Question

1a. What is the largest negative number that you could represent with 7 bits using signed magnitude? Show your work.

1b. What is the largest positive number that you could represent with 7 bits using 2’s complement? Show your work.

1c. Add the following two binary numbers 10110111 + 1011 = :Show your work.

1d. Solve the following decimal notation equation using 8-bit binary numbers and 2’s complement notation: 69 - 7 = :Show your work

Answer 1

Solution:-

1a The range of integers in sign magnitude form is:- - (2 ^(n-1) -1) to + (2^(n-1) -1)

Therefore the largest negative number that can be represented in sign magnitude with 7 bits is :-

Take n = 7, therefore - (2^(7-1)-1) = - ( 64-1) = -63.

1b. The range of integers in sign magnitude form is:- - (2 ^(n-1) ) to + (2^(n-1) -1)

Therefore take n = 7.

So largest positive integer that can be represented in 2's complement using 7 bits is = + (2^(7-1) -1 ) = 63.

1c. To add the 2 binary bits, we must know the following:-

a. 1 + 0 = 1 with carry 0

b. 0 + 0 = 0 with carry 0

c. 0 + 1 = 1 with carry 0

d. 1 + 1 =0 with carry forward as 1.

e. 1 + 1 + 1 = 1 with carry as 1

f. 1 + 1 + 0 = 0 with carry as 1.

Our binary numbers are:- A= 10110111 and  B= 1011.

Here number B is of lesser length than A. Therefore we append 0's in front of B to make it of equal length as of A. Appending 0's will not cause any effect on the number.

updated numbers are :- A= 10110111 and B = 00001011

Now when we add the 2 binary numbers, we start addition of each bit pair from right to left by traversing.

For example- Take last bits of each of A and B which are 1 and 1. Therefore 1 + 1 = 0 with carry=1.

So this carry will be forward towards one distance left side.

ADDITION:-

A = 1 0 1 1 0 1 1 1

B = 0 0 0 0 1 0 1 1

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C = 1 1 0 0 0 0 1 0 - Result/Sum

1d. We are given a substraction of 69 - 7 and we have to show it in binary substraction and use 2's complement notation in it.

Lets convert 69 into the binary format of 8 bits.

Idea - divide 69 by 2 and you will get a remainder, save it. Then divide the quotient we got while dividng 69 by 2, by 2 and save the remainder and repeat this until we are not able to divide.

Therefore binary representation of 69 = 0 1 0 0 0 1 0 1 (8bits)

Binary representation of 7 = 0 0 0 0 0 1 1 1 (8 bits).

Now we have to subtract 7 from 69 in binary.

Therefore we know that :- X - Y = X + (-Y)

Therefore we have to find the 2's complement of 7 which is = 1's complement of binary 7 + 1

Now 1's complement of binary 7 is = invert each of the bits of binary 7, therefore 1's complement of binary 7 is= 1 1 1 1 1 0 0 0

Lets now add binary representation of 69 and 2's complement of binary 7.

Therefore , using the previous addition concept we discussed above,

69 - 7 = 0 1 0 0 0 1 0 1 + 1 1 1 1 1 0 0 0

0 1 0 0 0 1 0 1

1 1 1 1 1 0 0 0

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0 0 1 1 1 1 0 1 (leftmost carry will be discarded). -Result.

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