19.5. Signed and Unsigned Integral Types

19.5. Signed and Unsigned Integral Types
	Chapter 19. Types and Expressions

[ fromfile: twoscomplement.xml id: twoscomplement ]

This section explains the differences between signed and unsigned integral types.

The underlying binary representation of an object x of any integral type looks like this (assuming n-bit storage):
```
d_n-1d_n-2...d₂d₁d₀
   
```
where each d_i is either 0 or 1.
The computation of the decimal equivalent value of x depends on whether x is an unsigned or signed type.

If x is unsigned, the decimal equivalent value is

     d_n-1*2^n-1 + d_n-2*2^n-2 +...+ d₂*2² + d₁*2¹ + d₀*2⁰

The largest (positive) value that can be expressed by an unsigned integer is, therefore,
```
2ⁿ - 1 = 1*2^n-1 + 1*2^n-2 +...+ 1*2² + 1*2¹ + 1*2⁰
   
```

If x is signed, the decimal equivalent value is

     d_n-1*-(2^n-1) + d_n-2*2^n-2 +...+ d₂*2² + d₁*2¹ + d₀*2⁰

The largest (positive) value that can be expressed by a signed integer is
```
2^n-1 - 1 = 0*-(2^n-1) + 1*2^n-2 +...+ 1*2² + 1*2¹ + 1*2⁰
   
```
This is called "two's complement" representation.
To determine the representation of the negative of a signed integer,
1. Compute the "one's complement" of the number (i.e., replace each bit with its complement).
2. Add 1 to the "one's complement" produced in the first step.

Suppose that you have a tiny system that uses only 8 bits to represent a number. On this system, the largest unsigned integer would be

11111111 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255

But that same number, interpreted as a signed integer, would be

11111111 = -128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = -1


19.4. Enumerations		19.5.1. Exercises: Signed and Unsigned Integral Types