Prasad Abstraction 1

VEDIC MATHEMATICS : Abstraction

T. K. Prasadhttp://www.cs.wright.edu/~tkprasad

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Numbers : Representation and Operations

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What are numbers?

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Numbers as Abstraction

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More on AbstractionsFruits

Planes

VegetablesFighter Planes

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Mathematics and Sciences

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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

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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Abstractions Invariants

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Numbers vs Numerals

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Symbols for numbers : Egyptian NumeralsHieroglyphs : 3500 BC

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                                                             = 12,425 birds

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Symbols for numbers : Roman Numerals

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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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(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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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).