198 CHAPTER 4 Systems of Numeration Subtraction Subtraction can also be performed in bases other than base 10. Always remember that when you “borrow,” you borrow the amount of the base given in the subtraction problem. For example, if subtracting in base 5, when you borrow, you borrow 5. If subtracting in base 12, when you borrow, you borrow 12. Example 5 Subtracting in Base 5 Subtract − 3032 1004 5 5 Solution We will perform the subtraction in base 10 and convert the results to base 5. Since 4 is greater than 2, we must borrow one group of 5 from the preceding column. This action gives a sum of + 5 2, or 7, in base 10. Now we subtract 4 from 7; the difference is 3. We complete the problem in the usual manner. The 3 in the second column becomes a − = 2, 2 0 2. In the third column, − = 0 0 0. Finally, in the fourth column, − = 3 1 2. − 3032 1004 2023 5 5 5 7 Now try Exercise 19 In the next example, we will subtract base 12 numerals. Recall from Section 4.3 that we use the capital letter A to represent the number ten and the capital letter B to represent the number eleven in base 12. Example 6 Subtracting in Base 12 Subtract 97A 4B8 12 12 − Solution In base 12, A represents 10. Therefore, in the units column we have − = 10 8 2. Next, in base 12, B represents 11. Therefore, in the next column we must subtract 11 from 7. Since 11 is greater than 7, borrowing is necessary. We must borrow one group of 12 from the preceding column. We then have a sum of + 12 7, or 19. We can now subtract 11 from 19 and the difference is 8. Since we borrowed 1 from the far-left column, the 9 becomes 8, and − = 8 4 4. 97A 4B8 482 12 12 12 − The difference is 482 . 12 7 Now try Exercise 29 Multiplication Multiplication can also be performed in bases other than base 10. Doing so is helped by forming a multiplication table for the base desired. Suppose we want to determine the product of × 4 3 . 5 5 In base × 10, 4 3 means there are four groups of three units. Similarly, in a base 5 system, × 4 3 5 5 means there are four groups of three units, or +++++++++++ (111) (111) (111) (111) Regrouping the 12 units above into groups of five gives ++++ + ++++ + + (11111) (11111) (11) or two groups of five, and two units. Thus, × = 4 3 22 . 5 5 5 Did You Know? Ever-Changing Numerals 12th century 13th century 15th century About 1429 About 1300 20th century Computer* * Times New Roman font 1 2 3 4 5 6 7 8 9 0 During the Middle Ages, Western Europeans were reluctant to give up Roman numerals in favor of Hindu–Arabic numerals. The rapid expansion of trade and commerce during the fifteenth century, however, caused the need for quicker systems of calculation. The invention of movable type in 1450 also ensured a certain consistency in the way numerals were depicted, yet we still find ways to alter them.
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