Answer :
A. the binary representation of 8125 is 0b1111111011101.
B. This is the binary representation of -8125 using 16 bits.
C. the result of the binary operation 100112 × 1012 is 0b1010011.
a) To convert decimal number 8125 to binary, we can use the division-by-2 method.
8125 / 2 = 4062 remainder 1
4062 / 2 = 2031 remainder 0
2031 / 2 = 1015 remainder 1
1015 / 2 = 507 remainder 1
507 / 2 = 253 remainder 1
253 / 2 = 126 remainder 1
126 / 2 = 63 remainder 0
63 / 2 = 31 remainder 1
31 / 2 = 15 remainder 1
15 / 2 = 7 remainder 1
7 / 2 = 3 remainder 1
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1
So, the binary representation of 8125 is 0b1111111011101.
b) To find the 16-bit representation of -8125, we can use the 2's complement method. First, we represent 8125 in binary using 15 bits (since 2^14 is the largest power of 2 that is less than 8125):
0b1111111011101
Then, we flip all the bits:
0b0000000100010
Finally, we add 1 to get the 2's complement:
0b0000000100011
This is the binary representation of -8125 using 16 bits.
c) To perform the binary operation 100112 × 1012, we can use the standard multiplication algorithm taught in elementary school.
10011
x 101
-------
10011
00000
10011
-------
1010011
So, the result of the binary operation 100112 × 1012 is 0b1010011.
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