Determine the decimal values of the following 1's complement numbers:

[shamieh]

Determine the decimal values of the following 1's complement numbers:

So i understand that if the left most bit number is a 1 it is a negative, and if it is a 0 it is poisitive. But my question is why do they start out with -511 when \(\displaystyle 2^9\) is obviously -512. Why are they adding 1 to it initially?

1011100111 = -511 + 128 + 64 + 32 + 4 + 2 + 1 = -280

 

[shamieh]

After looking at it some more, it looks like I would say -511 + [numbers here] and I would + 1 initially, I guess that 's what "1's complacent means", similarly when it 's "2's complacent" it looks like I would not +1, but would still have a negative number if my left most bit began with a 1.

Decimal values (1’s complement)
(a) 0111011110 = 256 + 128 + 64 + 16 + 8 + 4 + 2 = 478
(b) 1011100111 = -511 + 128 + 64 + 32 + 4 + 2 + 1 = -280

Decimal values (2’s complement)
(a) 0111011110 = 256 + 128 + 64 + 16 + 8 + 4 + 2 = 478
(b) 1011100111 = -512 + 128 + 64 + 32 + 4 + 2 + 1 = -281

 

[MarkFL]

The way I look at one's complement is to first find the unsigned value:

\(\displaystyle (1011100111)_2=1+2+4+32+64+128+512=743\)

Now count the number of binary digits, which is 10, and so subtract $2^{10}-1=1023$:

\(\displaystyle 743-1023=-280\)

 

[shamieh]

Wow that way is so much easier then what the book teaches.

 

[LudusRex]

1100011001 = -511 + 256 + 128 + 2 + 1 = -124

I would like to clarify that the 1's complement representation is a way to represent negative numbers in binary form. In this representation, the leftmost bit is used as a sign bit, with 0 representing a positive number and 1 representing a negative number. The remaining bits represent the magnitude of the number.

Now, to address your question about why the numbers start with -511 instead of -512, it is important to understand that in 1's complement representation, the range of numbers that can be represented is from -(2^n - 1) to (2^n - 1) where n is the number of bits. So, for a 10-bit number, the range would be from -511 to 511.

In the examples given, we can see that the first bit is always 1, indicating a negative number. So, to find the decimal value, we need to subtract 2^n from the binary number. In this case, 2^9 is subtracted from the binary number, hence we get -511 as the starting value.

I hope this explanation helps clarify your doubts. As a scientist, it is important to understand the fundamental principles and reasoning behind mathematical representations, rather than just memorizing formulas or patterns.