prasadabstraction1 vedic mathematics : abstraction t. k. prasad tkprasad
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Prasad Abstraction 1
VEDIC MATHEMATICS : Abstraction
T. K. Prasadhttp://www.cs.wright.edu/~tkprasad
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Prasad Abstraction 2
Numbers : Representation and Operations
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Prasad Abstraction 3
What are numbers?
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Prasad Abstraction 4
Numbers as Abstraction
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Prasad Abstraction 5
More on AbstractionsFruits
Planes
VegetablesFighter Planes
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Prasad Abstraction 6
Mathematics and Sciences
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Prasad Abstraction 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
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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 = 0Prasad Abstraction 8
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Prasad Abstraction 9
Abstractions Invariants
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Prasad Abstraction 10
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Prasad Abstraction 11
Numbers vs Numerals
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Prasad Abstraction 12
Symbols for numbers : Egyptian NumeralsHieroglyphs : 3500 BC
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Prasad Abstraction 13
= 12,425 birds
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Prasad Abstraction 14
Symbols for numbers : Roman Numerals
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Prasad Abstraction 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
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Prasad Abstraction 16
(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.
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Prasad Abstraction 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).