CS473 - Base Conversion Review answers

Number conversions from decimal

Decimal → binary

a)

Old	New	Bit
107	53	1
53	26	1
26	13	0
13	6	1
6	3	0
3	1	1
1	0	1
Answer		01101011

b) One way to convert a negative number is to calculate the binary version of the absolute value and then take the 2's complement (which is found by adding 1 to the 1's complement):

Old	New	Bit
39	19	1
19	9	1
9	4	1
4	2	0
2	1	0
1	0	1
Absolute		00100111
1's complement		11011000
Answer		11011001

c)

Old	New	Bit
58	29	0
29	14	1
14	7	0
7	3	1
3	1	1
1	0	1
Answer		00111010

d) Another way to convert a negative number is to realize that in 8-bit arithmetic, x+256 = x; in this case, -109 + 256 = 147:

Old	New	Bit
147	73	1
73	36	1
36	18	0
18	9	0
9	4	1
4	2	0
2	1	0
1	0	1
Answer		10010011

Decimal → hexademical

e)

Old	New	Radix
1342	83	14
83	5	3
5	0	5
Answer		053e

f) Using a technique similar to b):

Old	New	Radix
11234	702	2
702	43	14
43	2	11
2	0	2
Absolute		2db2
1's complement		d41d
Answer		d41e

g)

Old	New	Radix
8695	543	7
543	33	f
33	2	1
2	0	2
Answer		21f7

h) Using a technique similar to d) (-1234 = -1234 + 65536 = 64302):

Old	New	Radix
64302	4018	14
4018	251	2
251	15	11
15	0	15
Answer		fb2e

Number base conversions to decimal

Binary → decimal

i)

Old	Bit	New
0	1	1
2	0	2
4	0	4
8	1	9
18	0	18
36	1	37
74	0	74
148	1	149
Answer		149 - 256 = -107

j)

Old	Bit	New
0	0	0
0	1	1
2	0	2
4	1	5
10	0	10
20	1	21
42	0	42
84	1	85
Answer		85

k)

Old	Bit	New
0	1	1
2	1	3
6	0	6
12	0	12
24	1	25
50	1	51
102	0	102
204	0	204
Answer		204 - 256 = -52

l)

Old	Bit	New
0	0	0
0	0	0
0	0	0
0	0	0
0	0	0
0	1	1
2	1	3
6	0	6
Answer		6

Hexadecimal → decimal

m)

Old	Radix	New
0	10	10
160	11	171
2736	12	2748
43968	13	43981
Answer		43981 - 65536 = -21554

n)

Old	Radix	New
0	1	1
16	2	18
288	3	291
4656	4	4660
Answer		4660

o)

Old	Radix	New
0	10	10
160	15	175
2800	1	2801
44816	2	44818
Answer		44818 - 65536 = -20718

p)

Old	Radix	New
0	0	0
0	10	10
160	1	161
2576	11	2587
Answer		2587

Binary arithmetic

      111
  01100101
+ 00010011
  --------
  01111000

2. -01110110 = 10001001 + 1 = 10001010

     1
  01101100
+ 10001010
  --------
  11110110

3.

  1111111111101100 (first term * 00000001)
+ 1111111111011000 (first term * 00000010)
  ----------------
  1111111111000100
+ 1111111110110000 (first term * 00000100)
  ----------------
  1111111101110100
+ 1111110110000000 (first term * 00100000)
  ----------------
  1111110011110100

  00101011
+ 11101010 (divisor*00000010, negated)
  --------
  00010101
+ 11110101 (divisor*00000001, negated)
  --------
  00001010 (remainder)

quotient is 00000011