4.5 Early Computational Methods 203 Suppose that last week you worked at your job in a movie theater for 27 hours at a rate of $8 per hour. How much money did you make? You may determine the answer by multiplying 27 8 × “by hand.” 27 8 216 5 × Although most of us would use this method, it is not the only method for multiplying two numbers together. Early civilizations used various other methods. In this section, we will study three other methods of multiplication. Early Computational Methods SECTION 4.5 LEARNING GOALS Upon completion of this section, you will be able to: 7 Multiply whole numbers using duplation and mediation. 7 Multiply whole numbers using lattice multiplication. 7 Multiply whole numbers using Napier’s rods. Why This Is Important Studying other methods of multiplication can help us better understand how traditional multiplication works. In addition, there is a link between multiplication methods used by early civilizations and the way microprocessors used in computers and electronic devices multiply. Understanding other multiplication methods helps us gain a better understanding of computing in other bases. Duplation and Mediation The first method of multiplication we will study is duplation and mediation. This method is still used in parts of Russia today. Duplation refers to doubling one number, and mediation refers to halving one number. The duplation and mediation method is similar to a method used by the ancient Egyptians as described on the Rhind Papyrus. Example 1 illustrates this method of multiplication. Example 1 Using Duplation and Mediation Multiply 39 23 × using duplation and mediation. Solution Write 39 and 23 with a dash between the two numbers. Divide the number on the left, 39, by 2, drop the remainder and place the quotient, 19, under the 39. Double the number on the right, 23, to obtain 46 and place it under the 23. You will then have the following number pairs. 39—23 19—46 Research Activities 72. Computer Numeration Systems Write a report on how computers use the binary (base 2), octal (base 8), and hexadecimal (base 16) numeration systems. 73. Computer Subtraction One method used by computers to perform subtraction is the “end around carry method.” Do research and write a report explaining, with specific examples, how a computer performs subtraction by using the end around carry method. Charnsitr/Shutterstock Milkos/123RF
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