Bits, Bytes, and Integers

In computers, everything consists of bits. By encoding sets of bits in various ways, they made meanings:

instructions
and data(numbers, sets, strings, etc..)

Boolean Algebra

and A & B
or A | B
not ~A
xor A ^ B

in C

Shift (<<, >>)
- Left Shift(<<) Zero fill on right
- Right Shift(>>) Logical Shift: zero fill with 0's on left Arithmetic shift Relicate most significant bit on left

#include <stdio.h>

int main() {
  int a = 0x7fffffff;
  int as = a << 1;
  printf("shl of %d: %d(%08x)\n", a, as, as);
  

  unsigned b = 0x7fffffff;
  unsigned bs = b << 1;
  printf("shl of %u: %u(%08x)\n", b, bs, bs);
}

while ! [ -f 1_1.out ]; do sleep .1; done; ./1_1.out

Integers

Representation & Encoding

for w bits data x

B2U(X)=\sum_{i=0}^{w-1} x_i 2^{i}\quad B2T(X)=-x_{w-1}*2^{w-1} + \sum_{i=0}^{w-2}{x_i 2^i}

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


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

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