CS 473 HW 1 - Solutions
Sample solutions to cs473 HW1, Fall 2002
1.a. Old Bit New b. Old Bit New
----------- -----------
.25 0 .5 .125 0 .25
.5 1 .0 .25 0 .5
.5 1 .0
Solution is 0.01 Solution is .001
c. Old Bit New d. Old Bit New
----------- --------------
.375 0 .75 .5625 1 .125
.75 1 .5 .125 0 .25
.5 1 .0 .25 0 .5
.5 1 .0
Solution is .011 Solution is .1001
2.a. Old Bit New a. Old Bit New
------------- --------------
0 0 0 1 .5
0 1 .5 .5 1 .75
.5 0 .25 .75 0 .375
.25 1 .625 .375 1 .6875
Solution is .625 Solution is .6875
c. Old Bit New d. Old Bit New
------------- --------------
0 1 .5 0 1 .5
.5 1 .75 .5 1 .75
.75 1 .875 .75 0 .375
.875 0 .4375 .375 0 .1875
Solution is .4375 Solution is .1875
3.a. 40700000 (hex)
0100000001110-0 (binary)
0 10000000 1110-0 (divided into fields)
Sign: 0 means positive
Exponent: 128 - 127 = 1
Significand: 1.111
1
1.111 x 2 = 11.11
Converting integer to decimal.
Old Bit New
-------------
0 1 1
2 1 3
Converting fraction to decimal
Old Bit New
-------------
0 1 .5
.5 1 .75
Solution is 3.75
b. c1560000 (hex)
11000001010101100-0 (binary)
1 10000010 10101100-0 (fields)
Sign: 1 means negative
Exponent: 130 - 127 = 3
Significand: 1.101011
3
1.101011 x 2 = 1101.011
Converting integer to decimal
Old Bit New
-------------
0 1 1
2 1 3
6 0 6
12 1 13
Converting fraction to decimal
Old Bit New
-------------
0 1 .5
.5 1 .75
.375 0 .375
Solution is -13.375
c. 00000000 (hex)
Special case: solution is 0.
d. 3f800000 (hex)
0011111110000-0 (binary)
0 01111111 0000-0 (fields)
Sign: 0 means positive
Exponent: 127 - 127 = 0
Significand: 1.0
This one can be give by inspection from here: 1.0
4.a. 37.25
Convert integer part to binary
Old New Bit
-----------
37 18 1
18 9 0
9 4 1
4 2 0
2 1 0
1 0 1
100101
Convert fraction part to binary
(this is problem 1.a)
0.01
0 5
Significand: 100101.01 x 2 = 1.0010101 x 2
Sign: positive implies 0
Exponent: 01111111 + 101 = 10000100
0 10000100 00101010-0 (fields)
0100 0010 0001 0101 0-0 (binary)
42150000 (hexadecimal)
Solution is 42150000
b. -9.125
Integer: 9 -> 1001 (part of 4.a table)
Fraction: .001 (problem 1.b)
1001.001
0 3
Significand: 1001.001 x 2 = 1.001001 x 2
Sign: negative implies 1
Exponent: 01111111 + 11 = 10000010.
1 10000010 0010010-0 (fields)
1100 0001 0001 0010 0-0 (binary)
c1120000 (hexadecimal)
Solution is c1120000
c. 0.375
Integer: 0
Fraction: .011 (problem 1.c)
0.011
0 -2
Significand: .011 x 2 = 1.1 x 2
Sign: positive implies 0
Exponent: 01111111 - 10 = 01111101
0 01111101 10-0 (fields)
0011 1110 1100 0-0 (binary)
3ec00000 (hex)
Solution is 3ec00000
d. -15.75
Integer:
Old New Bit
-----------
15 7 1
7 3 1
3 1 1
1 0 1
1111
Fraction:
Old Bit New
-----------
.75 1 .5
.5 1 .0
.11
0 3
Significand: 1111.11 x 2 = 1.11111 x 2
Sign: negative implies 1
Exponent: 01111111 + 11 = 10000010
1 10000010 111110-0 (fields)
1100 0001 0111 1100-0 (binary)
c17c0000 (hex)
Solution is c17c0000
5.a. 41100000 / 40400000
41100000 (hex)
0100000100010-0 (binary)
0 10000010 0010-0 (fields)
Sign: 0
Exponent: 10000010
Significand: 1.001
40400000 (hex)
0100000001000-0 (binary)
0 10000000 1000-0 (fields)
Sign: 0
Exponent: 10000000
Significand: 1.1
Subtract exponents:
10000010
-10000000
---------
10
+01111111
---------
10000001
Divide significands:
0.11
-------------
1.1)1.001
0.0
----
1.00
.11
----
0.011
.011
----
.000
Renormalize:
0.11 -> 1.1
10000001
- 1
--------
10000000
Combine:
0 10000000 10-0 (fields)
0100 0000 0100-0 (binary)
40400000 (hex)
Solution is 40400000
b. 41080000 * 40e80000
41080000 (hex)
0100 0001 0000 10000-0 (binary)
0 10000010 00010000-0 (fields)
Sign: 0
Exponent: 10000010
Significand: 1.0001
40e80000 (hex)
0100 0000 1110 1000 0-0 (binary)
0 10000001 11010000-0 (fields)
Sign: 0
Exponent: 10000001
Significand: 1.1101
Add exponents:
10000010
+10000001
---------
100000011
-01111111
---------
10000100
Multiply significands
1.1101
1.0001
------
1.1101
11101
-----------
1.11101101
(Result is already normalized)
Combine:
0 10000100 11101101 (fields)
0100 0010 0111 0110 1 (binary)
42768000 (hex)
Solution is 42768000
c. 3fe00000 + 3f000000
3fe00000 (hex)
0011 1111 1110 0-0 (binary)
0 01111111 1100-0 (fields)
Sign: 0
Exponent: 01111111
Significand: 1.11
3f000000 (hex)
0011 1111 0-0 (binary)
0 01111110 0-0 (fields)
Sign: 0
Exponent: 01111110
Significand: 1.0
Adjust exponents: shift second operand
Exponent: 01111111
Significand: 0.1
Add significands
1.11
+0.1
-----
10.01
Normalize:
1.001
Exponent: 01111111 + 1 = 10000000
Combine:
0 10000000 001 0-0 (fields)
0100 0000 0001 0-0 (binary)
40100000 (hex)
Solution is 40100000
d. 40700000 - 40200000
40700000 (hex)
0100 0000 0111 000 (binary)
0 10000000 111000 (fields)
Sign: 0
Exponent: 10000000
Significand: 1.111
40200000 (hex)
0100 0000 0010 0000 0-0 (binary)
0 10000000 0100000 0-0 (fields)
Sign: 0
Exponent: 10000000
Significand: 1.01
Exponents are already aligned.
Subtract significands:
1.111
1.01
-----
0.101
Renormalize
0.101 -> 1.01
Exponent: 10000000 - 1 = 01111111
Combine:
0 01111111 010-0 (fields)
0011 1111 1010-0 (binary)
3fa00000 (hex)
Solution is 3fa00000
Last modified: Tue Oct 15 12:19:07 MDT 2002