# Bits, Bytes, and Integers

## Integers

### Representation

### Conversion

**Mapping Between `signed` and `unsigned`**

Maintain bit pattern

so the conversion of negative value of the signed or value over the T_MAX results different interpreted value.

### Expanding, Truncating

**expanding of $w$-bit signed integer**

$X = x_{w-1}x_{w-2}\cdots x_0$
where $s = x_{w-1}$

expanding it to $w+k$-bit signed int:

$X = s_{k}s_{k-1} \cdots s_1 x_{w-1}\cdots x_0,\quad \text{where}\, s_i = s$

**truncation of $w$-bit signed integer**

first $k$ bit change

## C Puzzle

```c

int x = foo();
int y = bar();

unsigned ux = x;
unsigned uy = y;
```

* `x < 0` -> `((x*2) < 0)`
  * `false`: underflow
* `ux >= 0`
  * it's converted to `ux >= 0u` so `true` for all ux
* `x & 7 == 7` -> `(x << 30) < 0`
  * `true`
* `ux > -1`
  * `always false`, `-1 = uint_max`
* `x > y` -> `-x < -y`
  * `false`
* `x * x >= 0`
  * `false`
* `x > 0 && y > 0` -> `x + y > 0`
* `x >= 0` -> `-x <= 0`
* `x <= 0` -> `-x >= 0`
* `(x|-x) >> 31 == -1`
* `ux >> 3 == ux/8`
* `x >> 3 == x/8`
* `x & (x-1) != 0`
  * `true`