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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