4.5

1. 4.5 Early 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 galley method, and by Napier rods.

3. Duplation and Mediation • Duplation and mediation uses a pairing method for multiplication. Example: Multiply 17  22 using duplation and mediation. Solution: Write 17 and 22 with a dash to separate. Divide the number on the left in half, drop the remainder and place the quotient under the 17. Double the number on the right, and place it under the 22. 17 – 22 8 – 44

4. 17 – 22 8 – 44 Continue this process until a 1 appears in the left hand column. 17 – 22 8 – 44 4 – 88 2 – 176 1 – 352 Cross out all the even numbers in the left-hand column and the corresponding numbers in the right-hand column. 17 – 22 8 – 44 4 – 88 2 – 176 1 – 352 Duplation and Mediation continued

5. Duplation and Mediation continued • Now, add the remaining numbers in the right-hand column, obtaining 22 + 352 = 374. • To check 17  22 = 374

7. The Lattice Method • The Lattice method is also referred to as the gelosia method. • This method uses a rectangle split into columns and rows with each newly-formed rectangle split in half by a diagonal.

8. Example: The Lattice Method Multiply 426  65. Solution: • Construct a rectangle consisting of 3 columns and 2 rows. • Place the 3-digit number above the boxes and the 2-digit number on the right of the boxes. • Place a diagonal in each box. • Complete by multiplying the number on the top of the box by the number on the right of the box and inserting the tens digit of the result above the diagonal and the units digit below the diagonal.

9. Add the numbers along the diagonals. The number is read down the left-hand column and along the bottom, as shown by the arrow. The answer is 27,690. 4 6 2 1 3 2 6 2 4 2 6 3 2 1 5 7 0 0 0 6 9 0 Example: The Lattice Method continued

11. Homework p. 203 # 5 – 20 Must show work.

12. Napier Rods • John Napier developed in the 17th century. • Napier rods, proved to be one of the forerunners of the modern-day computer. • Napier developed a system of separate rods numbered 0 through 9 and an additional strip for an index, numbered vertically 1 through 9. • Each rod is divided into 10 blocks. Each block below contains a multiple of a the number in the first block, with a diagonal separating its digits. The units are placed to the right of the diagonals and the tens digits to the left.

13. 2 8 4 6 1 2 4 1 8 2 4 7 0 4 Example: Napier Rods Multiply 6  284, using Napier rods. Solution: Line up the rods 2, 8, 4, using 6 as the index. To obtain the answer, add along the diagonals as in the lattice method. Thus, 6  284 = 1704.