Part B - Computations

Types

Select appropriate types for storing program variables and constants
Describe the internal representations defined by different types

"The fundamental purpose of a type system is to prevent the occurrence of
execution errors during the running of a program."
(Cardelli, L. 2004)

A typed programming language uses a type system to interpret the bit streams in memory.  C is a typed programming language.  A type is the rule that defines how to store values in memory and which operations are admissible on those values.  A type defines the number of bytes available for storing values and hence the range of possible values.  We use different types to store different information.  The relation between types and raw memory is illustrated in the figure below.

This chapter describes the four most common types in the C language and the ranges of values that these types allow.  This chapter concludes by describing how to allocate memory for variables by identifying their contents using a type.

Arithmetic Types

The four most common types of the C language for performing arithmetic calculations are:

• char
• int
• float
• double

A char occupies one byte and can store a small integer value, a single character or a single symbol:

 char 1 Byte

An int occupies one word and can store an integer value.  In a 32-bit environment, an int occupies 4 bytes:

 int (32-bit environment) 1 Byte 1 Byte 1 Byte 1 Byte

A float typically occupies 4 bytes and can store a single-precision, floating-point number:

 float 1 Byte 1 Byte 1 Byte 1 Byte

A double typically occupies 8 bytes and can store a double-precision, floating-point number:

 double 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte

Size Specifiers

Size specifiers adjust the size of the int and double types.

Type int Size Specifiers

Specifying the size of an int ensures that the type contains a minimum number of bits.  The three specifiers are:

• short
• long
• long long

A short int (or simply, a short) contains at least 16 bits:

 short 1 Byte 1 Byte

A long int (or simply, a long) contains at least 32 bits:

 long 1 Byte 1 Byte 1 Byte 1 Byte

A long long int (or simply, a long long) contains at least 64 bits:

 long long 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte

The size of a simple int is no less than the size of a short.

Type double Size Specifier

The size of a long double depends on the environment and is typically at least 64 bits:

 long double 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte

Specifying the long double type only ensures that it contains at least as many bits as a double.  The C language does not require a long double to contain a minimum number of bits.

const Qualifier

Any type can hold a constant value.  A constant value cannot be changed.  To qualify a type as holding a constant value we use the keyword const.  A type qualified as const is unmodifiable.  That is, if a program instruction attempts to modify a const qualified type, the compiler will report an error.

Representing Values

Hardware manufacturers distinguish intergal types from floating-point types and represent integral data and floating-point data differently.

• integral types: char int
• floating-point types: float double

Integral Types

C stores the char and int data in equivalent binary form.  Binary form represents the value stored exactly.  To learn how to convert between decimal and binary representation refer to the appendix entitled Data Conversions

Characters and Symbols

C stores characters and symbols in char types. Since characters and symbols have no intrinsic binary representation, the host platform provides the collating sequence for associating each character and symbol with a unique integer value.  C stores the integer value from this sequence as the representative of the character or symbol.

The two popular collating sequences are ASCII and EBCDIC.  ASCII is more popular.  ASCII represents the letter A by the bit pattern 010000012, that is the hexadecimal value 0x41, that is the decimal value 65.  EBCDIC represents the letter A by the bit pattern 110000012, that is the hexadecimal value 0xC1, that is the decimal value 193.

ASCII and EBCDIC are not compatible.  The symbol order in ASCII differs from that in EBCDIC.  In ASCII, the digits precede the letters, while in EBCDIC, the letters precede the digits.  If we sort symbolic information that contains digits and letters, we will obtain different results under each sequence.

Neither ASCII nor EBCDIC contain enough values to represent most of the characters and symbols in the world languages.  The Unicode standard, which is compatible with ASCII, provides a much more comprehensive collating system.  We use the ASCII collating sequence throughout these notes.

Negative Values

There are three schemes for storing negative integers:

• 2's complement notation (most popular)
• 1's complement notation
• sign magnitude notation

All three represent non-negative values identically.  Under the 2's complement rule, there is only one representation of 0 and separate addition and subtraction circuits in the ALU are unnecessary.

To obtain the 2's complement of an integer, we

• flip the bits

For example, we represent the integer -92 by 101001002

 Bit # 7 6 5 4 3 2 1 0 92 => 0 1 0 1 1 1 0 0 Flip Bits 1 0 1 0 0 0 1 1 Add 1 0 0 0 0 0 0 0 1 -92 => 1 0 1 0 0 1 0 0

Floating-Point Data

Floating-point types store tiny as well as huge values by decomposing the values into three distinct components: a sign, an exponent and a significand.  The C language leaves the implementation details to the hardware manufacturer.

The most popular model is the IEEE (I-triple-E or Institute of Electrical and Electronics Engineers) Standard 754 for Binary and Floating-Point Arithmetic.  Under IEEE 754, a float has 32 bits, consisting of one sign bit, an 8-bit exponent and a 23-bit significand (or mantissa):

 float 1 Byte 1 Byte 1 Byte 1 Byte s exponent significand

Under IEEE 754, a double occupies 64 bits, has one sign bit, an 11-bit exponent and a 52-bit significand:

 double 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte 1 Byte s exponent significand

Since the number of bits in the significand is limited, the float and double types cannot store all possible floating-point values exactly.  That is, the floating-point types store values approximately.

Value Ranges

The number of bytes allocated for a type determines the range of values that that type can store.

Integral Types

The ranges of values for the integral types are shown below.  Ranges for some types depend on the execution environment:

 Type Size Min Max char 8 bits -128 127 char 8 bits 0 255 short >=16 bits -32,768 32,767 int 2 bytes -32,768 32,767 int 4 bytes -2,147,483,648 2,147,483,647 long >=32 bits -2,147,483,648 2,147,483,647 long long >=64 bits -9,223,372,036,854,775,808 9,223,372,036,854,775,807

Note that the limits on both a char and an int vary with the execution environment.

Floating-Point Types

The limits on a float and double depend on the execution environment:

 Type Size Significant Min Exponent Max Exponent float minimum 6 -37 37 float typical 6 -37 37 double minimum 10 -37 37 double typical 15 -307 307 long double typical 15 -307 307

Note that both the number of significant digits and the range of the exponent are limited.  The limits on the exponent are in base 10.

Variable Declarations

We store program data in variables  A declaration associates a program variable with a type.  The type identifies the properties of the variable.

In C, a declaration takes the form

` [const] type identifier [= initial value];`

The brackets denote an optional part of the syntax.

We select a meaningful name for the identifier and optionally set the variable's initial value.  We conclude the declaration with a semi-colon, making it a complete statement.

For example,

 ``` char children; int nPages; float cashFare; const double pi = 3.14159265;```

Multiple Declarations

We may group the identifiers of variables that share the same type within a single declaration by separating the identifiers by commas.

For example,

 ``` char children, digit; int nPages, nBooks, nRooms; float cashFare, height, weight; double loan, mortgage;```

Naming Conventions

We may select any identifier for a variable that satisfies the following naming conventions:

• starts with a letter or an underscore (_)
• contains any combination of letters, digits and underscores (_)
• contains less than 32 characters (some compilers allow more, others do not)
• is not be a C reserved word

Reserved Words

The C language reserves the following words for its own use:

``` auto       _Bool      break     case
char       _Complex   const     continue
default    restrict   do        double
else       enum       extern    float
for        goto       if        _Imaginary
inline     int        long      register
return     short      signed    sizeof
static     struct     switch    typedef
union      unsigned   void      volatile
while```

For upward compatibility with C++, we avoid using the following C++ reserved words

``` asm              export           private          throw
bool             false            protected        true
catch            friend           public           try
class            mutable          reinterpret_cast typeid
const_cast       namespace        static_cast      typename
delete           new              template         using
dynamic_cast     operator         this             virtual
explicit                                           wchar_t```

Exercises

• Study the ASCII table and note the representations used for the digits '0' through '9', the upper-case letters 'A' through 'Z' and the lower-case letters 'a' through 'z'
• Complete the exercises on Data Types
• Select eight different variables, two of each type.  Ensure that the type of each variable is the best suited type for that variable and write the declaration statements for your set of variables