Memory and Numerical Systems: Comprehensive Study Guide

Introduction to Memory and Numerical Systems

• The main objective of this lesson is to understand the inherent limits of a computer system.
• It aims to identify specific scenarios and conditions where these computational or storage limits are likely to be encountered during processing.

Information Representation and the Binary Digit (Bit)

• Every computer processor is constructed from millions of miniature switches known as transistors.
• These switches function in two states: they can be turned either "on" or "off".
• Consequently, it is logical for a computer to represent data and perform computations using a number system with only two digits.
• Binary Number System: A system that utilizes only two digits, 0 and 1.
• Bit: This term stands for "binary digit". Bits correspond directly to the on/off positions of the switches within the computer processor.
  • A bit value of $0$ generally represents the "off" state.
  • A bit value of $1$ generally represents the "on" state.

Physical and Logical Organization of Memory

• Storage devices utilize physical materials with two distinct states to represent the smallest unit of information:

  • RAM (Random Access Memory): Memory cells are either electrically charged or cleared.
  • Magnetic Storage: Utilizes areas that are either magnetized or non-magnetized.
  • Optical Storage: Uses two different levels of light reflectance (pits and lands) to represent data.

• Memory Hierarchy and Categorization:

  • Bit: The binary digit (0 or 1).
  • Byte: A group of bits used to encode a single character. 1 Byte is equal to $8$ bits. The byte is the smallest addressable location in memory, and computer operations typically occur at the byte level.
  • Word: A collection of bytes. A "word" is defined as the number of bits a computer's Central Processing Unit (CPU) can handle or process in a single operation. This is highly machine-dependent and varies based on architecture:
  • Early computers: Used $8$-bit words.
  • Modern computers: Typically use $32$-bit or $64$-bit words.
  • Doubleword: Consists of $4$ bytes ($32$ bits).
  • Quadword: Consists of $8$ bytes ($64$ bits).

• Instruction Structure: Computer instructions are composed of operation codes (op-codes) and memory addresses.

• Computer Classification: Computers are often classified by their word size (e.g., $16$-bit, $32$-bit, and $64$-bit machines). Word size acts as a primary indicator of computer performance and the number of bits used to represent addresses in Main Memory (RAM).

Computer Memory Classification: Primary vs. Secondary

• Primary Memory: This is the memory accessed directly by the CPU.

  • Cache Memory (CPU Registers): Supplementary memory that temporarily stores frequently used instructions and data for quicker processing.
  • Main Memory: Divided into volatile and non-volatile types.
  • RAM (Random Access Memory): A volatile type of memory (data is lost when power is off) used to store information the computer is actively using for quick access by applications. It includes:
    • SRAM (Static RAM): Built using flip-flops.
    • DRAM (Dynamic RAM): Built using capacitors.
  • ROM (Read Only Memory): A non-volatile type of memory that permanently stores data. It contains the essential programming needed to start a PC (boot-up) and performs major input/output tasks. Variants include:
    • PROM (Programmable Read-Only Memory).
    • EPROM (Erasable Programmable Read-Only Memory).
    • EEPROM (Electrically Erasable Programmable Read-Only Memory).

• Secondary (Auxiliary) Memory: External memory used for long-term storage of data and programs. It can be fixed or removable.

  • Sequential Access: Examples include Magnetic Tape.
  • Magnetic Disk: Includes Hard Disk Drives (HDD) and Floppy Disk Drives.
  • Optical Disk:
  • CD: Capacity of approximately $700\,\text{MB}$.
  • DVD: Capacity of approximately $8.4\,\text{GB}$.
  • Blu-Ray Disk: Capacity of approximately $50\,\text{GB}$.
  • Flash Memory: Includes Memory Cards and Pen Drives.

Memory Measurement Units and Capacities

• Memory is measured in increasing orders of magnitude, traditionally based on powers of $2$ ($2^{10} = 1024$):
  • $1\,\text{Bit}$ = Binary Digit.
  • $8\,\text{Bits}$ = $1\,\text{Byte (B)}$.
  • $1024\,\text{Bytes}$ = $1\,\text{Kilo Byte (KB)}$ ($2^{10}\,\text{Bytes}$).
  • $1024\,\text{KB}$ = $1\,\text{Mega Byte (MB)}$ ($2^{20}\,\text{Bytes}$).
  • $1024\,\text{MB}$ = $1\,\text{Giga Byte (GB)}$.
  • $1024\,\text{GB}$ = $1\,\text{Tera Byte (TB)}$.
  • $1024\,\text{TB}$ = $1\,\text{Peta Byte (PB)}$.
  • $1024\,\text{PB}$ = $1\,\text{Exa Byte (EB)}$.
  • $1024\,\text{EB}$ = $1\,\text{Zetta Byte (ZB)}$.
  • $1024\,\text{ZB}$ = $1\,\text{Yotta Byte (YB)}$.

Numerical Systems: Decimal vs. Binary

• Decimal Numerical System:

  • The standard system for mathematical notation.
  • Base-10 system: Uses $10$ digits ($0, 1, 2, 3, 4, 5, 6, 7, 8, 9$).
  • Positional Notation: The position of the digit determines its value (units, tens, hundreds, etc.).
  • Example: The number $378$ represents $3$ hundreds, $7$ tens, and $8$ units.
  • Exponential Expression: $378 = 3 \times 10^2 + 7 \times 10^1 + 8 \times 10^0$.

• Binary Numerical System:

  • Used to understand how computers perform computations and code every instruction.
  • Base-2 system: Uses only two digits ($0$ and $1$).
  • Numbers are represented through powers of $2$.
  • Example Conversion (Binary to Decimal):
  • $1101_2 = 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$
  • $1101_2 = 8 + 4 + 0 + 1 = 13_{10}$
  • Notation: A subscript $2$ (e.g., $1101_2$) or the suffix "B" (e.g., $1101B$) indicates the number is in base-2.