Section 4.3 Other Bases

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 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 = 2× 10 3 + 3× 10 2 + 5× 10 1 +(8× 10 0 )

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.

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.

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.

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

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.

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.

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.

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. If a number is written without a subscript, we assume it is base 10. 123 means 12310.

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.

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

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

Example 3: 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 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.

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:

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:

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: