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Section 4.3 Other Bases

Section 4.3 Other Bases. Positional Values. The positional values in the Hindu-Arabic numeration system are … 10 5 , 10 4 , 10 3 , 10 2 , 10, 1 The positional values in the Babylonian numeration system are …, (60) 4 , (60) 3 , (60) 2 , 60, 1. Positional Values.

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Section 4.3 Other Bases

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  1. Section 4.3Other Bases

  2. 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

  3. Positional Values • To help students understand place value in our base 10 system, we have students write a given number in expanded form. • Example: Write 2,358 in expanded form. • 2,358 = 2×1000 + 3×100 + 5×10 + 8

  4. Positional Values and Bases • Any counting number greater than 1 may be used as a base for a positional-value numeration system. If a positional-value system has base b, then its positional values will be …, b4, b3, b2, b, 1 and the numerals used in the system include the counting numbers from 0 to b.

  5. Examples • The positional values in a base 8 system are …, 84, 83, 82, 8, 1 and the numerals used in the system include 0, 1, 2, 3, 4, 5, 6, and 7. • The positional values in a base 2 system are …, 24, 23, 22, 2, 1 and the numerals used in the system are 0 and 1.

  6. Other Base Numeration Systems • Base 10 is almost universal. • Base 2 is used in some groups in Australia, New Guinea, Africa, and South America. • Bases 3 and 4 is used in some areas of South America. • Base 5 was used by primitive tribes in Bolivia, who are now extinct. • Base 6 is used in Northwest Africa.

  7. Other Base Numeration Systems • Base 6 also occurs in combination with base 12, the duodecimal system. • Our society has remnants of other base systems: • 12: 12 inches in a foot, 12 months in a year, a dozen, 24-hour day, a gross (12 × 12) • 60: Time - 60 seconds to 1 minute, 60 minutes to 1 hour; Angles - 60 seconds to 1 minute, 60 minutes to 1 degree

  8. Other Base Numeration Systems • Computers and many other electronic devices use three numeration systems: • Binary – base 2 • Uses only the digits 0 and 1. • Can be represented with electronic switches that are either off (0) or on (1). • All computer data can be converted into a series of single binary digits. • Each binary digit is known as a bit.

  9. Other Base Numeration Systems • Octal – base 8 • Eight bits of data are grouped to form a byte • American Standard Code for Information Interchange (ASCII) code. • The byte 01000001 represents A. • The byte 01100001 represents a. • Other characters representations can be found at www.asciitable.com.

  10. Other Base Numeration Systems • Hexadecimal – base 16 • Used to create computer languages: • HTML (Hypertext Markup Language) • CSS (Cascading Style Sheets). • Both are used heavily in creating Internet web pages. • Computers easily convert between binary (base 2), octal (base 8), and hexadecimal (base 16) numbers.

  11. 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. • 1235represents a base 5 numeral. • 1236represents a base 6 numeral. • If a number is written without a subscript, we assume it is base 10. • 123 means 12310.

  12. Changing a Number from Another Base to Base 10 • To change a numeral from another base to base 10, multiply each digit by its respective positional value, then find the sum of the products.

  13. Example 1: • Convert 3256to base 10. • Solution: • 3256= (3 × 62) + (2 × 6) + (5 × 1) • = (3 × 36) + (2 × 6) + (5 × 1) • = 108 + 12 + 5 • = 125

  14. Example 2: • Convert 50328 to base 10. • Solution: • 50328 = (5 × 83) + (0 × 82) + (3 × 8) + (2 × 1) • = (5 × 512) + (0 × 64) + (3 × 8)+ (2 × 1) • = 2560 + 0 + 24 + 2 • = 2586

  15. Example 3: • Convert 1100102to 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

  16. Converting from Base 10 to Other Bases • List the numerals used for the new base. • List the place values. • Divide the base 10 numeral by the highest possible place value of the new base. • Divide the remainder by the next smaller place value of the new base. • Repeat this procedure until you divide by 1. • The answer is the set of quotients listed from left to right.

  17. Example 4: • Convert 6 to base 2. Solution: The numerals used in a base 2 system are 0 and 1. The place values are: …, 24, 23, 22, 2, 1 or …, 16, 8, 4, 2, 1 For 6, the highest possible place value that you can divide by is 4, or 22. Thus, we have:

  18. Example 5: • Convert 53 to base 4. Solution: The numerals used in a base 4 system are 0, 1, 2, 3. The place values are: …, 44, 43, 42, 4, 1 or …, 256, 64, 16, 4, 1 For 53, the highest possible place value that you can divide by is 16, or 42. Thus, we have:

  19. Example 6: • Convert 347 to base 3. Solution: The numerals used in a base 3 system are 0, 1, 2. The place values are: …, 36, 35, 34, 33, 32, 3, 1 or …, 729, 243, 81, 27, 9, 3, 1 For 347, the highest possible place value that you can divide by is 243, or 35. Thus, we have:

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