196 CHAPTER 4 Systems of Numeration Let’s now look at addition in a base 5 system. In base 5, the only digits are 0, 1, 2, 3, and 4, and the positional values are , 5, 5,5,5,1 or , 625, 125, 25, 5, 1 4 3 2 … … What is the sum of 4 3? 5 5 + We can consider this to mean + + + + (1 1 1 1) + + (1 1 1). We can regroup the seven 1’s into one group of five and two units as + + + + + + (1 1 1 1 1) (1 1). Thus, the sum of + = 4 3 12 5 5 5 (circled in Table 4.8, the base 5 addition table). Recall that 125 means × + × (1 5) (2 1). We can use this same procedure in obtaining the remaining values in the base 5 addition table. MATHEMATICS TODAY Speaking to Machines Besides our familiar base 10 numeration system, one of the most significant numeration systems is the binary, or base 2, numeration system. Most modern electronic devices use binary to process information and “talk” to one another. When a computer receives a command or data, each character must first be converted into a binary numeral for the computer to understand and use it. The binary system is also used to record data, including audio and video, on Blu-ray discs (BDs), digital video discs (DVDs), compact discs (CDs), and optical disc drives. Sound and images are stored in a binary system of pits and “lands” (nonpits). To play the disc, a laser beam tracks along the spiral and recognizes a pit as a 0 and a land as a 1. The binary sequence of 0s and 1s is then converted back into sound and images. Why This Is Important The recording of information on discs is just one of the many examples of how other numeration systems are used in technology. Solution Begin by adding the digits in the right-hand, or units, column. From previous discussion, and as can be seen in Table 4.7, + = 1 1 10 . 2 2 2 Place the 0 under the units column and carry the 1 to the 2’s column, the second column from the right. 2 2 2 1 1 1 0 1 1 1 1 0 3 2 1 2 ↓ ↓ ↓ ↓ + Place value of columns Now add the three digits in the 2’s column, + + 1 0 1 . 2 2 2 Treat it as + + (1 0 ) 1 . 2 2 2 Therefore, add + 1 0 2 2 to get 1 ,2 then add + 1 1 2 2 to get 10 .2 Place the 0 under the 2’s column and carry the 1 to the 22 column (the third column from the right). 1 1 0 1 1 1 1 0 0 1 1 2 + Now add the three 1’s in the 22 column to get + + = + = (1 1)1 10 1 11. 2 2 2 2 2 2 Place the 1 under the 22 column and carry the 1 to the 23 column (the fourth column from the right). + 1 1 0 1 1 1 1 1 00 1 1 1 2 Now add the two 1’s in the 23 column, + = 1 1 10 . 2 2 2 Place the 10 as follows. + 1 1 0 1 1 1 1 1 0 1 0 0 1 1 1 2 Therefore, the sum is 10100 .2 7 Now try Exercise 13 Table 4.8 Base 5 Addition Table + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 410 2 2 3 41011 3 3 4101112 4 4 10 11 12 13 Example 2 Use the Base 5 Addition Table Add + 34 23 5 5 Solution First determine from Table 4.8 that + 4 3 5 5 is 12 .5 Record the 2 and carry the 1 to the 5’s column. 3 4 2 3 2 1 5 5 5 +
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