Conversion Between Number Bases

Transcription

1 Conversion Between Number Bases MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018

2 General Number Bases Bases other than 10 are sometimes used in numeration systems. Base 2, 8, and 16 (respectively binary, octal, and hexadecimal ) are used around computers. When we refer to a number expressed in a base other than 10, we will subscript the number with the base. 243 five = In base-b we may only use the digits {0, 1, 2,..., b 1}.

3 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count

4 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count

5 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count

6 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count

7 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count

8 Counting in Base-b In base-10 the digits are {0, 1, 2,..., 9} and we count In base-6 the digits are {0, 1, 2,..., 5} and we count

9 Expanded Form Example = (2 5 2 ) + (4 5 1 ) + (3 5 0 ) = = 73

10 Expanded Form Example = (2 5 2 ) + (4 5 1 ) + (3 5 0 ) = = 73 Example = (7 8 3 ) + (6 8 2 ) + (5 8 1 ) + (4 8 0 ) = = 3778

11 Converting Between Bases To convert from base b to base 10: 1. Start with the first digit on the left and multiply by the base b. 2. Add the next digit and multiply again by the base b, and so on. 3. Add the last digit on the right (but do not multiply by the base).

12 Examples (1 of 2) Use the change of base algorithm to covert 345 to base-10 form (we know the answer will be 345).

13 Examples (1 of 2) Use the change of base algorithm to covert 345 to base-10 form (we know the answer will be 345) = 30

14 Examples (1 of 2) Use the change of base algorithm to covert 345 to base-10 form (we know the answer will be 345) = = 34

15 Examples (1 of 2) Use the change of base algorithm to covert 345 to base-10 form (we know the answer will be 345) = = = 340

16 Examples (1 of 2) Use the change of base algorithm to covert 345 to base-10 form (we know the answer will be 345) = = = = 345

17 Examples (2 of 2) Use the change of base algorithm to covert to base-10 form.

18 Examples (2 of 2) Use the change of base algorithm to covert to base-10 form. 3 8 = 24

19 Examples (2 of 2) Use the change of base algorithm to covert to base-10 form. 3 8 = = 28

20 Examples (2 of 2) Use the change of base algorithm to covert to base-10 form. 3 8 = = = 224

21 Examples (2 of 2) Use the change of base algorithm to covert to base-10 form. 3 8 = = = = 229

22 Examples Convert each of the following numbers into decimal (base-10) form. Use your i>clicker to submit your responses

23 Examples Convert each of the following numbers into decimal (base-10) form. Use your i>clicker to submit your responses

24 Examples Convert each of the following numbers into decimal (base-10) form. Use your i>clicker to submit your responses

25 Examples Convert each of the following numbers into decimal (base-10) form. Use your i>clicker to submit your responses

26 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division.

27 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6.

28 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder 6 789

29 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder

30 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder

31 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder

32 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder

33 Converting from Base-10 Converting from decimal (base-10) to another base can be done by repeated division. Example Convert 789 to base 6. Base Dividend Remainder Thus = 789. Note that we read the remainders from bottom to top.

34 Another Conversion Convert 789 to base 4.

35 Another Conversion Convert 789 to base 4. Base Dividend Remainder 4 789

36 Another Conversion Convert 789 to base 4. Base Dividend Remainder

37 Another Conversion Convert 789 to base 4. Base Dividend Remainder

38 Another Conversion Convert 789 to base 4. Base Dividend Remainder

39 Another Conversion Convert 789 to base 4. Base Dividend Remainder

40 Another Conversion Convert 789 to base 4. Base Dividend Remainder

41 Another Conversion Convert 789 to base 4. Base Dividend Remainder Thus = 789. Remember to read the remainders from bottom to top.

42 Examples Perform the following conversions. Use your i>clicker to submit your responses. 5. Convert 935 to base 8. Remember: numbers expressed in base b use only the digits {0, 1, 2,..., b 1}.

43 Examples Perform the following conversions. Use your i>clicker to submit your responses. 5. Convert 935 to base Convert to base 9. Remember: numbers expressed in base b use only the digits {0, 1, 2,..., b 1}.

44 Examples Perform the following conversions. Use your i>clicker to submit your responses. 5. Convert 935 to base Convert to base Convert to base 7. Remember: numbers expressed in base b use only the digits {0, 1, 2,..., b 1}.

45 Examples Perform the following conversions. Use your i>clicker to submit your responses. 5. Convert 935 to base Convert to base Convert to base Convert to base 4. Remember: numbers expressed in base b use only the digits {0, 1, 2,..., b 1}.

46 Hexadecimal (Base 16) Numbers The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}.

47 Hexadecimal (Base 16) Numbers The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Digit Name Base-10 Value Digit Name Base-10 Value 0 zero 0 8 eight 8 1 one 1 9 nine 9 2 two 2 A alpha 10 3 three 3 B bravo 11 4 four 4 C charlie 12 5 five 5 D dog 13 6 six 6 E echo 14 7 seven 7 F fox 15

48 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}.

49 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal.

50 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Base Dividend Remainder

51 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Base Dividend Remainder

52 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Base Dividend Remainder =E

53 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Base Dividend Remainder =E =F

54 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Base Dividend Remainder =E =F

55 Example The hexadecimal digits are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. Convert to hexadecimal. Thus 3FE3 16 = Base Dividend Remainder =E =F

56 Conversion Between Base 2 and Base 8 Base-10 Base-8 Base

57 Conversion Between Base 2 and Base 16 Base-10 Base-16 Base A B C D E F

58 Example (1 of 4) Convert to binary (base-2).

59 Example (1 of 4) Convert to binary (base-2). 1. Write each octal (base-8) digit as a three-digit binary (base-2) number =

60 Example (1 of 4) Convert to binary (base-2). 1. Write each octal (base-8) digit as a three-digit binary (base-2) number. 2. Group all the binary digits = = =

61 Example (2 of 4) Convert to octal (base-8).

62 Example (2 of 4) Convert to octal (base-8). 1. Starting on the right, make groups of three binary digits = =

63 Example (2 of 4) Convert to octal (base-8). 1. Starting on the right, make groups of three binary digits = = Convert each group of three binary digits into a single octal digit = =

64 Example (3 of 4) Convert F4 16 to binary (base-2).

65 Example (3 of 4) Convert F4 16 to binary (base-2). 1. Write each hexadecimal (base-16) digit as a four-digit binary (base-2) number. F4 16 =

66 Example (3 of 4) Convert F4 16 to binary (base-2). 1. Write each hexadecimal (base-16) digit as a four-digit binary (base-2) number. 2. Group all the binary digits. F4 16 = F4 16 =

67 Example (4 of 4) Convert 37 8 to hexadecimal (base-16).

68 Example (4 of 4) Convert 37 8 to hexadecimal (base-16). 1. Write each octal (base-8) digit as a three-digit binary (base-2) number =

69 Example (4 of 4) Convert 37 8 to hexadecimal (base-16). 1. Write each octal (base-8) digit as a three-digit binary (base-2) number = Group all the binary digits into groups of four digits = =

70 Example (4 of 4) Convert 37 8 to hexadecimal (base-16). 1. Write each octal (base-8) digit as a three-digit binary (base-2) number = Group all the binary digits into groups of four digits = = Convert each group of four binary digits to a hexadecimal digit = 1F 16