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Section 4.5 Early Computation Methods

Section 4.5 Early Computation Methods. Early Civilizations. Early civilizations used a variety of methods for multiplication and division. Multiplication was performed by duplation and mediation, by the lattice method, and by Napier’s rods. Using Duplation and Mediation to Find a Product.

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Section 4.5 Early Computation Methods

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  1. Section 4.5Early Computation Methods

  2. Early Civilizations • Early civilizations used a variety of methods for multiplication and division. • Multiplication was performed by duplation and mediation, by the lattice method, and by Napier’s rods.

  3. Using Duplation and Mediation to Find a Product • Example 1: Multiply 19 × 17 using duplation and mediation. Solution 19 – 17 9 – 34 4 – 68 2 – 136 1 – 272

  4. Using Duplation and Mediation to Find a Product • Example 2: Multiply 26 × 18 using duplation and mediation. Solution 26 – 18 13 – 36 6 – 72 3 – 144 1 – 288

  5. The Lattice Method • The Lattice method is also referred to as the gelosia method. • The name comes from the use of a grid, or lattice, when multiplying two numbers.

  6. The Lattice Method This method uses a rectangle split into columns and rows with each newly-formed rectangle split in half by a diagonal.

  7. Lattice Multiplication Example: Multiply 312 × 75 using lattice multiplication. Solution 3 1 2 2 0 1 7 1 7 4 1 0 1 5 5 5 0

  8. Napier’s Rods • John Napier developed this method in the early 1600s. • Napier’s rods proved to be one of the forerunners of the modern-day computer.

  9. Napier’s Rods • Napier developed a system of separate rods numbered 0 through 9 and an additional rod for an index, numbered vertically 1 through 9. Each rod is divided into 10 blocks. Each block below contains a multiple of the number in the first block, with a diagonal separating its digits. Units digits are placed to the right of the diagonals, tens digits to the left.

  10. Napier’s Rods

  11. Example 4: Using Napier’s Rods to Multiply Two- and Three-Digit Numbers Multiply 48 × 365 using Napier’s rods. Solution • 48 × 365 = (40 + 8) × 365 • (40 + 8)×365=(40 × 365) + (8 × 365) • To find 40 × 365, determine 4 × 365 • and multiply the product by 10 • To evaluate 4 × 365, set up Napier’s rods for 3, 6, and 5 with index 4

  12. Example 4: Using Napier’s Rods to Multiply Two- and Three-Digit Numbers Solution 3 6 5 1 2 2 4 1 2 4 0 4 6 0 Evaluate along the diagonals 4 × 365 = 1460

  13. Example 4: Using Napier’s Rods to Multiply Two- and Three-Digit Numbers Solution 40 × 365 = 1460 × 10 = 14,600 48 × 365 = (40 × 365) + (8 × 365) = 14,600 + 2920 = 17,520

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