Method $\mathbf{1}:$ The Place Value (Positional Weight) Method
In $2$'s complement, the most significant bit (MSB) has a negative weight, while all other bits have their standard positive weights. For an $8$-bit number, the weights from left to right are:
$-128,\ 64,\ 32,\ 16,\ 8,\ 4,\ 2,\ 1$.
Matching our binary number to these weights:
$\mathbf{1} \times (-128) = -128$
$\mathbf{0} \times 64 = 0$
$\mathbf{0} \times 32 = 0$
$\mathbf{1} \times 16 = 16$
$\mathbf{0} \times 8 = 0$
$\mathbf{1} \times 4 = 4$
$\mathbf{1} \times 2 = 2$
$\mathbf{0} \times 1 = 0$
Now, sum the values:
$-128 + 16 + 4 + 2 = \mathbf{-106}$
Method $\mathbf{2}:$ The Invert and Add $\mathbf{1}$ Method (Finding the Magnitude)
Because the leftmost bit is a 1, we know the number is negative. To find out its exact magnitude, we can reverse the $2$'s complement process:
Step $\mathbf{1}:$ Find the $\mathbf{1}$’s complement
Invert every bit (change $0$s to $1$s, and $1$s to $0$s).
Original$: 10010110$
Inverted$: 01101001$
Step $\mathbf{2}:$ Add $\mathbf{1}$
Add $1$ to the inverted result to find the positive magnitude.
$$01101001 + 1 = \mathbf{01101010}$$
Step $\mathbf{3}:$ Convert the result to decimal
Convert the positive binary number $01101010$ to decimal:
$$64 + 32 + 8 + 2 = 106$$
Since we knew from the beginning that the original number was negative, we add the negative sign back, giving us $\mathbf{-106}$.
