Question

3) Convert following decimal to 8-bit signed numbers in hexadecimal, use two’s-complement for signed integer

127d, -20d, -128d, -1d

4) Convert the 16-bit signed numbers to the decimal

C0A3h, 3AECh, 0101 1001 0111b, 1011 0101 1001 0111b

please solve the problems step by step. It would be of great help.

Answer 1

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3) Convert following decimal to 8-bit signed numbers in hexadecimal

Steps for converting decimal to hexadecimal number(unsigned numbers):

1. Start dividing the decimal number by 16.

2. Note down the remainder in decimal and in hexadecimal.

3. Again divide the quotient by 16.

4. Repeat step 2 and 3 until quotient is equal to 0.

5. The hexadecimal value is the sequence of the remainders in hex from the last to first.

HEXADECIMAL	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
DECIMAL	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

a) 127 d=7F in hexadecimal

DIVISION	RESULT	REMAINDER	REMAINDER (in HEX)
127 / 16	7	15	F
7/ 16	0	7	7
ANSWER			7F

Steps for converting signed decimal to hexadecimal

1. First convert the decimal number to two's complement signed binary.

2. Then convert that binary value to hexadecimal.

b) -20d

Step1: Convert 20₁₀ to binary:

Division by 2	Quotient	Remainder	Bit #
20/2	10	0	0
10/2	5	0	1
5/2	2	1	2
2/2	1	0	3
1/2	0	1	4

So 20₁₀ = 010100₂

Fill the value to 16 bits by adding 10 leading zeroes on the left

Thus 20₁₀ = 0000 0000 0001 0100₂

Step2: Convert to 1’s complement by inverting the bits

Result=1111 1111 1110 1011

Step 3: Find 2’s complement by adding 1 to the above result

Result=1111 1111 1110 1011+1=1111 1111 1110 1100

Step 4: Convert this binary value to hexadecimal. For that group the bits into groups of four, then convert each group to its hexadecimal equivalent

1111 1111 1110 1100b=FFEC in hexadecimal

b) -128d

Step1: Convert 128₁₀ to binary:

Division by 2	Quotient	Remainder	Bit #
128/2	64	0	0
64/2	32	0	1
32/2	16	0	2
16/2	8	0	3
8/2	4	0	4
4/2	2	0	5
2/2	1	0	6
1/2	0	1	7

So 128₁₀ = 1000 0000₂

Fill the value to 16 bits by adding 8 leading zeroes on the left.

Thus 128₁₀ = 0000 0000 1000 0000₂

Step2: Convert to 1’s complement by inverting the bits

Result=1111 1111 0111 1111

Step 3: Find 2’s complement by adding 1 to the above result

Result=1111 1111 0111 1111+1=1111 1111 1000 0000

Step 4: Convert this binary value to hexadecimal. For that group the bits into groups of four, then convert each group to its hexadecimal equivalent

1111 1111 1000 0000=FF80 in hexadecimal

d) -1d

Step1: Convert 1₁₀ to binary:

Division by 2	Quotient	Remainder	Bit #
1/2	0	1	0

So 1₁₀ = 1₂

Fill the value to 16 bits by adding 15 leading zeroes on the left

Thus 1₁₀ = 0000 0000 0000 0001₂

Step2: Convert to 1’s complement by inverting the bits

Result=1111 1111 1111 1110

Step 3: Find 2’s complement by adding 1 to the above result

Result=1111 1111 1111 1110+1=1111 1111 1111 1111

Step 4: Convert this binary value to hexadecimal. For that group the bits into groups of four, then convert each group to its hexadecimal equivalent

1111 1111 1111 1111b=FFFF in hexadecimal

4) Convert the 16-bit signed numbers to the decimal

a. C0A3h=49315d

Ch=12d

Ah=10d

C0A3=12*16³+0*16²+10*16¹+3*16⁰

        =12*4096+0+10*16+3*1

       =49152+0+160+3

       =49315

b. 3AECh=15084d

Ch=12d

Eh=14d

Ah=10d

3AEC=3*16³+10*16²+14*16¹+12*16⁰

        =3*4096+10*256+14*16+12*1

       =12288+2560+224+12

       =15084

c.0101 1001 0111b=1431d

0101 1001 0111=(0*2¹¹)+(1*2¹⁰)+(0*2⁹)+(1*2⁸)+(1*2⁷)+(0*2⁶)+(0*2⁵)+(1*2⁴)+(0*2³)+(1*2²)+(1*2¹)+(1*2⁰)

=0+1024+0+256+128+0+0+16+0+4+2+1

=1431

d.

1011 0101 1001 0111b=46487d

1011 0101 1001 0111b=

(1*2¹⁵)+(0*2¹⁴)+(1*2¹³)+(1*2¹²)+(0*2¹¹)+(1*2¹⁰)+(0*2⁹)+(1*2⁸)+(1*2⁷)+(0*2⁶)+(0*2⁵)+(1*2⁴)+(0*2³)+(1*2²)+(1*2¹)+(1*2⁰)

=32768+0+8192+4096+0+1024+0+256+128+0+0+16+0+4+2+1

=46487