Section 4.3. Other Bases. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

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1 Section 4.3 Other Bases

2 INB Table of Contents Date Topic Page # May 14, 2014 Section 4.3 Examples 8 May 14, 2014 Section 4.3 Notes

3 What You Will Learn Converting base 10 numerals to numerals in other bases Converting numerals in other bases to base 10 numerals 4.3-3

4 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,

5 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 positionalvalue system has base b, then its positional values will be, b 4, b 3, b 2, b,

6 Positional Values The positional values in a base 8 system are, 8 4, 8 3, 8 2, 8, 1 The positional values in a base 2 system are, 2 4, 2 3, 2 2, 2,

7 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

8 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 4.3-8

9 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

10 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 represents A. The byte represents a. Other characters representations can be found at

11 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

12 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

13 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 represents a base 5 numeral represents a base 6 numeral. The value of is not the same as the value of Base 10 numerals can be written without a subscript: 123 means

14 Bases Less Than 10 The symbols that represent the base itself, in any base b, are 10 b represents = = = 5 To change a numeral from one base to base multiply each digit by its respective positional value 2. find the sum of the products

15 Example 1: Converting from Base 5 to Base 10 Convert to base

16 Units Digits in Different Bases Notice that 3 5 has the same value as 3 10, since both are equal to 3 units. That is,3 5 = If n is a digit less than the base b, and the base b is less than or equal to 10, then n b = n

17 Example 3: Converting from Base 2 to Base 10 Convert to base

18 Converting Base 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. 2. Then divide the remainder by the next smaller power of the new base and record this quotient. 3. Repeat this procedure until the remainder is less than the new base. 4. The answer is the set of quotients listed from left to right, with the remainder on the far right

19 Example 5: Converting from Base 10 to Base 3 Convert 273 to base

20 Example 5: Converting from Base 10 to Base 3 Solution

21 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

22 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

23 Example 7: Converting to and from Base 16 Convert 7DE 16 to base

24 Example 7: Converting to and from Base 16 Convert 6713 to base

25 Example 7: Converting to and from Base 16 Solution Thus 6713 = 1A

26 Convert to a Hindu-Arabic numeral. a. 26 b. 65 c. 47 d. 59 Copyright 2009 Pearson Education, Inc. Slide 4-32

27 Convert to a Hindu-Arabic numeral. a. 140 b. 502 c. 872 d Copyright 2009 Pearson Education, Inc. Slide 4-33

28 Convert to a Hindu-Arabic numeral. a. 17 b. 33 c. 39 d. 57 Copyright 2009 Pearson Education, Inc. Slide 4-34

29 Convert to a Hindu-Arabic numeral. a. 171 b. 207 c. 233 d. 521 Copyright 2009 Pearson Education, Inc. Slide 4-35

30 Convert 57 to base 2. a b c d Copyright 2009 Pearson Education, Inc. Slide 4-36

31 Convert 2034 to base 12. a b c d Copyright 2009 Pearson Education, Inc. Slide 4-37

32 Convert 103 to base 7. a b c d Copyright 2009 Pearson Education, Inc. Slide 4-38

33 Convert 798 to base 5. a b c d Copyright 2009 Pearson Education, Inc. Slide 4-39