VEDIC MATHEMATICS : Abstraction

1
VEDIC MATHEMATICS Abstraction

• T. K. Prasad
• http//www.cs.wright.edu/tkprasad

2
Numbers Representation and Operations
3
What are numbers?
4
Numbers as Abstraction
5
More on Abstractions
Fruits
Planes
Fighter Planes
Vegetables
6
Mathematics and Sciences

• Mathematics and Sciences deal with abstractions
  of our understanding about the world.
• Reasoning about Change Learning Invariants

7
Abstraction Another Example

• Five little monkeys jumping on the bed.
• One fell down and broke its crown (head).
• How many monkeys will now jump on the bed?
• 5 1 4

8
Invariants Axioms

• For natural numbers m and n
• 0 m m
• n m m n
• n (m1 m2) n m1 n m2
• For every integer i,
• there exists an additive inverse j,
• such that i j 0

9
Abstractions
Invariants
10
(No Transcript)
11
Numbers vs Numerals
12
Symbols for numbers Egyptian Numerals
Hieroglyphs 3500 BC
13
                                                
                                                
                                        12,425
birds
14
Symbols for numbers Roman Numerals
15
Symbols for numbers Roman Numerals

• Alphabet I, V, X, L, C, D, M,
• Numerals
• I, II, III, IV, V, VI, VII, VIII, IX, X
• 1, 2, 3, 4, 5, 6, 7, 8,
  9, 10
• XI, XII, XIII, XIV, XV, XVI, XVII, XVIII,
  XIX, XX
• 11, 12, 13, 14, 15, 16, 17,
  18, 19, 20
• L, C, D, M, LXIII, XL,
  MCMLXIX
• 50, 100, 500, 1000, 63, 40,
  1969

16
(contd)

• XIX
• Is this 11 10 21? Or, is this 10 9 19?
• 19
• If smaller digit follows larger digit, the digits
  are added. If the smaller digit precedes larger,
  the smaller is subtracted from the larger.
• A symbol representing 10x may not precede any
  symbol larger than 10x1. For example,
• C cannot be preceded by I or V, but only by X
  (or, by a symbol equal to or larger than C).
• The number ninety-nine is XCIX, not IC or ILL.

17
Summary Two Lessons

• Fixed Alphabet for numerals Use a fixed set of
  symbols to represent all numbers.
• Unambiguous representation for number Each
  numeral must have a unique meaning (that is, a
  unique number associated with it).