Lecture on Binary Mathematics in Ancient Indian Texts

Introduction

• The lecture focuses on a work from the second century BCE which lays the foundations for binary mathematics.
• Pingala's "Chanda Shastra" is a key text discussing binary numbers and combinatorial problems in the context of Sanskrit poetry (Chandas).

Binary Representation of Metrical Patterns

• Chandas: Analyzed using two syllables, Laghu (1) and Guru (0).
• Metrical patterns can be transformed into binary sequences.
• Binary Sequence: Composed of 1s and 0s, e.g., 1001 or 0100.

Pingala's Combinatorial Problems

Operations in Chanda Shastra

1. Prastara

   • Generates all possible metrical patterns or binary sequences of a specified length.
   • Example: For three digits, sequences range from 000 to 111.

2. Sankhya

   • Process to find the total number of binary sequences in a Prastara.

3. Nashta

   • Identifies the binary sequence for a given row number in an array.
   • Example: For row 12 in a four-digit sequence, determines the specific sequence.

4. Udhistha

   • Reverse of Nashta; finds the row number for a given binary sequence.
   • Example: Find the row for sequence 1000.

5. Lagakriya

   • Finds binary sequences in the array with a specific number of 1s and 0s.

6. Adva Yoga

   • Determines the space or area needed for the array or Prastara.

Generation of Binary Arrays

• Algorithm:
  1. Start with sequences 0 and 1.
  2. At each iteration, replicate the current array and add a column.
  3. Populate columns with 0s and 1s in a structured manner.
  4. Continue replicating and adding rows progressively.

Nashta Algorithm Example

• Illustrates finding a binary sequence for a row number using division and sequence placement.

Uddiyasta Algorithm

• Reverse process: Start with a binary sequence to find the row number.
• Uses a process of doubling and subtracting to trace back to the row.

Combinatorial Problem: Lagakriya

• Determines the count of binary sequences with a specific number of 1s.
• Based on binomial coefficients (nCr).
• Pingala's Varna Meru: An algorithm equivalent to Pascal's Triangle, used for generating binomial coefficients.

Historical Context

• Pascal rediscovered Pingala's concepts in 1655 CE, long after Pingala's time (2nd century BCE).
• Varaha Mihira's "Brihasa Mita" (550 CE) further demonstrates combinatorial calculations.

Conclusion

• The lecture highlights ancient Indian contributions to binary mathematics and combinatorial methods, which were foundational to later developments in mathematics.
• Upcoming topic: Magic squares in mathematics.

Note: This lecture provides insights into the historical development of binary mathematics and its applications in ancient Indian texts.