CS107
Spring 2004
Converting from decimal to base n
Idea: find the largest power of n that can
represent this number
April 2, 2004
More fun with number systems
Binary arithmetic
Procedure:
Examples
100810 = 130135
Check: 1*54 + 3*53 + 1*5 + 3 = 1008
1/5 = 0 rem 1
replace number by quotient
repeat until quotient is 0
Convert 496 to base 8:
8/5 = 1 rem 3
40/5 = 8 rem 0
201/5 = 40 rem 1
remainder = next leftmost digit
Examples (cont.)
1008/5 = 201 rem 3
Convert 1008 to base 5:
divide number by n
496/8 = 62 rem 0
62/8 = 7 rem 6
7/8 = 0 rem 7
49610 = 7608
Check: 7*64 + 6*8 = 448 + 48 = 496
convert from base n to decimal
convert from decimal to base m
Octal to binary: Replace each digit with its 3digit binary equivalent
Example: Convert 332 from base 5 to base 3
3325 = 92, 92 = 101023
4568 = 302, 302 = 100101110
10101001 = 169, 169 = A9
3616 to binary: 0011 0110
101010101 to hex: 1 0101 0101 = 15516
11011010 to hex: 1101 1010 = DC16
368 = 011 110
101010100 = 101 010 100 = 5248
10101010101010 = 10 101 010 101 010 = 252528
Same procedure as decimal arithmetic:
4AC16 to binary: 0100 1000 1010
175348 = 001 111 101 011 100
Arithmetic in other bases
Converting from hex to binary, and
vice versa
Same procedure as octal, only use groups of 4
digits
4568 = 100 101 110
Binary to octal: Group into 3 digits, then convert
each group to octal
Example: Convert 10101001 to hex
Example: Convert 4568 to binary
Converting from octal to binary, and
vice versa
Procedure:
Converting from base n to base m
add up digits in the same position in both numbers
if the sum is less than n-1, digit in position of new
number = sum
otherwise, digit in position of new number = n – sum,
and add 1 to the two digits in the next position over to
the left
3325 + 2045 = 10415
check: 458 + 1038 = 37 + 67 = 104 = 1508
10101010 + 10011010 = 101000100
check: 3325 + 2045 = 92 + 54 = 146 = 10415
458 + 1038 = 1508
Examples:
check: 170 + 154 = 324 = 101000100
CS107
Spring 2004
More fun with number systems
Binary arithmetic
April 2, 2004
Idea: find the largest power of n that can
represent this number
Procedure:
Converting from decimal to base n
divide number by n
remainder = next leftmost digit
replace number by quotient
repeat until quotient is 0
Examples
Convert 1008 to base 5:
1008/5 = 201 rem 3
201/5 = 40 rem 1
40/5 = 8 rem 0
8/5 = 1 rem 3
1/5 = 0 rem 1
100810 = 130135
Check: 1*54 + 3*53 + 1*5 + 3 = 1008
Examples (cont.)
Convert 496 to base 8:
496/8 = 62 rem 0
62/8 = 7 rem 6
7/8 = 0 rem 7
49610 = 7608
Check: 7*64 + 6*8 = 448 + 48 = 496
Converting from base n to base m
Procedure:
convert from base n to decimal
Example: Convert 332 from base 5 to base 3
4568 = 302, 302 = 100101110
Example: Convert 10101001 to hex
3325 = 92, 92 = 101023
Example: Convert 4568 to binary
convert from decimal to base m
10101001 = 169, 169 = A9
Octal to binary: Replace each digit with its 3digit binary equivalent
4568 = 100 101 110
175348 = 001 111 101 011 100
368 = 011 110
Binary to octal: Group into 3 digits, then convert
each group to octal
Converting from octal to binary, and
vice versa
101010100 = 101 010 100 = 5248
10101010101010 = 10 101 010 101 010 = 252528
Same procedure as octal, only use groups of 4
digits
Converting from hex to binary, and
vice versa
4AC16 to binary: 0100 1000 1010
3616 to binary: 0011 0110
101010101 to hex: 1 0101 0101 = 15516
11011010 to hex: 1101 1010 = DC16
Arithmetic in other bases
#
"
!
Same procedure as decimal arithmetic:
add up digits in the same position in both numbers
if the sum is less than n-1, digit in position of new
number = sum
otherwise, digit in position of new number = n – sum,
and add 1 to the two digits in the next position over to
the left
3325 + 2045 = 10415
check: 458 + 1038 = 37 + 67 = 104 = 1508
10101010 + 10011010 = 101000100
)
(
check: 3325 + 2045 = 92 + 54 = 146 = 10415
458 + 1038 = 1508
'
&
%
$
Examples:
check: 170 + 154 = 324 = 101000100