Section 4.3 Other Bases

# Section 4.3 Other Bases

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## Section 4.3 Other Bases

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1. Section 4.3Other Bases

2. What You Will Learn • Converting base 10 numerals to numerals in other bases • Converting numerals in other bases to base 10 numerals

3. 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

4. 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

5. 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

6. 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.

7. 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.

8. 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.

9. 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

10. Try this Convert the following to base 10

11. 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.

12. 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

13. 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.

14. 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.

15. 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

16. Try This Convert 52 to base 4

17. 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.

18. 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.

19. 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

20. 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.

21. Example 7: Converting to and from Base 16 Solution Thus 6713 = 1A3916.

22. Try This Convert 2731 to base 12

23. Homework