Question

2. There are 26 characters in the alphabet we use for writing English. What is the least number of bits needed to give each character a unique bit pattern?
How many bits would we need to distinguish between upper- and lowercase versions of all 26 characters? (12 points)

3. What is the largest positive number one can represent in an 8-bit 2's complement code? Write your result in binary and decimal.

a. What is the greatest magnitude negative number one can represent in an 8-bit 2's complement code? Write your result in binary and
decimal.



b. What is the largest positive number one can represent in n-bit 2's complement code? 


c. What is the greatest magnitude negative number one can represent in n-bit 2's complement code? 


4. Convert the following 2's complement binary numbers to decimal. (9 points)
   a. 1010
   

b. 01011010


            c. 11111110


Answer 1

QUESTION : There are 26 characters in the alphabet we use for writing English. What is the least number of bits needed to give each character a unique bit pattern?

SOLUTION : No. of combinitions that can be represented by n bits is 2^n.

1 bit = 2^1 combinitions = 2, but we need 26 unique combinitions only so 1 bit is not enough.

2 bits= 2^2 combinitions = 4, but we need 26 unique combinitions only so 2 bits are not enough.

3 bit = 2^3 combinitions = 8, but we need 26 unique combinitions only so 3 bits are not enough.

4 bit = 2^4 combinitions = 16, but we need 26 unique combinitions only so 4 bits are not enough.

5 bit = 2^5 combinitions = 32, we can definitely represt 26 unique combinitons .

So from above we can clearly say that the 5 bits are needed to represent 26 characters.

QUESTION : How many bits would we need to distinguish between upper- and lowercase versions of all 26 characters?

SOLUTION : In the previous question we ONLY considered either upper case or ONLY lower case letters are required. If both are required together then total different combinitions are 52. On calculating from the above formula we get the answer as 6 bits .

QUESTION : What is the largest positive number one can represent in an 8-bit 2's complement code? Write your result in binary and decimal.

SOLUTION : Two write a negative number in 2's complement we first write the positive number and then convert it to negative.

Steps to get representation of a negative number:

1) Write the corresponding positive number.

2) Going from right to left invert every bit after the first " 1".

Ex. Represent -2 in 8 bit 2's complement.

2 in 8 bit 2's complement is 00000010

Therefore -2 will be = 11111110

Since out of 8 bits we can use only 7, therefor a total of 2^7 numbers can be represented which is 128 positive numbers , which is 0 to 127.

DECIMAL = 127

BINARY = 01111111

QUESTION :  What is the greatest magnitude negative number one can represent in an 8-bit 2's complement code? Write your result in binary and
decimal.

SOLUTION : Just like we did above, we are left with 128 other combinitions which can all be used to represent 128 negative numbers, which is from -1 to -128.

Therefore the magnitude of the maximum negative number that we can represent is 128.

DECIMAL = -128

BINARY = 10000000

QUESTION : What is the largest positive number one can represent in n-bit 2's complement code?

SOLUTION : From previously solved questions we can deduce that, for n bits:

Total positive numbers = 2^ (n -1) i.e from ) 0 to 2^(n-1) - 1

Therefore, maximum positive number = 2^(n-1) - 1

QUESTION : What is the greatest magnitude negative number one can represent in n-bit 2's complement code?

SOLUTION : From previously solved questions we can deduce that, for n - bits:

We can have a total of 2^(n-1) negative numbers from -1 to - 2^(n-1).

Therefore magnitude of the highest negative number is -2^(n-1).

QUESTION : Convert the following 2's complement binary numbers to decimal. (9 points)
a. 1010 b. 01011010 c.11111110

SOLUTION :

a. Since the first bit is 1, there it is negative. so we will first calculate the two's complement which is =  0110 and it's value is 6. Therefore the original number was -6.

b. Since the first bit is 0, therefore the number must be positive and we can directly convert it from binary to decimal which on calculating we get to be as 90.

c. Since the first bit is 1, there it is negative. so we will first calculate the two's complement which is = 00000010 , which is equal to 2. Therefore the original number was -2.