but will be covered in more detail later.

Negative Numbers and Two's Complement:

  1. Two's Complement Representation:

    • A standard method to represent negative numbers.
    • To represent a negative number, complement all the bits (invert 0s to 1s and 1s to 0s) and then add 1.

  2. MSB (Most Significant Bit):

    • In the two's complement system, the MSB (most significant bit) indicates the sign of the number.
    • MSB = 0: Positive number.
    • MSB = 1: Negative number.

  3. Converting Negative Numbers:

    • Example: For a 64-bit number, find its complement, invert all bits, and then add 1.
    • Negative numbers are computed by inverting bits (1’s complement) and then adding 1.
    • The final step is to check the MSB and apply the relevant steps to determine if the number is negative.

  4. Binary Representation:

    • Positive binary numbers are represented as they are.
    • Negative binary numbers involve taking the two's complement.

  5. Range of Positive and Negative Numbers:

    • For a 64-bit system, the range of positive and negative numbers is determined by the number of bits.
    • The largest positive number that can be represented is 2^63 - 1.
    • The largest negative number is -2^63.

  6. Understanding Binary to Decimal Conversion:

    • For positive numbers, convert the binary representation directly.
    • For negative numbers (identified by MSB = 1), apply the two’s complement process and multiply by 2^(n-1) (where n is the number of bits) to get the final negative value.

  7. Special Observations:

    • The MSB is crucial in identifying if the number is positive or negative.
    • Positive numbers are directly represented, while negative numbers follow the two’s complement approach.

  8. Range of Numbers in RV64 Architecture:

    • Positive numbers: Start from 0 up to 2^63 - 1.
    • Negative numbers: Start from -2^63 to -1.

Application in Integer Instructions (RISC-V Architecture):

  1. Handling Integers in RISC-V:
    • RISC-V follows these principles for integer instructions.
    • The system distinguishes between positive and negative numbers using the MSB.
    • Integer arithmetic instructions work with signed and unsigned numbers by leveraging the two's complement system.