Number Systems and Binary Representation Notes

Introduction to Number Systems

  • Number systems are crucial for computer representation of data.
  • Binary Number System: Represents numbers using only two digits, 0 and 1.

Finite Precision Numbers

  • Limited digit representation (e.g., 3 decimal digits: 000 to 999) leads to:
    • Upper limit: Numbers larger than 999 cannot be represented.
    • Non-positive integers: Negative numbers cannot be expressed.
    • Rational numbers: Fractions (e.g., 1/2) are not representable.
    • Irrational numbers: (e.g., √2) cannot be exactly expressed.
    • Complex numbers: Cannot be represented using basic number systems.

Radix Number Systems

  • Radix: Refers to the base of a number system. Examples include:
    • Radix 2 – Binary
    • Radix 10 – Decimal

Conversion between Radix

  • Decimal to Other Systems: Understand how to convert a decimal number into binary, octal, and hexadecimal systems.

    • Example: Convert 2001 into binary, octal, hexadecimal.

  • Grouping for Conversion:

    • Hexadecimal: Group binary bits in sets of 4.
    • Octal: Group binary bits in sets of 3.

Methods of Conversion

  • Decimal to Binary via Successive Halving:

    • Example: Converting 1492 by repeatedly dividing by 2 to obtain remainders.

  • Binary to Decimal via Successive Doubling:

    • Example: Converting binary number 101110110111 to decimal by doubling the previous result and adding corresponding bits.

Representing Negative Numbers

  • Four Methods for negative number representation:
    1. Signed Magnitude:
    • Convert to binary; first bit indicates sign (0 for positive, 1 for negative).
    • Problem: Double representation of 0 (e.g., 0000 for +0 and 1000 for -0).
    1. One’s Complement:
    • Invert bits: 0s to 1s and vice versa.
    • Problem: Double representation of 0.
    1. Two’s Complement:
    • Invert bits and add 1. E.g., -3 is represented as 1101.
    • Advantage: No double representation of 0.
    1. Excess (or Bias) Representation:
    • Store number as value + $2^{(m-1)}$ for m bit numbers. E.g., for m=4, -3 is stored as 5 (0101).

Binary Arithmetic

  • Arithmetic operations (addition, subtraction, multiplication, division) can also be performed with binary numbers.
  • Binary Addition Table: Used in calculations.
  • Addition in One’s and Two’s complement needs to be practiced.

Further Topics

  • Characters representation in binary using ASCII and other formats will be discussed later in the course.
  • Practice in tutorials for binary operations will be available in subsequent weeks.