206 CHAPTER 4 Systems of Numeration Example 4 illustrates the procedure for multiplying numbers containing more than one digit, using Napier’s rods. Profile in Mathematics During the seventeenth century, the growth of scientific fields such as astronomy required the ability to perform often unwieldy calculations. Scottish mathematician John Napier (1550–1617) made great contributions toward solving the problem of computing these numbers. His inventions include simple calculating machines and a device for performing multiplication and division known as Napier’s rods. Napier also developed the theory of logarithms. 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 × = + × Write (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, and then evaluate along the diagonals, as indicated. INDEX 2 2 2 4 4 4 3 6 5 6 0 0 1 1 Therefore, 4 365 1460. × = Then 40 365 1460 10 14,600. × = × = 48 365 (40 365) (8 365) 14,600 2920 17,520 × = × + × = + = 7 8 365 2920 × = from Example 3 Now try Exercise 27 Exercises Warm Up Exercises In Exercises 1– 4, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. When determining the product 39 73, × using duplation and mediation, a) the 39 is first ________ by 2, and then Divided b) the 73 is ________. Doubled 2. When using duplation and mediation, when you divide the number on the left by two you disregard the ________. Remainder 3. When determining the product 327 45, × using lattice multiplication, first construct a rectangle consisting of ________ columns and ________ rows. Three, two 4. When using lattice multiplication, after using multiplication to complete the boxes in the rectangle, add the numbers along the ________. Diagonals Practice the Skills In Exercises 5–12, multiply using duplation and mediation. 5. 13 17 × 221 6. 18 22 × 396 7. 37 23 × 851 8. 138 41 × 5658 9. 35 236 × 8260 10. 96 53 × 5088 11. 93 93 × 8649 12. 49 124 × 6076 In Exercises 13–20, use lattice multiplication to determine the product. 13. 4 124 × 496 14. 6 18 × 108 15. 6 425 × 2550 16. 9 509 × 4581 17. 75 12 × 900 18. 47 259 × 12,173 19. 314 652 × 204,728 20. 634 832 × 527,488 In Exercises 21–28, multiply using Napier’s rods. 21. 3 48 × 144 22. 4 72 × 288 23. 5 81 × 405 24. 6 171 × 1026 25. 5 125 × 625 26. 7 3456 × 24,192 27. 19 6742 × 128,098 28. 75 125 × 9375 SECTION 4.5 Learning Catalytics Keyword: Angel-SOM-4.5 (See Preface for additional details.)
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