4.5 Early Computational Methods 205 Napier’s Rods The third method used to multiply numbers was developed from the lattice method by John Napier in the early 1600s and is known as Napier’s rods or Napier’s bones. This method is similar to that used in modern computers. Napier developed a system of separate rods (which were often made out of bones) numbered 0 through 9 and an additional rod for an index, numbered vertically 1 through 9 (Fig. 4.4). Each rod is divided into 10 blocks. Each block below the first block contains a multiple of the number in the first block, with a diagonal separating the digits. The units digits are placed to the right of the diagonals and the tens digits to the left. Example 3 explains how Napier’s rods are used to multiply numbers. Add the numbers along the diagonals, as shown with the blue shaded arrows in Fig. 4.3, starting with the bottom right diagonal. If the sum in a diagonal is 10 or greater, record the units digit below the rectangle and carry the tens digit to the next diagonal to the left. For example, when adding 4, 1, and 5 (along the second blue diagonal from the right), the sum is 10. Record the 0 below the rectangle and carry the 1 to the next blue diagonal. The sum of 1 1 7 0 5 + + + + is 14. Record the 4 and carry the 1 to the next blue diagonal. The sum of the numbers in the next blue diagonal is 1 0 1 1 + + + or 3. The answer is read down the left-hand column and along the bottom, as shown by the purple arrow in Fig. 4.3. Therefore, 312 75 23,400. × = 7 Now try Exercise 13 3 3 1 1 1 1 1 2 2 2 4 7 7 5 5 5 0 0 0 0 0 4 Figure 4.3 0 2 2 2 2 1 1 3 3 3 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 1 1 0 0 0 0 0 0 0 0 0 2 2 1 1 1 0 0 1 0 0 1 0 0 1 1 2 2 1 1 1 1 1 1 1 3 2 2 2 2 3 3 2 3 3 3 3 3 4 5 5 5 6 6 6 7 7 4 4 4 3 3 4 4 4 5 4 4 4 5 5 5 5 6 6 7 7 6 5 0 0 5 5 5 3 4 4 5 5 4 4 4 4 7 7 6 6 6 7 9 9 9 9 9 8 8 8 8 8 6 6 6 4 4 8 6 8 8 8 8 8 4 6 6 2 1 3 5 7 4 9 8 6 8 2 4 6 8 8 INDEX Figure 4.4 m Napier’s Rods Example 3 Using Napier’s Rods Multiply 8 365, × using Napier’s rods. Solution To multiply 8 365, × line up the rods 3, 6, and 5 to the right of the index 8, as shown in Fig. 4.5. Below the 3, 6, and 5 place the blocks that contain the products of 8 3, 8 6, × × and 8 5, × respectively. To obtain the answer, add along the diagonals as is done with the lattice method. INDEX 3 5 6 2 2 2 8 8 4 4 4 9 0 0 Thus, 8 365 2920. × = 7 Now try Exercise 21 INDEX 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 0 0 0 0 0 0 0 0 4 4 4 4 5 5 5 5 0 5 0 5 5 5 5 6 6 6 2 2 2 2 4 4 4 4 6 6 7 7 9 9 8 8 8 8 Figure 4.5 Photo Researchers/ Science History Images/ Alamy Stock Photo Instructor Resources for Section 4.5 in MyLab Math • Objective-Level Videos 4.5 • PowerPoint Lecture Slides 4.5 • MyLab Exercises and Assignments 4.5 • Chapter 4 Projects
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