Integers, or whole number from elemental mathematics, are the most common and
fundamental numbers used in the computers. It’s represented as
fixed-point numbers, contrast to floating-point numbers in the machine.
Today we are going to learn a whole bunch of way to encode it.

There are mainly two properties to make a integer representation different:

  1. Size, of the number of bits used.
    usually the power of 2. e.g. 8-bit, 16-bit, 32-bit, 64-bit.

  2. Signed or unsigned.
    there are also multiple schemas to encode a signed integers.

We are also gonna use the below terminologies throughout the post:

  • MSB: Most Significant Bit
  • LSB: Least Significant Bit

Prerequisite - printf Recap

We will quickly recap the integers subset of usages of printf.
Basically, we used format specifier to interpolate values into strings:

Format Specifier

%[flags][width][.precision][length]specifier

  • specifier
    • d, i : signed decimal
    • u : unsigned decimal
    • c : char
    • p: pointer addr
    • x / X : lower/upper unsigned hex
  • length
    • l : long (at least 32)
    • ll : long long (at least 64)
    • h : short (usually 16)
    • hh : short short (usually 8)
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using namespace std; 
int main()  {  
  cout << "Size of int = "<< sizeof(int) << endl;  
  cout << "Size of long = " << sizeof(long) << endl;  
  cout << "Size of long long = " << sizeof(long long);  
}  
Output in 32 bit gcc compiler: 4 4 8
Output in 64 bit gcc compiler: 4 8 8

inttypes.h from C99

Also in cppreference.com

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// signed int (d or i)
#define PRId8     "hhd"
#define PRId16    "hd"
#define PRId32    "ld"
#define PRId64    "lld"

// unsigned int (u)
#define PRIu8     "hhd"
#define PRIu16    "hd"
#define PRIu32    "ld"
#define PRIu64    "lld"

// unsigned hex
#define PRIx8     "hhu"
#define PRIx16    "hu"
#define PRIx32    "lu"
#define PRIx64    "llu"

// uintptr_t (64 bit machine word len)
#define PRIxPTR   "llx"

Unsigned Integers

The conversion between unsigned integers and binaries are trivial.
Here, we can represent 8 bits (i.e. a byte) as a hex pair, e.g.
255 == 0xff == 0b11111111.

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#include <stdint.h>    // uintN_t
#include <inttypes.h>  // PRI macros

uint8_t u8 = 255;
printf("0x%02" PRIx8 "\n", u8); // 0xff
printf(  "%"   PRId8 "\n", u8); // 255

Signed Integers

Signed integers are more complicated. We need to cut those bits to halves
to represent both positive and negative integers somehow.

There are four well-known schemas to encode it, according to
signed number representation of wikipedia.

Sign magnitude 原码

It’s also called “sign and magnitude”. From the name we can see how straightforward it is:
it’s basically put one bit (often the MSB) as the sign bit to represent sign and the remaining bits indicating
the magnitude (or absolute value), e.g.

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  binary   | sign-magn |  unsigned 
-----------|-----------|------------
0 000 0000 |    +0     |     0
0 111 1111 |    127    |    127
...
1 000 0000 |    -0     |    128
1 111 1111 |   -127    |    255

It was used in early computer (IBM 7090) and now mainly used in the
significand part in floating-point number

Pros:

  • simple and nature for human

Cons:

  • 2 way to represent zeros (+0 and -0)
  • not as good for machine
    • add/sub/cmp require knowing the sign
      • complicate CPU ALU design; potentially more cycles

Ones’ complement 反码

It form a negative integers by applying a bitwise NOT
i.e. complement of its positive counterparts.

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  binary   |  1s comp  |  unsigned 
-----------|-----------|------------
0000 0000  |     0     |     0
0000 0001  |     1     |     1
...
0111 1111  |    127    |    127
1000 0000  |   -127    |    128
...
1111 1110  |    -1     |    254 
1111 1111  |    -0     |    255

N.B. MSB can still be signified by MSB.

It’s referred to as ones’ complement because the negative can be formed
by subtracting the positive from ones: 1111 1111 (-0)

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  1111 1111       -0
- 0111 1111       127
---------------------
  1000 0000      -127

The benefits of the complement nature is that adding becomes simple,
except we need to do an end-around carry to add resulting carry
back to get the correct result.

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  0111 1111       127
+ 1000 0001      -126
---------------------
1 0000 0000        0
          1       +1     <- add carry "1" back
---------------------
  0000 0001        1

Pros:

  • Arithmetics on machien are fast.

Cons:

  • still 2 zeros!

Twos’ complement 补码

Most of the current architecture adopted this, including x86, MIPS, ARM, etc.
It differed with one’s complement by one.

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  binary   |  2s comp  |  unsigned 
-----------|-----------|------------
0000 0000  |     0     |     0
0000 0001  |     1     |     1
...
0111 1111  |    127    |    127
1000 0000  |   -128    |    128
1000 0001  |   -127    |    129
...
1111 1110  |    -2     |    254 
1111 1111  |    -1     |    255

N.B. MSB can still be signified by MSB.

It’s referred to as twos’ complement because the negative can be formed
by subtracting the positive from 2 ** N (congruent to 0000 0000 (+0)),
where N is the number of bits.

E.g., for a uint8_t, the sum of any number and it’s twos’ complement would
be 256 (1 0000 0000):

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1 0000 0000       256  = 2 ** 8
- 0111 1111       127
---------------------
  1000 0001      -127

Becuase of this, arithmetics becomes really easier, for any number x e.g. 127
we can get its twos’ complement by:

  1. ~x => 1000 0000 bitwise NOT (like ones’ complement)
  2. +1 => 1000 0001 add 1 (the one differed from ones’ complement)

Cons:

  • bad named?

Pros:

  • fast machine arithmatics.
  • only 1 zeros!
  • the minimal negative is -128

Offset binary 移码

It’s also called excess-K (偏移 K) or biased representation, where K is
the biasing value (the new 0), e.g. in excess-128:

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  binary   |  K = 128  |  unsigned 
-----------|-----------|------------
0000 0000  |   -128(-K)|     0
0000 0001  |   -127    |     1
...
0111 1111  |    -1     |    127
1000 0000  |     0     |    128  (K)
1000 0001  |     1     |    129
...
1111 1111  |    127    |    255

It’s now mainly used for the exponent part of floating-point number.

Type Conversion & Printf

This might be a little bit off topic, but I want to note down what I observed
from experimenting. Basically, printf would not perform an implicit type
conversion but merely interpret the bits arrangement of your arguments as you
told it.

  • UB! stands for undefined behaviors
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uint8_t u8 = 0b10000000; // 128
 int8_t s8 = 0b10000000; // -128

printf("%"PRIu8 "\n", u8);          // 128
printf("%"PRId8 "\n", u8);          // 128 (UB! but somehow it's got right)
printf("%"PRId8 "\n", (int8_t)u8);  // -128

printf("%"PRId8 "\n", s8);          // -128
printf("%"PRIu8 "\n", s8);          // 4294967168 (UB!)
printf("%"PRId8 "\n", (uint8_t)s8); // 128

printf("%"PRIxPTR "\n", s8);             // ffffff80
printf("%"PRIxPTR "\n", (uintptr_t)s8);  // ffffffffffffff80

Char & ASCII

Traditionally, char is represented in the computer as 8 bits as well. And
really, ASCII is only defined between 0 and 127 and require 7 bits.
(8-bit Extended ASCII is not quite well popularized and supported.)

It’s more complicated in extension such as Unicode nowadays, but we’ll ignore
it for future posts dedicated for char and string representation.

So how is a char different with a byte?

Well, the answer is whether a char is a signed char (backed by int8_t)
or a unsigned char (backed by uint8_t) is… implementaton-defined.
And most systems made it signed since most types (e.g. int) were signed
by default.

N.B. int is standard-defined to be equivalent to signed int. This is
not the case of char.

That’s why you often see such typedef such as:

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typedef unsigned char Byte_t;
typedef uint8_t byte_t;

to emphysize the nature of byte should be just plain, unsigned, bits.

References