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Section 4.5 Early Computation Methods

Learn about early computation methods used by ancient civilizations, such as duplation, mediation, the lattice method, and Napier's rods. Practice examples included.

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Section 4.5 Early Computation Methods

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  1. Section 4.5Early Computation Methods

  2. What You Will Learn • Other computation methods: • Duplation and mediation • Lattice method • Napier’s rods

  3. 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.

  4. Duplation and Mediation • Know as Russian Peasant Multiplication • Duplation refers to doubling a number • Mediation refers to halving a number • Similar to ancient Egyptian method described in Rhind Papyrus

  5. Example 1: Using Duplation and Mediation • Multiply 19 × 17 using duplation and mediation. Solution Write 19 and 17 with a dash between 19 – 17 Divide the number on the left, 19, by 2 Drop the remainder Place the quotient, 9, under the 19

  6. Example 1: Using Duplation and Mediation Solution 19 – 17 9 Double the number on the right, 17, to obtain 34 place it under the 17 19 – 17 9 – 34 Continue this process until 1 appears in the left-hand column

  7. Example 1: Using Duplation and Mediation Solution 19 – 17 9 – 34 4 – 68 2 – 136 1 – 272 Cross out all the even numbers in the left-hand column and the corresponding numbers in the right-hand column

  8. Example 1: Using Duplation and Mediation Solution 19 – 17 9 – 34 4 – 68 2 – 136 1 – 272 Add remaining numbers in right-hand column: 17 + 34 + 272 = 323 19 × 17 = 323

  9. Try This Multiply 27 x 13

  10. 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.

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

  12. Example 2: Using Lattice Multiplication Multiply 312 × 75 using lattice multiplication. Solution Construct a rectangle consisting of 3 columns and 2 rows • Place the 312 above the boxes • Place 75 on the right of the boxes • Place a diagonal in each box

  13. Example 2: Using Lattice Multiplication Solution 3 1 2 7 5

  14. Example 2: Using Lattice Multiplication Solution • Complete each box by multiplying the number on the top of the box by the number on the right of the box • Place the units digit of the product below the diagonal • Place the tens digit of the product above the diagonal

  15. Example 2: Using Lattice Multiplication Solution 3 1 2 2 0 1 7 1 7 4 1 0 1 5 5 5 0

  16. Example 2: Using Lattice Multiplication Solution Add diagonals as shown carry tens digits to next diagonal 3 1 2 2 0 1 2 7 1 7 4 1 0 1 3 5 5 5 0 4 0 0

  17. Example 2: Using Lattice Multiplication Solution Read the answer down the left-hand column and along the bottom 3 1 2 2 0 1 2 7 1 7 4 1 0 1 3 Therefore, 312 × 75 = 23, 400 5 5 5 0 4 0 0

  18. Try This Multiply 237 x 73

  19. Homework

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