Integer Formats

Sign-Magnitude Representation (Original Binary Data)

To represent a singed decimal number in binary format, it uses the highest bit as sign bit to determine whether the binary number is negative.

$0$ stands for positive number and $1$ means negative number.

One’s Complements (Radix-1 Complements)

• Positive Number: Same as original.
• Negative Number: Inverse all bits except the sign bit of a binary number.

Two’s Complements (Radix Complements)

• Positive Number: Same as original.
• Negative Number: Plus $1$ on one’s complements.

Range of Representation

Suppose the data has $n$ bits.

Both sign-magnitude representation and one’s complements have $+0$ and $-0$, so the range is $[-2^{n-1}-1, 2^{n-1}-1]$

While two’s complements has only one $0$, so the range is $[-2^{n-1}, 2^{n-1}-1]$

In conclusion:

Representation	Range
Sign-Magnitude Representation	$[-2^{n-1}-1, 2^{n-1}-1]$
One’s Complements	$[-2^{n-1}-1, 2^{n-1}-1]$
Two’s Complements	$[-2^{n-1}, 2^{n-1}-1]$

Application of Two’s Complements

When doing subtraction, like calculating $a-b$, it can be converted to calculating $a + (-b)$, where $-b$ is generated by its two’s complements.

Conversion Between Sign-Magnitude Representation and Two’s Complement

If the number is positive, then the sign-magnitude representation and the two’s complement are the same.

If the number is negative, then the MSB must be $1$. The conversion processes are the same:

Reverse all bits except the sign bit, then plus $1$.

For example, a sign-magnitude representation is $1101\;0001$. Its two’s complement is:

$$1101\;0001 \rightarrow 1010\;1110 + 1 \rightarrow 1010\;1111$$

Then, the two’s complement can be converted back to sign-magnitude representation with the same process:

$$1010\;1111 \rightarrow 1101\;0000 + 1 \rightarrow 1101\;0001$$

Specially, the two’s complement of $1000\;0000$ can not use this process. Since the range is $[-128, 127]$, its two’s complement is the same as its sign-magnitude representation, which is $1000\;0000$.