1. Section 4.5 Early 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
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: = 323
19 × 17 = 323

9. 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.

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

11. 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

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

13. 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

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

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

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

17. 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.

18. 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.

19. Napier’s Rods

20. 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

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

22. 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,
= 17,520