2. By PRADUMN KUMAR 1

3. Symbols for numbers: Hindu Arabic System 2 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

4. 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, … 3 INVENTION OF ZERO Attributed to Indian Mathematicians. In fact, zero thrust itself upon us in the context of positional number system!

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

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

7.  6 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.

8. 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} 7

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

10. Symbols for numbers : Hindi (Devanagari) Numerals 9

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

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

13. Exercises  1a) Local Values  1b-e) Verbalizing Numbers 12

14.  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 13 Large Numbers