VEDIC MATHEMATICS : Positional Number System

1. VEDIC MATHEMATICS : Positional Number System T. K. Prasad http://www.cs.wright.edu/~tkprasad Positional Number System

2. Symbols for numbers: Hindu Arabic System 1 11 12 13 14 15 16 17 18 19 2 1 · 11 12 13 14 15 16 17 18 19 2 · 1 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 1 1 ·· 1 · 1 1 · 2 1 · 3 1 · 4 1 · 5 1 · 6 1 · 7 1 · 8 1 · 9 1 1 · 1 2 3 4 5 6 7 8 9 1 Read · as blank. Need care when writing (aligning) numbers with blanks. Positional Number System

3. 1,2,3,4,5,6,7,8,9 1 ,11,12,… .. 1 , 1 1, 1 2, … 0,1,2,3,4,5,6,7,8,9 10,11,12,… .. 100, 101, 102, … Invention of Zero Attributed to Indian Mathematicians. In fact, zero thrust itself upon us in the context of positional number system! Positional Number System

4. Names of Zero Positional Number System

5. Vedic Mathematics • Vedas: Texts from Ancient India in Sanskrit • Mathematics: Study of quantity (how much?), change (how fast? ), space (shapes), … • Vedic Mathematics is the mathematical knowledge of ancient Hindus passed down through generations (initially verbally and later codified) as slokas (verses) in Sanskrit. Positional Number System

6. Discovery is finding out what exists but previously unknown Discovery of America by Christopher Columbus (of Spain) in October 1492 Discovery of Helium by Sir William Ramsay Ref: Sun, Balloon Discovery of Galaxies by Edwin P. Hubble Number zero was discovered. Invention is something you create out of imagination Invention of Car Invention of Wheel Invention of Computers Invention of Television Symbol zero was invented. Invention vs Discovery Positional Number System

7. Why base 10? • 10 fingers, 10 toes, …? • Alternatives • Base 60 (Babylonian Cunieform) • Upto 59 by crescent (weight 10) and wedge (weight 1) • "<< VVV" = 2*10 + 3*1 = 23 • "< VV<< VVV" = 12*60 + 23*1 = 743 • => 1 hour = 60 minutes, 1 minute = 60 seconds, • => 1 year ~ 360 days, 1 day = 24 hours • 24 = 1 x 2 x 3 x 4; 60 = 1 x 2 x 3 x 2 x 5 • Base 2 (Modern) • Two eyes, two ears, two hands, two legs, …? • Digital Computers based on “switches” {0, 1} Positional Number System

8. Dashavtar • Matsya (fish), • Kurma (tortoise), • Varaha (boar), • Narasimha (man-lion), • Vamana (dwarf man), • Parashuram • Ram (moral man), • Krishna (philosophical / practical man) • Balaram • Kalki Positional Number System

9. Symbols for numbers : Hindi (Devanagari) Numerals Positional Number System

10. Representing Numbers : Positional System • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • For larger numbers: • Problem: cannot keep inventing new symbols • Solution: use finite number of symbols with Position-based Weighting • 52 = 5 * 10 + 2 • 743 = 7 * 100 + 4 * 10 + 3 Positional Number System

11. Positions and Weights 43210 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 + 0 Positional Number System

12. Exercises • 1a) Local Values • 1b-e) Verbalizing Numbers Positional Number System

13. Large Numbers • Hard Disk Storage in 30 GB IPod • 30 x 1024 x 1024 x 1024 bytes (30 Giga bytes) = 32,212,254,720 bytes • Number of stars in our Milky Way Galaxy • About 100,000,000,000 (100 billion) • Googol • One followed by 100 zeros • Googolplex • One followed by Googol zeros Positional Number System