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

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 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.

12. 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
Base 10 numerals can be written without a subscript: 123 means

13. Bases Less Than 10
The symbols that represent the base itself, in any base b, are 10b.
105 represents 5
105 = 1 × × 1 = = 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.

14. Example 1: Converting from Base 5 to Base 10
Convert 2435 to base 10.
Solution
2435 = (2 × 52) + (4 × 5) + (3 × 1)
= (1 × 25) + (4 × 5) + (3 × 1) =
= 73

15. 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.

16. Example 3: Converting from Base 2 to Base 10
Convert to base 10.
Solution
= (1 × 25) + (1 × 24)
+ (0 × 23) + (0 × 22) (1 × 2) + (0 × 1)
= (1 × 32) + (1 × 16) + (0 × 8) + (0 × 4) + (1 × 2) + (0 × 1)
= = 50

17. 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.

18. Example 5: Converting from Base 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.

19. Example 5: Converting from Base 10 to Base 3
Solution

20. Example 5: Converting from Base 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) =

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 7DE16 to base 10.
Solution
7DE16 =(7 × 162) + (D × 16) + (E × 1)
= (7 × 256) + (13 × 16) + (14 × 1) =
= 2014

24. 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.

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