Question

2) Show the decimal equivalent of each of the numbers if they are interpreted as (4 answers): 11001101 01101001 a. Unsigned binary b. Signed binary

Answer 1

Answer:

Signed: -4 -5 +6 -1

Unsigned: 12 13 6 9

Explanation:

We are given with binary number i.e. 1100110101101001. First of all we will break this binary number into sets of 4 starting from the right side of the binary number. First set will be 1001, second will be 0110, third wil be 1101, fourth will be 1100.

Basic concept of converting binary numbers into decimal numbers:

256 128 64 32 16 8  4  2  1  

0      1      1    0    1  0  0  1  1

Add the number written above each of the binary number if its 1 and ignore if its 0. Starting from the left side 0 represents 256 so, we will ignore it. 1 represents 128 so we will consider it and so on.

128+64+16+2+1 = 211

011010011 is the binary of 211.

For signed binary, if the last number of the set is 0 the, it is a postive number. For unsigned binary, if the last number of the set is negative then, it is a negative number.

For signed binary:

1100    1101    0110   1001

-4        -5      +6       -1

Note:

We are not adding these numbers because in the question it is specified to give 4 answers.

For unsigned binary:

1100    1101    0110   1001

12        13      6       9

Note:

We are not adding these numbers because in the question it is specified to give 4 answers.

Answer 2

Answer:

11001101 (as unsigned binary) is 205 in decimal

11001101 (as signed binary) is -51 in decimal

01101001 (as unsigned binary) is 105 in decimal

01101001 (as signed binary) is still 105 in decimal.

Explanation:

a.  11001101

1 => Treating as unsigned binary

Since the number is unsigned, the usual direct conversion to decimal is sufficient. i.e

11001101 = 1 x $2^{7}$ + 1 x $2^{6}$ + 0 x $2^{5}$ + 0 x $2^{4}$ + 1 x $2^{3}$ + 1 x $2^{2}$ + 0 x $2^{1}$ + 1 x $2^{0}$

11001101 = 128 + 64 + 0 + 0 + 8 + 4 + 0 + 1

11001101 = 205 (in decimal)

Therefore 11001101 (as unsigned binary) is 205 in decimal

2 => Treating as signed binary

Signed binary using 2's complements dictates that the most significant bit (leftmost bit) in a binary number represents the sign of the bit. If the most significant bit is 0, then the number is positive. If it is 1, the number is negative.

Since the most significant bit of the number (11001101) is 1, then the number is negative.

Therefore to convert it to its decimal counterpart;

i. flip all its bits by changing all 1s to 0s and all 0s to 1s as follows

=> 11001101 = 00110010

ii. add 1 to the result above as follows

=> 00110010 + 1 = 00110011

iii. now convert the result to the decimal representation as follows

00110011 = 0 x $2^{7}$ + 0 x $2^{6}$ + 1 x $2^{5}$ + 1 x $2^{4}$ + 0 x $2^{3}$ + 0 x $2^{2}$ + 1 x $2^{1}$ + 1 x $2^{0}$

00110011 = 0 + 0 + 32 + 16 + 0 + 0 + 2 + 1

00110011 = 51 (in decimal)

Therefore, 11001101 (as signed binary) is -51 in decimal

b.  01101001

1 => Treating as unsigned binary

Since the number is unsigned, the usual direct conversion to decimal is sufficient. i.e

01101001 = 0 x $2^{7}$ + 1 x $2^{6}$ + 1 x $2^{5}$ + 0 x $2^{4}$ + 1 x $2^{3}$ + 0 x $2^{2}$ + 0 x $2^{1}$ + 1 x $2^{0}$

01101001 = 0 + 64 + 32 + 0 + 8 + 0 + 0 + 1

01101001 = 105 (in decimal)

Therefore 01101001 (as unsigned binary) is 105 in decimal

2 => Treating as signed binary

Signed binary using 2's complements dictates that the most significant bit (leftmost bit) in a binary number represents the sign of the bit. If the most significant bit is 0, then the number is positive. If it is 1, the number is negative.

Since the most significant bit of the number (01101001) is 0, then the number is positive and the usual conversion to decimal will suffice. i.e

01101001 = 0 x $2^{7}$ + 1 x $2^{6}$ + 1 x $2^{5}$ + 0 x $2^{4}$ + 1 x $2^{3}$ + 0 x $2^{2}$ + 0 x $2^{1}$ + 1 x $2^{0}$

01101001 = 0 + 64 + 32 + 0 + 8 + 0 + 0 + 1

01101001 = 105 (in decimal)

Therefore 01101001 (as signed binary) is still 105 in decimal.

Note: A positive binary number will have the same value (in decimal) whether it is treated as signed or unsigned.