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## Section 4.3 Other Bases

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**What You Will Learn**• Converting base 10 numerals to numerals in other bases • Converting numerals in other bases to base 10 numerals**Positional Values**• The positional values in the Hindu-Arabic numeration system are … 105, 104, 103, 102, 10, 1 • The positional values in the Babylonian numeration system are …, (60)4, (60)3, (60)2, 60, 1**Positional Values and Bases**• 10 and 60 are called the bases of the Hindu-Arabic and Babylonian systems, respectively. • Any counting number greater than 1 may be used as a base. If a positional-value system has base b, then its positional values will be …, b4, b3, b2, b, 1**Positional Values**• The positional values in a base 8 system are …, 84, 83, 82, 8, 1 • The positional values in a base 2 system are …, 24, 23, 22, 2, 1**Bases Less Than 10**• A place-value system with base b has • b distinct objects, one for zero and one for each numeral less than the base. • Base 6 system: 0, 1, 2, 3, 4, 5 • All numerals in base 6 are constructed from these 6 symbols. • Base 8 system: 0, 1, 2, 3, 4, 5, 6, 7 • All numerals in base 8 are constructed from these 8 symbols.**Bases Less Than 10**• A numeral in a base other than base 10 will be indicated by a subscript to the right of the numeral. • 1235 represents a base 5 numeral. • 1236 represents a base 6 numeral. • The value of 1235 is not the same as the value of 12310. • Base 10 numerals can be written without a subscript: 123 means 12310.**Bases Less Than 10**• The symbols that represent the base itself, in any base b, are 10b. • 105 represents 5 • 105 = 1 × 5 + 0 × 1 = 5 + 0 = 5 • To change a numeral from one base to base 10, multiply each digit by its respective positional value, then find the sum of the products.**Example 1: Converting fromBase 5 to Base 10**• Convert 2435 to base 10. • Solution • 2435 = (2 × 52) + (4 × 5) + (3 × 1) • = (1 × 25) + (4 × 5) + (3 × 1) • = 50 + 20 + 3 • = 73**Try this**Convert the following to base 10**Units Digits in Different Bases**• Notice that 35 has the same value as 310, since both are equal to 3 units. • That is,35 = 310. • If n is a digit less than the base b, and the base b is less than or equal to 10, then nb = n10.**Example 3: Converting fromBase 2 to Base 10**• Convert 1100102 to base 10. • Solution • 1100102 = (1 × 25) + (1 × 24) • + (0 × 23) + (0 × 22) + (1 × 2) + (0 × 1) • = (1 × 32) + (1 × 16) + (0 × 8) + (0 × 4) + (1 × 2) + (0 × 1) • = 32 + 16 + 0 + 0 + 2 + 0 = 50**Converting Base 10**• Divide the base 10 numeral by the highest power of the new base that is less than or equal to the given base 10 numeral and record this quotient. • Then divide the remainder by the next smaller power of the new base and record this quotient. • Repeat this procedure until the remainder is less than the new base. • The answer is the set of quotients listed from left to right, with the remainder on the far right.**Example 5: Converting fromBase 10 to Base 3**• Convert 273 to base 3. Solution The place values in the base 3 system are …, 36, 35, 34, 33, 32, 3, 1 or …, 729, 243, 81, 27, 9, 3, 1 Highest power of the base that is less than or equal to 273 is 35, or 243. Begin by dividing 273 by 243.**Example 5: Converting fromBase 10 to Base 3**Solution We can represent 273 as one group of 243, no groups of 81, one group of 27, no groups of 9, one group of 3, and no units. 273 = (1 × 243) + (0 × 81) + (1 × 27) + (0 × 9) + (1 × 3) + (0 × 1) = (1 × 35) + (0 × 34) + (1 × 33) + (0 × 32) + (1 × 3) + (0 × 1) = 1010103**Try This**Convert 52 to base 4**Bases Greater Than 10**• We will need single digit symbols to represent the numbers ten, eleven, twelve, . . . up to one less than the base. • In this textbook, whenever a base larger than ten is used we will use the capital letter A to represent ten, the capital letter B to represent eleven, the capital letter C to represent twelve, and so on.**Bases Greater Than 10**• For example, for base 12, known as the duodecimal system, we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and B, where A represents ten and B represents eleven. • For base 16, known as the hexadecimal system, we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.**Example 7: Converting to and from Base 16**Convert 7DE16 to base 10. Solution 7DE16 =(7 × 162) + (D × 16) + (E × 1) = (7 × 256) + (13 × 16) + (14 × 1) = 1792 + 208 + 14 = 2014**Example 7: Converting to and from Base 16**Convert 6713 to base 16. Solution The highest power of base 16 less than or equal to 6713 is 163, or 4096. If we obtain a quotient greater than nine but less than sixteen, we will use the corresponding letter A through F.**Example 7: Converting to and from Base 16**Solution Thus 6713 = 1A3916.**Try This**Convert 2731 to base 12