4.4 Perform Computations in Other Bases 195 You may recall learning how to add, subtract, multiply, and divide when you were in grade school. You may recall “carrying” when adding and “borrowing” when subtracting. For example, when performing + 75 18, you would add 5 and 8 to get 13. You would write down the 3 and “carry the 1.” When performing − 43 19 you would “borrow from the 4” so you could perform − 13 9 to get 4. In this section, we will use similar procedures when working with numerals that have bases other than base 10. SECTION 4.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Add numerals with bases other than base 10. 7 Subtract numerals with bases other than base 10. 7 Multiply numerals with bases other than base 10. 7 Divide numerals with bases other than base 10. Perform Computations in Other Bases Pressmaster/ Shutterstock Why This Is Important By working in other bases, we will gain a better understanding of our own base 10 place-value system. We also gain an understanding of bases such as base 2, base 8, and base 16, which are used in computers and electronic devices. Addition When computers perform calculations, they do so in base 2, the binary system. In this section, we will explain how to perform calculations in base 2 and other bases. In a base 2 system, the only digits are 0 and 1, and the place values are ,2 ,2 ,2 ,2,1 or ,16, 8, 4, 2,1 4 3 2 … … Suppose we want to add + 1 1 . 2 2 The subscript 2 indicates that we are adding in base 2. Remember that the answer to + 1 1 2 2 must be written using only the digits 0 and 1. The sum of + 1 1 2 2 is 10 ,2 which represents 1 group of two and 0 units in base 2. Recall that 102 means × + × (1 2) (0 1). If we wanted to determine the sum of + 10 1 , 2 2 we would add the digits in the right-hand, or units, column. Since + = 0 1 1 , 2 2 2 the sum of + = 10 1 11 . 2 2 2 We are going to work additional examples and exercises in base 2, so rather than performing individual calculations in every problem, we can construct and use an addition table for base 2, Table 4.7 ( just as we used an addition table in base 10 when we first learned to add in base 10). Table 4.7 Base 2 Addition Table Example 1 Adding in Base 2 Add + 1101 111 2 2 Research Activities 72. Duodecimal System Write a report on the use of the duodecimal (base 12) system of numeration. You may wish to contact the Dozenal Society (see the Did You Know? on page 191) for more information. 73. Numeration Systems Throughout history, various societies have used different bases for their numeration systems. Groups of people in North and South America, Africa, and Australia all used a variety of numeration systems. Choose several such groups and write a report on their numeration systems. 74. Papua New Guinea Today, the various groups of people in Papua New Guinea still use a variety of numeration systems. Write a report on the different systems used by these groups. + 0 1 0 0 1 1 1 10
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