# Arithmetic types

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(See also type for type system overview and the list of type-related utilities that are provided by the C library)

### Boolean type

  - _Bool (also accessible as the macro bool) - type, capable of holding one of the two values: 1 and 0 (also accessible as the macros true and false).

Note that conversion to _Bool does not work the same as conversion to other integer types: (bool)0.5 evaluates to 1, whereas (int)0.5 evaluates to ​0​.

(since C99)

### Character types

  - signed char - type for signed character representation.
  - unsigned char - type for unsigned character representation. Also used to inspect object representations (raw memory).
  - char - type for character representation. Equivalent to either signed char or unsigned char (which one is implementation-defined and may be controlled by a compiler commandline switch), but char is a distinct type, different from both signed char both unsigned char

Note that the standard library also defines typedef names wchar_t , char16_t, and char32_t (since C11) to represent wide characters.

### Integer types

  - short int (also accessible as short, may use the keyword signed)
  - unsigned short int (also accessible as unsigned short)
  - int (also accessible as signed int)
    This is the most optimal integer type for the platform, and is guaranteed to be at least 16 bits. Most current systems use 32 bits (see Data models below).
  - unsigned int (also accessible as unsigned), the unsigned counterpart of int, implementing modulo arithmetic. Suitable for bit manipulations.
  - long int (also accessible as long)
  - unsigned long int (also accessible as unsigned long)

  - long long int (also accessible as long long)
  - unsigned long long int (also accessible as unsigned long long)

(since C99)

Note: as with all type specifiers, any order is permitted: unsigned long long int and long int unsigned long name the same type.

The following table summarizes all available integer types and their properties:

Type specifier | Equivalent type | Width in bits by data model
--- | --- | ---
C standard | LP32 | ILP32 | LLP64 | LP64
short | short int | at least 16 | 16 | 16 | 16 | 16
short int
signed short
signed short int
unsigned short | unsigned short int
unsigned short int
int | int | at least 16 | 16 | 32 | 32 | 32
signed
signed int
unsigned | unsigned int
unsigned int
long | long int | at least 32 | 32 | 32 | 32 | 64
long int
signed long
signed long int
unsigned long | unsigned long int
unsigned long int
long long | long long int (C99) | at least 64 | 64 | 64 | 64 | 64
long long int
signed long long
signed long long int
unsigned long long | unsigned long long int (C99)
unsigned long long int

Besides the minimal bit counts, the C Standard guarantees that

  1 == sizeof(char) <= sizeof(short) <= sizeof(int) <= sizeof(long) <= sizeof(long long).

Note: this allows the extreme case in which bytes are sized 64 bits, all types (including char) are 64 bits wide, and sizeof returns 1 for every type.

Note: integer arithmetic is defined differently for the signed and unsigned integer types. See arithmetic operators, in particular integer overflows.

#### Data models

The choices made by each implementation about the sizes of the fundamental types are collectively known as data model. Four data models found wide acceptance:

32 bit systems:

  - LP32 or 2/4/4 (int is 16-bit, long and pointer are 32-bit)
    - Win16 API
  - ILP32 or 4/4/4 (int, long, and pointer are 32-bit);
    - Win32 API
    - Unix and Unix-like systems (Linux, Mac OS X)

64 bit systems:

  - LLP64 or 4/4/8 (int and long are 32-bit, pointer is 64-bit)
    - Win64 API
  - LP64 or 4/8/8 (int is 32-bit, long and pointer are 64-bit)
    - Unix and Unix-like systems (Linux, Mac OS X)

Other models are very rare. For example, ILP64 (8/8/8: int, long, and pointer are 64-bit) only appeared in some early 64-bit Unix systems (e.g. Unicos on Cray).

Note that exact-width integer types are available in <stdint.h> since C99

### Real floating types

C has three types for representing real floating-point values:

  - float - single precision floating point type. Matches IEEE-754 32 bit floating point type if supported.
  - double - double precision floating point type. Matches IEEE-754 64 bit floating point type if supported
  - long double - extended precision floating point type. Matches IEEE-754 extended floating-point type if supported, otherwise matches some non-standard extended floating-point type as long as its precision is better than double and range is at least as good as double, otherwise matches the type double. Some x86 and x86_64 implementations use the 80-bit x87 floating point type.

Floating-point types may support special values:

- infinity (positive and negative), see `INFINITY`
- the negative zero, -0.0. It compares equal to the positive zero, but is meaningful in some arithmetic operations, e.g. 1.0/0.0 == INFINITY, but 1.0/-0.0 == -INFINITY)
- not-a-number (NaN), which does not compare equal with anything (including itself). Multiple bit patterns represent NaNs, see nan, `NAN`. Note that C takes no special notice of signalling NaNs (specified by IEEE-754), and treates all NaNs as quiet.

Real floating-point numbers may be used with arithmetic operators + - / * and various mathematical functions from math.h. Both built-in operators and library functions may raise floating-point exceptions and set errno as described in `math_errhandling`

Floating-point expressions may have greater range and precision than indicated by their types, see FLT_EVAL_METHOD. Assignment, return, and cast force the range and precision to the one associated with the declared type.

Floating-point expressions may also be contracted, that is, calculated as if all intermediate values have infinite range and precision, see #pragma STDC FP_CONTRACT.

Some operations on floating-point numbers are affected by and modify the state of the floating-point environment (most notably, the rounding direction)

Implicit conversions are defined between real floating types and integer, complex, and imaginary types.

See Limits of floating point types and the math.h library for additional details, limits, and properties of the floating-point types.

### Complex floating types

Complex floating types model the mathematical complex numbers, that is the numbers that can be written as a sum of a real number and a real number multiplied by the imaginary unit: a + bi

The three complex types are

  - float _Complex (also available as float complex if <complex.h> is included)
  - double _Complex (also available as double complex if <complex.h> is included)
  - long double _Complex (also available as long double complex if <complex.h> is included)

Note: as with all type specifiers, any order is permitted: long double complex, complex long double, and even double complex long name the same type.

Run this code

    #include <complex.h>
    #include <stdio.h>
    int main(void)
    {
        double complex z = 1 + 2*I;
        z = 1/z;
        printf("1/(1.0+2.0i) = %.1f%+.1fi\n", creal(z), cimag(z));
    }

Output:

    1/(1.0+2.0i) = 0.2-0.4i

If the macro constant `__STDC_NO_COMPLEX__`(C11) is defined by the implementation, the complex types (as well as the library header `<complex.h>`) are not provided. | (since C11)

Each complex type has the same object representation and alignment requirements as an array of two elements of the corresponding real type (float for float complex, double for double complex, long double for long double complex). The first element of the array holds the real part, and the second element of the array holds the imaginary component.

    float a[4] = {1, 2, 3, 4};
    float complex z1, z2;
    memcpy(&z1, a, sizeof z1); // z1 becomes 1.0 + 2.0i
    memcpy(&z2, a+2, sizeof z2); // z2 becomes 3.0 + 4.0i

Complex numbers may be used with arithmetic operators + - * and /, possibly mixed with imaginary and real numbers. There are many mathematical functions defined for complex numbers in complex.h. Both built-in operators and library functions may raise floating-point exceptions and set errno as described in `math_errhandling`

Increment and decrement are not defined for complex types

Relational operators are not defined for complex types (there is no notion of "less than")

| This section is incomplete Reason: review other ops, link library

Implicit conversions are defined between complex types and other arithmetic types.

In order to support the one-infinity model of complex number arithmetic, C regards any complex value with at least one infinite part as an infinity even if its other part is a NaN, guarantees that all operators and functions honor basic properties of inifinities and provides cproj to map all infinities to the canonical one (see arithmetic operators for the exact rules)

Run this code

    #include <stdio.h>
    #include <complex.h>
    #include <math.h>
    int main(void)
    {
       double complex z = (1 + 0*I) * (INFINITY + I*INFINITY);
    // textbook formula would give
    // (1+i0)(∞+i∞) ⇒ (1×∞ – 0×∞) + i(0×∞+1×∞) ⇒ NaN + I*NaN
    // but C gives a complex infinity
       printf("%f + i*%f\n", creal(z), cimag(z));

    // textbook formula would give
    // cexp(∞+iNaN) ⇒ exp(∞)×(cis(NaN)) ⇒ NaN + I*NaN
    // but C gives  ±∞+i*nan
       double complex y = cexp(INFINITY + I*NAN);
       printf("%f + i*%f\n", creal(y), cimag(y));

    }

Possible output:

    inf + i*inf
    inf + i*nan

C also treats multiple infinities so as to preserve directional information where possible, despite the inherent limitations of the Cartesian representation:

multiplying the imaginary unit by real infinity gives the correctly-signed imaginary infinity: i × ∞ = i∞. Also, i × (∞ – i∞) = ∞ + i∞ indicates the reasonable quadrant

| This section is incomplete Reason: wording

### Imaginary floating types

Imaginary floating types model the mathematical imaginary numbers, that is numbers that can be written as a real number multiplied by the imaginary unit: bi The three imaginary types are

  - float _Imaginary (also available as float imaginary if <complex.h> is included)
  - double _Imaginary (also available as double imaginary if <complex.h> is included)
  - long double _Imaginary (also available as long double imaginary if <complex.h> is included)

Note: as with all type specifiers, any order is permitted: long double imaginary, imaginary long double, and even double imaginary long name the same type.

Run this code

    #include <complex.h>
    #include <stdio.h>
    int main(void)
    {
        double imaginary z = 3*I;
        z = 1/z;
        printf("1/(3.0i) = %+.1fi\n", cimag(z));
    }

Output:

    1/(3.0i) = -0.3i

A compiler that defines __STDC_IEC_559_COMPLEX__ is recommended, but not required to support imaginary numbers. POSIX recommends checking if the macro _Imaginary_I is defined to identify imaginary number support. | (since C99) (until C11)
Imaginary numbers are supported if __STDC_IEC_559_COMPLEX__ is defined. | (since C11)

Each of the three imaginary types has the same object representation and alignment requirement as its corresponding real type (float for float imaginary, double for double imaginary, long double for long double imaginary).

Note: despite that, imaginary types are distinct and not compatible with their corresponding real types, which prohibits aliasing.

Imaginary numbers may be used with arithmetic operators + - * and /, possibly mixed with complex and real numbers. There are many mathematical functions defined for imaginary numbers in complex.h. Both built-in operators and library functions may raise floating-point exceptions and set errno as described in `math_errhandling`

Increment and decrement are not defined for imaginary types

| This section is incomplete Reason: review other ops, link library

Implicit conversions are defined between imaginary types and other arithmetic types.

The imaginary numbers make it possible to express all complex numbers using the natural notation x + I*y (where I is defined as _Imaginary_I). Without imaginary types, certain special complex values cannot be created naturally. For example, if I is defined as _Complex_I, then writing 0.0 + I*INFINITY gives NaN as the real part, and CMPLX(0.0, INFINITY) must be used instead. Same goes for the numbers with the negative zero imaginary component, which are meaningful when working with the library functions with branch cuts, such as csqrt: 1.0 - 0.0*I results in the positive zero imaginary component if I is defined as _Complex_I and the negative zero imaginary part requires the use of CMPLX or conj.

Imaginary types also simplify implementations; multiplication of an imaginary by a complex can be implemented straightforwardly with two multiplications if the imaginary types are supported, instead of four multiplications and two additions.

(since C99)

### Keywords

`char`, `int`, `short`, `long`, `signed`, `unsigned`, `float`, `double`. `_Bool`, `_Complex`, `_Imaginary`

### Range of values

The following table provides a reference for the limits of common numeric representations. As the C Standard allows any signed integer representation, the table gives both the minimum guaranteed requirements (which correspond to the limits of one's complement or sign-and-magnitude) and the limits of the most commonly used implementation, two's complement. All popular data models (including all of ILP32, LP32, LP64, LLP64) use two's complement representation, though.

Type | Size in bits | Format | Value range
--- | --- | --- | ---
Approximate | Exact
character | 8 | signed (one's complement) | -127 to 127
signed (two's complement) | -128 to 127
unsigned | 0 to 255
integral | 16 | signed (one's complement) | ± 3.27 · 104 | -32767 to 32767
signed (two's complement) | -32768 to 32767
unsigned | 0 to 6.55 · 104 | 0 to 65535
32 | signed (one's complement) | ± 2.14 · 109 | -2,147,483,647 to 2,147,483,647
signed (two's complement) | -2,147,483,648 to 2,147,483,647
unsigned | 0 to 4.29 · 109 | 0 to 4,294,967,295
64 | signed (one's complement) | ± 9.22 · 1018 | -9,223,372,036,854,775,807 to 9,223,372,036,854,775,807
signed (two's complement) | -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807
unsigned | 0 to 1.84 · 1019 | 0 to 18,446,744,073,709,551,615

floating
point

32

IEEE-754

- min subnormal:
  ± 1.401,298,4 · 10-45
- min normal:
  ± 1.175,494,3 · 10-38
- max:
  ± 3.402,823,4 · 1038

- min subnormal:
  ±0x1p-149
- min normal:
  ±0x1p-126
- max:
  ±0x1.fffffep+127

64

IEEE-754

- min subnormal:
  ± 4.940,656,458,412 · 10-324
- min normal:
  ± 2.225,073,858,507,201,4 · 10-308
- max:
  ± 1.797,693,134,862,315,7 · 10308

- min subnormal:
  ±0x1p-1074
- min normal:
  ±0x1p-1022
- max:
  ±0x1.fffffffffffffp+1023

Note: actual (as opposed to guaranteed minimal) ranges are available in the library headers <limits.h> and <float.h>

### See also

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