1. PrasadPositional Number System1 VEDIC MATHEMATICS : Positional Number System T. K. Prasad http://www.cs.wright.edu/~tkprasad

2. PrasadPositional Number System2 Symbols for numbers: Hindu Arabic System Read · as blank. 1 2 3 4 5 6 7 8 9 1 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 · Need care when writing (aligning) numbers with blanks. 1 11 12 13 14 15 16 17 18 19 2 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1

3. PrasadPositional Number System3 Invention of Zero 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, … Attributed to Indian Mathematicians. In fact, zero thrust itself upon us in the context of positional number system!

4. PrasadPositional Number System4 Names of Zero Hindu Sunya ( “ void ” / “ empty ” ) Arabic Sifir ( “ vacant ” ) RomanCifra LatinZephirum or Zephyrum EnglishZero, Cipher

5. PrasadPositional Number System5 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.

6. PrasadPositional Number System6 Invention vs Discovery 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.

7. PrasadPositional Number System7 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}

8. PrasadPositional Number System8 Dashavtar 1.Matsya (fish),Matsya 2.Kurma (tortoise),Kurma 3.Varaha (boar),Varaha 4.Narasimha (man-lion),Narasimha 5.Vamana (dwarf man),Vamana 6.ParashuramParashuram 7.Ram (moral man),Ram 8.Krishna (philosophical / practical man)Krishna 9.BalaramBalaram 10.Kalki

9. PrasadPositional Number System9 Symbols for numbers : Hindi (Devanagari) Numerals

10. PrasadPositional Number System10 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

11. PrasadPositional Number System11 Positions and Weights TEN- THOUSANDS THOUSANDSHUNDREDSTENSUNITS 43210 = 4 * 10,000 + 3 * 1,000 + 2 * 100 + 1 * 10 + 0

12. PrasadPositional Number System12 Exercises 1a) Local Values 1b-e) Verbalizing Numbers

13. PrasadPositional Number System13 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