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+# 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`