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Unformatted text preview: COMPSCI 102 Discrete Mathematics for Computer Science Ancient Wisdom: On Raising A Number To A Power Lecture 12 a 1 5 1 5 The Egyptians used decimal numbers but multiplied and divided in binary Egyptian Multiplication a x b By Repeated Doubling b has nbit representation: b n1 b n2 …b 1 b Starting with a, repeatedly double largest number so far to obtain: a, 2a, 4a, …., 2 n1 a Sum together all the 2 k a where b k = 1 a b = b 2 a + b 1 2 1 a + b 2 2 2 a + … + b n1 2 n1 a 2 k a is in the sum if and only if b k = 1 b = b 2 + b 1 2 1 + b 2 2 2 + … + b n1 2 n1 Wait! How did the Egyptians do the part where they converted b to binary? They used repeated halving to do base conversion! Egyptian Base Conversion Output stream will print right to left Input X; repeat { if (X is even) then print 0; else {X := X1; print 1;} X := X/2 ; } until X=0; Sometimes the Egyptians combined the base conversion by halving and multiplication by doubling into a single algorithm Binary for 13 is 1101 = 2 3 + 2 2 + 2 70*13 = 70*2 3 + 70*2 2 + 70*2 70 x 13 Rhind Papyrus [1650 BC] 70 140 280 560 13 6 3 1 * 70 350 910 Doubling Halving Odd? Running Total * * 5 x 30 5 10 20 40 30 15 7 3 * 30 70 Doubling Halving Odd? Running Total * * 80 1 150 * 10 184 / 17 Rhind Papyrus [1650 BC] Doubling Powers of 2 Check 17 34 68 136 1 2 4 8 * * 184 = 17*8 + 17*2 + 14 184/17 = 10 with remainder 14 This method is called “Egyptian Multiplication / Division” or “Russian Peasant Multiplication / Division” × 1101 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Standard Binary Multiplication = Egyptian Multiplication We can view numbers in many different, but corresponding ways Representation: Representation: Understand the relationship between Understand the relationship between different representations of the same different representations of the same information or idea information or idea 1 2 3 Our story so far… Induction is how we define and manipulate mathematical ideas Induction has many guises Master their interrelationship • Formal Arguments • Loop Invariants • Recursion • Algorithm Design • Recurrences Our story so far… Abstraction Abstraction : : Abstract away the inessential Abstract away the inessential features of a problem or solution features of a problem or solution = Let’s Articulate a New One: b:=a 8 b:=a*a b:=b*a b:=b*a b:=b*a b:=b*a b:=b*a b:=b*a b:=a*a b:=b*b...
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 Fall '09
 Multiplication, Binary numeral system, Rhind Papyrus, addition chain

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