Lecture on Numbering Systems

Introduction to Number Bases

• Decimal system (base 10) is most common but not necessarily optimal.
• Other bases might be better suited for certain tasks.

Dozenal System (Base 12)

• Counts by twelves, not tens.
• Requires two extra digits for 10 and 11. Common notations include:
  • ">↊<" or ">T<" for ten.
  • ">↋<" or ">E<" for eleven.
• Some contexts already use base 12 (e.g., counting eggs, inches).
• Vocabulary example: one, two, three… ten, eleven, dozen (for 12).

Advantages of Dozenal

1. Compact Representation:
   • Larger numbers require fewer digits in dozenal compared to decimal.
   • Example: Up to a gross (144) in dozenal is represented shorter.
2. Factor Richness:
   • Twelve has more factors (1, 2, 3, 4, 6, 12).
   • Easier division due to multiple factors.
   • Example: Last digit of numbers tells divisibility by many factors.
3. Simpler Fractions:
   • Easier to write fractions like 1/2, 1/3, 1/4.
   • Example: 1/2 in dozenal is .6, 1/3 is .4, 1/4 is .3.

Challenges of Dozenal

• More complex representation of fifths: 1/5 in dozenal is .2497... (recurring).

Seximal System (Base 6)

• Introduced by Jan Misali as a more favorable number system.
• Counts by sixes, retaining more simplicity.

Advantages of Seximal

1. Small Base:
   • Simplifies numbers with fewer digits.
   • Example: Point where seximal outpaces decimal is at 10,000.
2. Ease of Arithmetic:
   • Simple addition, subtraction, and multiplication with smaller digits.
   • Only need to memorize a few one-digit multiplication results.
3. Factor Efficiency:
   • Six is highly composite (SHCN), making division easy.
   • Example: Factors of six: 1, 2, 3, 6.
4. Finger Counting:
   • Efficient for visual counting using two hands (one hand up to five, then swap).
   • Enables counting higher without confusion.

Division and Rational Numbers in Seximal

• Division tests and fractional representation are simpler.
• Example: 1/2 is .3, 1/3 is .2, 1/4 is .13.
• Handles fifths better than dozenal.
• Complex fractions like sevenths are simpler.

Conclusion

• Seximal is practical for both small and large numbers.
• Encouragement to explore more via resources like seximal.net.