1. PrasadAbstraction1 VEDIC MATHEMATICS : Abstraction T. K. Prasad http://www.cs.wright.edu/~tkprasad

2. PrasadAbstraction2 Numbers : Representation and Operations

3. PrasadAbstraction3 What are numbers?

4. PrasadAbstraction4 Numbers as Abstraction

5. PrasadAbstraction5 More on Abstractions Fruits Planes Vegetables Fighter Planes

6. PrasadAbstraction6 Mathematics and Sciences Mathematics and Sciences deal with abstractions of our understanding about the world. Reasoning about Change : Learning Invariants

7. PrasadAbstraction7 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 * (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 PrasadAbstraction8

9. PrasadAbstraction9 AbstractionsInvariants

10. PrasadAbstraction10

11. PrasadAbstraction11 Numbers vs Numerals

12. PrasadAbstraction12 Symbols for numbers : Egyptian Numerals Hieroglyphs : 3500 BC

13. PrasadAbstraction13 = 12,425 birds

14. PrasadAbstraction14 Symbols for numbers : Roman Numerals

15. PrasadAbstraction15 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. PrasadAbstraction16 (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.

17. PrasadAbstraction17 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).