1.2b Signed Integers

Signed Integer

A signed integer is a data type in computer science and mathematics that can represent both positive and negative whole numbers by assigning $1$ or $0$ to the leftmost bit to indicate the sign of the integer. Signed integers can therefore represent numbers between $-2^{N-1}-1$ and $2^{N-1}-1$ inclusive, where $N$ represents the number of bits representing the integer.

Intuition

Let’s start with formulating the range of signed integers. We previously said that unsigned integers can represent numbers in the range $[0,2^N-1]$. However, when using signed integers, two issues arise:

1. There is a -0 and +0! We can see this with a quick example: as the leftmost bit is allocated to the number sign, 0b1000 and 0b0000 are the same number!
2. Similarly, because the leftmost bit takes up space, we can only utilize $N-1$ bits for our signed integer.
   Combining the two facts, we get our desired range: $[-2^{N-1}-1,2^{N-1}-1]$.

We also arise into something else here: adding a positive and a negative number always leads to a negative number!