Question

Computers typically do not implement integer subtraction directly, but instead using addition and negation.

To accomplish negation, they often use the formula −x = (x ˆ 11 . . . 1) + 1, which can be 3 implemented using bitwise XOR and addition.

The string 11 . . . 1 here represents the all 1’s string of length n, where n is the width of the CPU.

The addition is done modulo 2n. Verify that −x ≡ (x ˆ 11 . . . 1) + 1 (mod 2n) for n = 4 and x = 3. Show your work

Answer 1

Let

for x = 3

-x = -3

Finding -x using negation and addition:-

given, n=4 and x = 3

representing 3 in binary 11

Since, the width of CPU is 4 ( n = 4 )

So 3 can be represented as 0011 (width is 4 by adding 2 zeros before rightmost bit does not change the notation of 3)

0011 is same as 11

the negation of 0011 is 1100 [ negation is represented as '~' , where negation means changing 0's & 1's as 1's & 0's]

and addition of 1 gives:

~0011

1100 [representing 1 as 0 & 0 as 1]

+ 1 [a simple addition]

result= 1101

Therefore, -3 is represented as 1101 using negation and addition

Finding -x using the given formula for negation:-

-x = (x^11..1) + 1

where ^ represents bitwise xor operation

and the string 11..1 here represents the all 1's string of length n, where n is the width of the CPU and also the addition is done by using modulo 2n.

finally, the formula becomes

-x = (x^11..1) + 1 (mod 2n.)

given, n=4 and x = 3

Right hand side (R.H.S) :-

x = 3 -> in binary 3 = 11

doing xor with 1111 (since n =4 , string contains 4 1's )

11 same as 0011 (preceding zeros doesn't change the number)

Truth table for xor operation:-

input 1   input 2   output
0   0   0
0   1   1
1   0   1
1   1   1
0011

^ 1111

1100 (from the truth table of xor)

+ 1 [ 1 (mod 2n.) = 1 (mod 24 ) { given, n=4}= 1 (mod 16 ) = 1 ]

result = 1101

from the formula, result = 1101

Left hand side(L.H.S) -x = -3

Since, from the result of using negation and addition -3 is equal to 1101

By equating L.H.S = R.H.S

Therefore,

-3 = 1101
Hence verified that,

-x = (x^11..1) + 1 (mod 2n.)
where the string 11..1 represents the string 11..1 of length n,

and n is the width of the CPU.