VEDIC MATHEMATICS Abstraction T K Prasad http www
VEDIC MATHEMATICS : Abstraction T. K. Prasad http: //www. cs. wright. edu/~tkprasad Prasad Abstraction 1
Numbers : Representation and Operations Prasad Abstraction 2
What are numbers? Prasad Abstraction 3
Numbers as Abstraction Prasad Abstraction 4
More on Abstractions Fruits Planes Fighter Planes Vegetables Prasad Abstraction 5
Mathematics and Sciences • Mathematics and Sciences deal with abstractions of our understanding about the world. • Reasoning about Change : Learning Invariants Prasad Abstraction 6
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 Prasad Abstraction 7
Invariants : Axioms For natural numbers m and n: 0+m = m n+m = m+n n * (m 1 + m 2) = n * m 1 + n * m 2 For every integer i, there exists an additive inverse j, such that i + j = 0 Prasad Abstraction 8
Abstractions Prasad Invariants Abstraction 9
Prasad Abstraction 10
Numbers vs Numerals Prasad Abstraction 11
Symbols for numbers : Egyptian Numerals Hieroglyphs : 3500 BC Prasad Abstraction 12
= 12, 425 birds Prasad Abstraction 13
Symbols for numbers : Roman Numerals Prasad Abstraction 14
Symbols for numbers : Roman Numerals • Alphabet: { I, V, X, L, C, D, M, …} • Numerals: I, III, IV, V, VII, VIII, IX, X 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 XI, XIII, XIV, XVI, 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 Prasad Abstraction 15
(cont’d) • 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 10^x may not precede any symbol larger than 10^{x+1}. 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. Prasad Abstraction 16
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). Prasad Abstraction 17
- Slides: 17