2.12. Value Types and Conversions

2.12.1. Type int

A variable is associated with a space in memory. This space has a fixed size associated with the type of data. The int and double types are examples of value types, which means that this memory space holds an encoding of the complete data for the value of the variable. The fixed space means that an int cannot be a totally arbitrary integer of an enormous size. In fact an int variable can only hold an integer in a specific range. See Data Representation for the general format of the underlying encoding in bits.

An int is held in a memory space of 32 bits, so it can have at most \(2^{32}\) values, and the encoding is chosen so about half are positive and half are negative: An int has maximum value \(2^{31} - 1 = 2147483647\) and a minimum value of \(-2^{31} = -2147483648\). The extreme values are also named constants in C#, int.MaxValue and int.MinValue.

In particular this means int arithmetic does not always work. What is worse, it fails silently:

csharp> int i = int.MaxValue;
csharp> i;
2147483647
csharp> i + 5;
-2147483644

Add two positive numbers and get a negative number! Getting such a wrong answer is called overflow. Be very careful if you are going to be using big numbers! Note: with addition, overflow will give the wrong sign, but the sign may not give such a clue if another operation overflows, like multiplication.

2.12.2. Type long

Most everyday uses of integers fit in this range of an int, and many modern computers are designed to operate on an int very efficiently, but sometimes you need a larger range. Type long uses twice as much space.

The same kind of silent overflow errors happen with long arithmetic, but only with much larger numbers.

When we get to Arrays, you will see that a program may store an enormous number of integers, and then the total space may be an issue. If some numbers fit in a long, but not an int, long must be used, taking us twice the space of an array of int elements. If all the integers have even more limited ranges, they might be stored in the smaller space of a short or a byte. We will not further discuss or use types short or byte in this book. Here we will only use the integral types int and long.

2.12.3. Type double

A double is also a value type, stored in a fixed sized space. There are even more issues with double storage than with an int: Not only do you need to worry about the total magnitude of the number, you also need to choose a precision: There are an infinite number of real values, just between 0 and 1. Clearly we cannot encode for all of them! As a result a double has a limited number of digits of accuracy. There is also an older type float that takes up half of the space of a double, and has a smaller range and less accuracy. This at least gives a reason for the name double: double the storage space of a float.

To avoid a ridiculously large number of trailing 0’s, a big double literal can be expressed using a variant of scientific notation:

1.79769313486232E+308 means \(1.7976931348623157(10^{308})\)

C# does not have the typography for raised exponents. Instead literal values can use the E to mean “times 10 to the power”, and the E is followed by and exponent integer that can be positive or negative. The whole double literal may not contain any embedded blanks. Internally these numbers are stored with powers of 2, not 10: See Data Representation.

Arithmetic with the double type does not overflow silently as with the integral types. We show behavior that could be important if you do scientific computing with enormous numbers: There are values for Infinity and Not a Number, abbreviated NaN. See them used in csharp:

csharp> double x = double.MaxValue;
csharp> x;
1.79769313486232E+308
csharp> double y = 10 * x;
csharp> y;
Infinity
csharp> y  + 1000;
Infinity
csharp> y  - 1000;
Infinity
csharp> 1000/y;
0
csharp> double z = 10 - y;
csharp> z;
-Infinity
csharp> double sum = y + z;
csharp> sum;
NaN
csharp> sum/1000;
NaN

Once a result gets too big, it gets listed as infinity. As you can see, there is some arithmetic allowed with a finite number and infinity! Still some operations are not legal. Once a result turns into NaN, no arithmetic operations change further results away from NaN, so there is a lasting record of a big error!

Note that Infinity, -Infinity and NaN are just representations when displayed as strings. The numerical constants are Double.PositiveInfinity, Double.NegativeInfinity, and Double.NaN.

Warning

There is no such neat system for showing off small inaccuracies in double arithmetic accumulating due to limited precision. These inaccuracies still happen silently.

2.12.4. Numeric Types and Limits

The listing below shows how the storage size in bits translates into the limits for various numerical types. We will not discuss or use short, byte or float further.

long
64 bits; range -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807
int
32 bits; range -2,147,483,648 to 2,147,483,647
short
16 bits; range -32,768 to 32,767
byte
8 bits; range 0 to 255 (no negative values)
double
64 bits; maximum magnitude: \(1.7976931348623157(10^{308})\); about 15 digits of accuracy
float
32 bits; maximum magnitude: \(3.402823(10^{38})\); about 7 digits of accuracy
decimal
128 bits; maximum value: 79228162514264337593543950335; 28 digits of accuracy; can exactly represents decimal values for financial operations; briefly discussed in optional Decimal Type.
char
See char as integer.

2.12.5. Casting

While the mathematical ideas of 42 and 42.0 are the same, C# has specific types. There are various places where numerical types get converted automatically by C# or explicitly by the programmer. A major issue is whether the new type can accurately represent the original value.

Going from int to double has no issue: Any int can be exactly represented as a double. Code like the following is fine:

csharp> int i = 33;
csharp> double d = i;
csharp> double x;
csharp> x = 11;
csharp> double z = i + 2.5;
csharp> ShowVars();
int i = 33
double d = 33
double x = 11
double z = 35.5

The double variable d is initialized with the value of an int variable. The double variable x is assigned a value using an int literal. The double variable z is initialized with the value of a sum involving both an int variable and a double literal. As we have discussed before in Arithmetic, the int is converted to a double before the addition operation is done.

The other direction for conversion is more problematic:

csharp> double d= 2.7;
csharp> int i;
csharp> i = d;
{interactive}(1,4): error CS0266: Cannot implicitly convert type
   'double' to 'int'.
An explicit conversion exists (are you missing a cast?)

The int i cannot accurately hold the value 2.7. Since the compiler does this checking, looking only at types, not values, this even fails if the the double happens to have an integer value:

csharp> double d = 2.0;
csharp> int i = d;
{interactive}(1,4): error CS0266: Cannot implicitly convert type
   'double' to 'int'.
An explicit conversion exists (are you missing a cast?)

If you really want to possibly lose precision and convert a double to an int result, you can do it, but you must be explicit, using a cast as the csharp error messages suggest.

csharp> double d= 2.7;
csharp> int i;
csharp> i = (int)d;
csharp> i;
2

The desired result type name in parentheses (int) is a cast, telling the compiler you really intend the conversion. Look what is lost! The cast does not round to the nearest integer, it truncates toward 0, dropping the fractional part, .7 here.

Rounding is possible, but if you really want the int type, it takes two steps, because the function Math.Round does round to a mathematical integer, but leaves the type as double! To round d to an int result we could use:

csharp> i = (int)Math.Round(d);
csharp> i;
3

You can also use an explicit cast from int to double. This is generally not needed, because of the automatic conversions, but there is one place where it is important: if you want double division but have int parts. Here is a quick artificial test:

csharp> int sum = 14;
csharp> int n = 4;
csharp> double avg = sum/n;
csharp> avg;
3

Oops, integer division. Instead, continue with:

csharp> avg = (double)sum/n;
csharp> avg;
3.5

We get the right decimal answer.

This is a bit more subtle than it may appear: The cast to double, (double) is an operation in C# and so it has a precedence like all operations. Casting happens to have precedence higher than any arithmetic operation, so the expression is equivalent to:

avg = ((double)sum)/n;

On the other hand, if we switch the order the other way with parentheses around the division:

csharp> avg = (double)(sum/n);
csharp> avg;
3

then working one step at a time, (sum/n) is integer division, with result 3. It is the 3 that is then cast to a double (too late)!

See the appendix Precedence of Operators, listing all C# operations discussed in this book.

2.12.6. Type char

The type for an individual character is char. A char literal value is a single character enclosed in single quotes, like 'a' or '$'. The literal for a single quote character itself and the literal for a newline use escape codes, like in String Special Cases: The literals are '\'' and '\n' respectively.

Be careful to distinguish a char literal like 'A' from a string literal "A".

Char as integer: Though the char type has character literals and prints as a character, internally a char is a type of integer, stored in 16 bits, with the correspondence between numeric codes and characters given by the Unicode standard. Unicode allows special symbol characters and alphabets of many languages. We will stick to the standard American keyboard for these characters.

Besides different alphabets, Unicode also has characters for all sorts of symbols: emoticons, chess pieces, advanced math.... See http://www.unicode.org/charts. All the symbols can be represented as escape codes in C#, starting with \u followed by 4 hexadecimal digits. For example \u262F produces a yin-yang symbol.

We mention the char type being numeric mostly because of errors that you can make that would otherwise be hard to figure out. This code does not concatenate the char symbols:

csharp> Console.WriteLine('A' + '-');
110

What? We mentioned that modern computers are set up to easily work with the int type. In arithmetic with smaller integral types the operands are first automatically converted to type int. An int sum is an int, and that is what is printed.

You can look at the numeric values inside a char with a cast!

csharp> (int)'A';
65
csharp> (int)'-';
45

So the earlier 110 is correct: 65 + 45 = 110.

For completeness: It is also possible to cast from small int back to char. This may be useful for dealing with the alphabet in sequence (or simple classical cryptographic codes):

csharp> 'A' + 1;
66
csharp> (char)('A' + 1);
'B'

The capital letter one place after A is B.

2.12.7. Type Boolean or bool

There is one more very important value type, that we introduce here for completeness, though we will not use it until Decisions. Logical conditions are either true or false. The type with just these two values is Boolean, or bool for short. The type is named after George Boole, who invented what we now call Boolean algebra. Though it seemed like a useless mathematical curiosity when Boole invented it, a century later Boolean algebra turned out to be at the heart of the implementation of computer hardware.

Note

The Boolean literals are true and false, with no quotes around them.

With quotes they would be of type string, not Boolean!

2.12.8. Overflow to Positive Exercise

We gave an example above in Type int, adding two positive int values and clearly having an error, since the result was negative. Declare and initialize two positive int variables x and y. Experiment with the initializations so

  1. Their product is too big to fit in an int AND
  2. The wrong overflow result for x*y is positive, not negative.