section 4.3 other bases

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 4.3

Other Bases


What You Will Learn

Converting base 10 numerals to numerals in other bases

Converting numerals in other bases to base 10 numerals

4.3-2


Positional ValuesThe 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

4.3-3


Positional Values and Bases10 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, 14.3-4


Positional ValuesThe 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

4.3-5


Other Base Numeration SystemsBase 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.

4.3-6


Other Base Numeration SystemsBase 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

4.3-7


Other Base Numeration SystemsComputers and many other electronic devices use three numeration systems:Binary – base 2Uses 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.

4.3-8


Other Base Numeration SystemsOctal – base 8Eight bits of data are grouped to form a byteAmerican 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.

4.3-9


Other Base Numeration SystemsHexadecimal – base 16Used 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.

4.3-10


Bases Less Than 10A place-value system with base b hasb distinct objects, one for zero and one for each numeral less than the base.Base 6 system: 0, 1, 2, 3, 4, 5All numerals in base 6 are constructed from these 6 symbols.Base 8 system: 0, 1, 2, 3, 4, 5, 6, 7All numerals in base 8 are constructed from these 8 symbols.

4.3-11


Bases Less Than 10A 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.

4.3-12


Bases Less Than 10The symbols that represent the base itself, in any base b, are 10b.105 represents 5105 = 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.

4.3-13


Example 1: Converting fromBase 5 to Base 10Convert 2435 to base 10.

Solution2435 = (2 × 52) + (4 × 5) + (3 × 1)

= (1 × 25) + (4 × 5) + (3 × 1) = 50 + 20 + 3 = 73

4.3-14


Units Digits in Different BasesNotice 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.

4.3-15


Example 3: Converting fromBase 2 to Base 10Convert 1100102 to base 10.

Solution1100102 = (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

4.3-16


Converting Base 10Divide 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.

4.3-17


Example 5: Converting fromBase 10 to Base 3Convert 273 to base 3.SolutionThe place values in the base 3 system are

…, 36, 35, 34, 33, 32, 3, 1or …, 729, 243, 81, 27, 9, 3, 1Highest power of the base that is less than or equal to 273 is 35, or 243.Begin by dividing 273 by 243.

4.3-18


Example 5: Converting fromBase 10 to Base 3Solution

4.3-19


Example 5: Converting fromBase 10 to Base 3SolutionWe 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

4.3-20


Bases Greater Than 10We 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.

4.3-21


Bases Greater Than 10For 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.

4.3-22


Example 7: Converting to and from Base 16Convert 7DE16 to base 10.

Solution7DE16 =(7 × 162) + (D × 16) + (E × 1)

= (7 × 256) + (13 × 16) + (14 × 1)

= 1792 + 208 + 14= 2014

4.3-23


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

SolutionThe 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.4.3-24


Example 7: Converting to and from Base 16Solution

Thus 6713 = 1A3916.

4.3-25

section 4.3 other bases

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