Decimal value..

Question

What is the decimal value of this 64-bit two’s complement number?

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111000₂

_{Just want to see the methodology thanks!}

Answer 1

Just as a review, the reasons for using two-s complement notation are (1) only one representation of 0 [not +0 and -0] and (2) use less hardware [e.g., adders]

Here are the values that may be stored in a 4-bit register::

Decimal Number	Two-s Complement
-8	1000
-7	1001
-6	1010
-5	1011
-4	1100
-3	1101
-2	1110
-1	1111
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111

Numbers from -8 to +7 may be represented (note: since there is only one 0, an extra negative number may be stored)..

To convert from binary to two's complement, take the NOT (the 1's complement) of the number and add 1..

To convert from two's complement to binary, take the NOT (the 1's complement) of the number and add 1.

Yes, doing the same operation saves extra circuitry.

The two's complement of 5 (that is, 0101) is 1010 + 1 or 1011.

The binary value of 1011 is 0100 (i.e., the NOT, or one's complement) + 1 or 0101, which is decimal 5.

This problem:

Binary value in computer:

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111000₂

NOT (one's complement)

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000111₂

Add +1

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00001000₂

Convert to decimal and append the negative sign :

-8

Note that one's complement values consider the MSB (Most Significant Bit) [the left-most bit] and get a correct result (see the 4-bit table above).

Answer 2

I can already see that the number is -8 ... but why?

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111000

convert to hex bytes ff ff ff ff ff ff ff f8

invert hex bytes 00 00 00 00 00 00 00 07

7 + 1 = 8

And 7 plus 1 equals 8, but we were representing a negative number in 2's compliment form -8