Question

Add the following 2's complement binary numbers. Also express the answer in decimal. a. 01+ 1011b. 11+ 01010101c. 0101+ 110d. 01+10

Answer 1

Answer:

(a) 01 + 1011

Convert the two numbers into their 4-bit representation.

=> 01 = 0001

=> 1011

Addition of the two numbers (in their 2's complement form) gives the following:

0 0 0 1       ------------------> + 1 (in decimal)

1  0 1  1      ------------------> - 5 (in decimal)

1  1  0 0    -------------------> - 4 (in decimal)

Explanation for (a)

01 or 0001

Since the most significant bit is 0 (leftmost bit), it shows that it is a positive number. Therefore, it can be directly converted to its decimal representation as follows:

0001 = 0 x $2^{3}$ + 0 x $2^{2}$ + 0 x $2^{1}$ + 1 x $2^{0}$ = 1

0001 is + 1 in decimal.

1011

Its most significant bit(MSB) is 1 showing that it is negative.

Flipping all its bits and adding 1 to the result, will convert it to it's positive counterpart.

i.e 1011 => 0100 + 1 = 0101.

Converting the result (0101) to decimal gives

0101 = 0 x $2^{3}$ + 1 x $2^{2}$ + 0 x $2^{1}$ + 1 x $2^{0}$ = 5

1011 is -5 in decimal

01 + 1011 or 0001+ 1011 = 1100

The result (1100), of the addition of the two numbers, is a negative number as the MSB is 1.

Convert it to it's positive counterpart.

i.e 1100 => 0011 + 1 = 0100.

Converting the result (0100) to decimal gives

0100 = 0 x $2^{3}$ + 1 x $2^{2}$ + 0 x $2^{1}$ + 0 x $2^{0}$ = 4

Therefore, the result, 1100 is -4 in decimal.

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(b) 11 + 01010101

Convert the two numbers into their 8-bit representation since one of them is in 8-bit.

PS: Taking 11, its MSB is 1 showing that it is a negative number. Also, it's 8-bit representation will mean pre-padding it with ones(1s) rather than zeros as follows. In other words, pre-padding a 2's complement  number is done using its sign bit.

=> 11 = 11111111

=> 01010101

Addition of the two numbers (in their 2's complement form) gives the following:

  1  1  1  1  1  1  1  1       ------------------> - 1 (in decimal)

  0 1  0 1  0  1 0 1      ------------------> + 85 (in decimal)

1 0 1  0 1  0  1 0 0    ------------------->  + 84 (in decimal)

Discard the carry-out bit (1) making the result 01010100

Explanation for (b)

11 or 11111111

Its MSB is 1 showing that it is negative.

Convert it to it's positive counterpart.

11111111 => 00000000 + 1 = 00000001

Converting the result (00000001) to decimal gives 1

11 or 11111111 is  - 1 in decimal.

01010101

Its MSB is 0 showing that it is also positive. Now, convert to decimal

01010101 = 85

01010101 is + 85 in decimal.

11111111 + 01010101 = 01010100

The result (01010100), is also positive as the MSB is 0. Now, convert to decimal.

01010100 = 84

Therefore the result (01010100) is + 84 in decimal.

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(c) 0101 + 110

Convert the two numbers into their 4-bit representations.

=> 0101 = 0101

=> 110 = 1110

Addition of the two numbers (in their 2's complement form) gives the following:

     0 1 0 1       ------------------> + 5 (in decimal)

     1  1 1  0      -----------------> - 2 (in decimal)

  1 0 0 1  1    -------------------> + 3 (in decimal)

Dicard the carry-out bit (leftmost bit) to get 0011 as result.

Explanation for (c)

0101

Since the MSB is 0 (leftmost bit), it shows that it is a positive number. Now, convert to decimal.

0101 =  5

0101 is +5 in decimal.

110 or 1110

Its MSB is 1 showing that it is negative.

Convert it to it's positive counterpart.

i.e 1110 => 0001 + 1 = 0010.

Converting the result (0010) to decimal gives

0010 =  2

1110 is -2 in decimal

0101 + 110 = 0101+ 1110 = 0011

The result (0011), is a positive number as the MSB is 0.

Converting the result (0011) to decimal gives

0011 = 3

Therefore, the result, 0011 is 3 in decimal.

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(d) 01 + 10

Convert the two numbers into their 4-bit representation.

=> 01 = 0001

=> 10 = 1110

Addition of the two numbers (in their 2's complement form) gives the following:

0 0 0 1       ------------------> + 1 (in decimal)

1  1  1  0      ------------------> - 2 (in decimal)

1  1  1  1    -------------------> - 1 (in decimal)

Explanation for (d)

01 or 0001

Since the MSB is 0 (leftmost bit), it shows that it is a positive number. Now convert to decimal.

0001 = 1

0001 is + 1 in decimal.

1110

Its MSB is 1 showing that it is negative.

Convert it to it's positive counterpart.

i.e 1110 => 0001 + 1 = 0010.

Converting the result (0010) to decimal gives

0010  = 2

1110 is -2 in decimal

01 + 1011 or 0001+ 1110 = 1111

The result (1111), is a negative number as the MSB is 1.

Convert it to it's positive counterpart.

i.e 1111 => 0000 + 1 = 0001.

Converting the result (0001) to decimal gives

0001 = 1

Therefore, the result, 1111 is -1 in decimal.