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Homework Solutions
Problem 1.a.
379 / 2 = 189 rem 1
= 94 rem 1
= 47 rem 0
= 23 rem 1
= 11 rem 1
= 5 rem 1
= 2 rem 1
= 1 rem 0
= 0 rem 1
Listing remainders from last to first gives, 101111011
Padded to 16 bits, 0000000101111011
Problem 1.b.
0000000101111011
==> 0000 0001 0111 1011
==> 017B
Problem 1.c.
Because the value is positive and because it requires less than 16 bits to represent,
the result is the same as in b.
Problem 1.d.
-476. Sign bit = 1 for a negative number. Magnitude:
476 / 2 = 238 rem 0
= 119 rem 0
= 59 rem 1
= 29 rem 1
= 14 rem 1
= 7 rem 0
= 3 rem 1
= 1 rem 1
=0 rem 1
111011100 for magnitude.
1 | padding | 111011100
==> 1000000111011100
==> 1000 0001 1101 1100
==> 81DC
Problem 1.e.
Because the value is positive and because it requires less than 16 bits to represent,
the result is the same as in b.
Problem 1.f.
2's-complement of -476:
magnitude 476: 111011100
pad to 16 bits: 0000000111011100
negative, so complement: 1111111000100011
then add 1: 1111111000100100
1111 1110 0010 0100
==> FE24
Problem 1.g.
14 + 127 = 141
So, the value +14 encodes as 141 in offset-127
(Note: offset-127 encoding uses an 8-bit result)
141 ==> 10001101 ==> 1000 1101 ==> 8D
Problem 2.a.
the unsigned binary integer 000012AF
0001 0010 1010 1111 ==> 1001010101111
==> 4096 + 512 + 128 + 32 + 8 + 4 + 2 + 1
= 4783
Problem 2.b.
the sign-magnitude binary integer 80001876
1000 0000 0000 0000 0001 1000 0111 0110
The leading 1 tells us that it's a 31-bit negative number with magnitude:
000 0000 0000 0000 0001 1000 0111 0110
1100001110110
==> 4096 + 2048 + 64 + 32 + 16 + 4 + 2 = 6262
So the value represented is -6262
Problem 2.c.
the 2's-complement binary integer FFFFFFFF
1111 1111 1111 1111 1111 1111 1111 1111
The leading 1 tells us that it's a 31-bit negative number, so complement and add 1:
complement: 0000 0000 0000 0000 0000 0000 0000 0000
add 1: 0000 0000 0000 0000 0000 0000 0000 0001
So the number represented is -1.
Problem 3.
To compute the sum, first convert to binary:
ABCD ==> 1010 1011 1100 1101
1234 ==> 0001 0010 0011 0100
Then add the bits:
1010 1011 1100 1101
+ 0001 0010 0011 0100
---------------------
1011 1110 0000 0001
Now convert back to hexadecimal:
1011 1110 0000 0001
B E 0 1
Thus, ABCD + 1234 = BE01 in hexadecimal.
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