Lecture Notes: Evolution of Number Systems

Introduction

• Numbers are fundamental symbols used for counting and recording values.
• The common symbols (0-9) allow representation of any rational number.

Historical Background

• Early counting methods included:
  • Using body parts or tally marks.
  • Limited as complexity of life increased.
• Different civilizations developed unique systems for higher numbers:
  • Greek, Hebrew, Egyptian Numerals: Extensions of tally marks.
  • Roman Numerals: Subtraction principle (e.g., IV for 4).

Positional Notation

• Emerged as a more efficient system:
  • Reuses symbols (0-9) based on position.
  • Developed independently by Babylonians, Ancient Chinese, Aztecs.
• Indian Mathematicians (8th century) perfected positional notation.
  • Spread to Europe through Arab traders and scholars.

The Decimal System

• Based on ten unique symbols (0-9).
• Each position indicates different powers of ten.
  • Example: 316 = 6x10^0 + 1x10^1 + 3x10^2.
• Inclusion of Zero:
  • Acts as a placeholder, avoiding confusion (e.g., 63 vs. 603).

Evolution of Digit Glyphs

• Symbols varied regionally before standardization.
• Current digits evolved from North African Maghreb glyphs.
• By the 15th century, the Hindu-Arabic numeral system became common.

Base Systems

• Most numbers use base ten due to practicality.
• Other base systems:
  • Base 20 (Vigesimal): Used by Aztecs.
  • Base 60 (Sexagesimal): Used by Babylonians, still seen in degrees and time.
  • Base 12 (Duodecimal): Benefits for fractions, seen in measurements (dozen, gross).
  • Base 2 (Binary): Used in digital devices; base 8 (octal) and base 16 (hexadecimal) for programming.

Conclusion

• The simplicity of using just ten symbols captures a vast quantity of information.
• Encouragement to consider different representations for large numbers.