Chapter 2 csc429 actual [Compatibility Mode] (1) (1)

CSC429: Computer Organization & Architecture

Chapter 2: Number System

  • Prepared by: Assoc. Prof. Ts. Dr. Norhaslinda Kamaruddin, UiTM Shah Alam

Overview

  • Introduction to number systems.

  • Key topics:

    • Numeric conversion between number bases

    • Performing arithmetic operations in different number bases

    • Alphanumeric representation

    • Binary Coded Decimal representation

    • Unsigned numbers

    • Representation for signed numbers

    • IEEE floating point representation

Why Not Base 10?

Base 10 Number Representation

  • Base 10 is cumbersome for electronic implementation:

    • Hard to store and transmit.

    • ENIAC (the first electronic computer) needed 10 vacuum tubes per digit.

    • Difficult to implement logical functions due to the complexity of carries.

  • Significant issues with floating point representation:

    • Cannot precisely represent values like $1.20 due to precision limits.

    • Example in scientific notation: 1.5213 X 10^4.

Binary Representation

Base 2 Number Representation

  • Converting numbers to binary:

    • 15213 in decimal to binary: 111011011011012.

    • 1.20 in decimal to binary: 1.0011001100110011...2.

    • Scientific notation: 1.5213 x 10^4 to binary: 1.11011011011012 x 213.

  • Advantages of binary representation:

    • Easy storage with bi-stable elements.

    • Reliable transmission over noisy wires.

Bit, Byte, and Word

Definitions

  • Bit: Smallest data unit, can be 0 or 1.

  • Byte: Group of 8 bits, can store values 0-255 (256 combinations).

  • Word: Size of data processor can handle at a time (common sizes: 8, 16, 32, 64 bits).

  • Illustrated representations of bit, byte, and word.

Number Bases Overview

Place Value

Decimal

Binary

Octal

Hexadecimal

100

10,000

0

00

0

10

1,000

1

1

1

1

100

10

2

2

-

...

...

...

...

Conversion Examples

  • Conversion charts and examples for different bases with emphasis on single and multi-digit conversions.

Numeric Conversion

Base 10 to Base 2

  • Methods for converting decimal to binary, including table and division methods. Examples:

    • 10510 to binary: 1101001.

    • 8710 to binary: 1010111.

Checking Conversions

  • Established rules to verify conversions based on digit patterns and counts.

Signed and Unsigned Numbers

Representation

  • Unsigned Numbers: Only positive values.

    • Examples:

      • 0 to 255 for 8-bit, 0 to 65535 for 16-bit.

  • Signed Numbers: Represent both positive and negative values.

    • Methods:

      1. Sign-and-magnitude

      2. 1's complement

      3. 2's complement

Examples

  • Converted examples and their binary representations of signed and unsigned integers.

1's Complement

  • Inversion technique for negative binary representation but has practical issues.

2's Complement

Conversion Technique

  1. Binary representation of the absolute value.

  2. Invert the bits (1's complement).

  3. Add 1 to the inverted binary number.

  • Common practical methods include arithmetic operations, especially for binary subtraction.

IEEE Floating Point Representation

Overview

  • Issues with integer representation lead to the floating point solution for fractions.

  • Adheres to the IEEE 754 standard:

    • IEEE short real: 32 bits (single precision)

    • 64 bits (double precision)

Conversion Steps

  • Conversion of decimal to IEEE format explained step-wise, including:

    1. Binary conversion

    2. Format conversion into mantissa and exponent with bias.

    3. Examples provided for conversions to proper IEEE format.

Summary

  • Comprehensive coverage of number systems relevant to computer organization and architecture with specific focus on binary representation, numeric conversions, and the techniques employed for signed and unsigned numbers.