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A detailed explanation of various number systems, including positional and non-positional systems, and their operations. ​

INTRODUCTION

  • Number systems represent how numbers are symbolized. ​
  • Different systems can represent the same quantity differently. ​
  • Importance of numbers in computers: precise information representation and fast processing. ​
  • Number systems are categorized into positional and non-positional systems. ​

POSITIONAL NUMBER SYSTEMS ​

  • In positional systems, the position of a symbol determines its value. ​
  • The value is calculated based on the base (radix) of the system. ​

DECIMAL SYSTEM ​

  • Base 10 system using symbols 0-9. ​
  • Each symbol’s value depends on its position. ​
  • Example: 7594 in decimal is calculated as 7×10^3 + 5×10^2 + 9×10^1 + 4×10^0.

BINARY SYSTEM ​

  • Base 2 system using symbols 0 and 1. ​
  • 1 bit represents 2 values; 4 bits represent 16 values. ​
  • 8 bits form a byte; 4 bits form a nibble. ​
  • General formula: For M values, log2M bits are needed. ​

BINARY ADDITION ​

  • Binary addition involves adding pairs of digits with carry propagation. ​
  • A binary addition table illustrates the process. ​

BINARY SUBTRACTION ​

  • Binary subtraction uses borrowing when needed. ​
  • A binary subtraction table is provided for reference. ​

Subtraction using complements ​

  • Subtraction can be performed using one’s or two’s complements. ​
  • One’s complement involves flipping bits; two’s complement adds 1 to the one’s complement. ​

BINARY MULTIPLICATION ​

  • Binary multiplication involves multiplying each digit of the multiplier with the multiplicand. ​
  • Partial products are summed to get the final result.

BINARY DIVISION ​

  • Binary division is similar to decimal division, following the same principles. ​

OCTAL SYSTEM

  • Base 8 system using symbols 0-7. ​

HEXADECIMAL SYSTEM ​

  • Base 16 system using symbols 0-9 and A-F (where A=10, B=11, etc. ​).

NONPOSITIONAL NUMBER SYSTEMS ​

  • Non-positional systems use fixed symbol values regardless of position. ​
  • Roman numerals are an example, where values are summed based on symbols present. ​

Number Systems – Lesson 2

Engineering Institute of Technology