High School

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a) Find the 15-bit binary number of 8125.

b) Hence, find the 16-bit representation of -8125.

c) Perform the binary operation \(10011_2 \times 101_2\).

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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