Prasad Positional Number System 1

VEDIC MATHEMATICS : Positional Number

SystemT. K. Prasad

http://www.cs.wright.edu/~tkprasad

Prasad Positional Number System 2

Symbols for numbers: Hindu Arabic System

Read · as blank.

1 2 3 4 5 6 7 8 91

1 · 1112131415161718192 ·

1 · · 1 · 11 · 21 · 31 · 41 · 51 · 61 · 71 · 81 · 91 1 ·

Need carewhen writing

(aligning) numbers

with blanks.

1 1112131415161718192

1 1 11 21 31 41 51 61 71 81 91 1

Prasad Positional Number System 3

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!

Prasad Positional Number System 4

Names of Zero

Hindu Sunya (“void”/“empty”)

Arabic Sifir (“vacant”)

Roman Cifra

Latin Zephirum or Zephyrum

English Zero, Cipher

Prasad 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.

Prasad Positional Number System 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.

Prasad 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}

Prasad Positional Number System 8

Dashavtar

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

practical man) 9. Balaram10. Kalki

Prasad Positional Number System 9

Symbols for numbers : Hindi (Devanagari) Numerals

Prasad 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

Prasad Positional Number System 11

Positions and WeightsTEN-

THOUSANDSTHOUSANDS HUNDREDS TENS UNITS

43210=

4 * 10,000+ 3 * 1,000+ 2 * 100+ 1 * 10

+ 0

Prasad Positional Number System 12

Exercises

• 1a) Local Values

• 1b-e) Verbalizing Numbers

Prasad 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