computer systems 2

ECS404: Computer Systems and Networks - Digital Representation

Week 2: Overview

• Topics Covered: Integer representation, binary encoding, hexadecimal, and two's complement.

• Learning Objectives:

  • Understand basic binary concepts: bits and bytes.

  • Learn the conversion between decimal, binary, and hexadecimal.

  • Understand two’s complement representation for signed integers.

    • Convert between binary two's complement and signed decimal.

    • Add and multiply in two's complement system.

    • Negate and test positivity for two's complement numbers.

Binary Representation

• Binary uses bits (binary digits: 0s and 1s).

• Byte: A group of 8 bits.

• Examples:

  • 0000 0101 represents the number 5.

  • 1111 1011 can represent the number -5 in two's complement.

  • 0111 1000 represents both the letter x and the number 120.

  • Interpretation is context-dependent (e.g., same bit patterns can mean different values).

Conversion Methods

• Binary can be converted to decimal and vice versa.

• Hexadecimal: Used to represent binary in a compact form.

• Key conversions include:

  • From binary to decimal using powers of 2.

  • From decimal to binary by breaking down the decimal number.

Two's Complement System

• Two’s complement simplifies arithmetic with signed numbers.

• Why Use Two's Complement?:

  • It eliminates the ambiguity of having two zeros (+0 and -0) as seen in sign and magnitude representation.

  • Using two's complement allows leveraging the same addition circuitry for both signed and unsigned integers.

• To Negate a Two's Complement Number:

  • Invert bits and add 1.

Understanding Number Types

• Whole Numbers: Positive and Negative (e.g., -3, -2, 0, 1, 2, 3).

• Fractions: e.g., 1/4, -5/6.

• Real Numbers: E.g., π ≈ 3.14159.

• Complex Numbers: E.g., i.

Java Data Representations

• Java's 8 Primitive Data Types:

  • byte: 8-bit signed two’s complement.

  • short: 16-bit signed.

  • int: 32-bit signed.

  • long: 64-bit signed.

  • float: 32-bit IEEE floating point.

  • double: 64-bit IEEE floating point.

  • boolean and void types are also included.

Endianness

• Endianness Concepts:

  • Big Endian: High-order bytes come first.

  • Little Endian: Low-order bytes come first.

  • Most computers use Little Endian for internal representation, while network protocols often use Big Endian.

Summary of Key Concepts

• Binary (base 2) is fundamental for computer systems.

• Understanding bits, bytes, and how to convert between different representations is crucial for programming and computer science.

• Arithmetic with integers can use similar algorithms across different systems by leveraging representations such as two's complement.