Section 4.5 Early Computation Methods

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

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.

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

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

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

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

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

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.

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

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

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

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

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

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

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

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.

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.

Napier’s Rods

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

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

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