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Floating point numerical information

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• Previously discussed
  • Recall that:

    A byte is a memory cell consisting of 8 switches and can store a binary number (See: click here ) Although the computer stores binary numbers, we can interpret them as decimal number through a conversion operation: The computer can combine adjacent bytes in the RAM memory to form larger memory cells Although a computer memory can only store numbers, we can use an encoding scheme to represent any kind of information (E.g.: marital status information:   0 = single, 1 = married, and so on...           gender information:              0 = male, 1 = female) We must have context information to interpret a number that is stored in the RAM memory (E.g., we cannot interpret the number 0 stored in RAM memory unless we know kind of information it is.)

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• Storing information inside a computer
  • A Java computer program uses variables to store information

  • There are different kinds (types) of variables

  • In this webnote, we will study the floating point variables

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• Variables (in general)
  • Variable in any programming language

    A variable in a computer program is in fact one or more memory cells These memory cells jointly store the encoding of a number A variable in a computer program stores a number Each variable has a data type The data type allows the computer program to use the appropriate encoding (interpretation) scheme to interpret the number stored in the variable

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  • Analogy of a variable

    A variable (in Computer Science) is like a piece of paper with a tag attached to it. The tag represents the data type of the variable (the value of the tag cannot be changed) You can only write one (binary) number on the piece of paper. You can erase the number and write a new number on the paper; but you can never write more than one number on the paper at any time.

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• Floating point numbers
  • A floating point number is a number where the number of decimal digits before and after the decimal point is not fixed (i.e., the decimal point "floats")

  • Examples:

    1234.5 1 decimal digits after the decimal point 0.087 3 decimal digits after the decimal point 3.1415 5 decimal digits after the decimal point

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  • Floating point numbers can also be written in the exponent form:

    0.31415e1 (= 0.31415×101 = 3.1415) 31.415e-1 (= 31.415×10-1 = 3.1415)

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  • A peek into CS255 (Computer Architecture) material:

    A floating point number is in fact represented by 2 numbers: A mantissa (e.g., the 0.31415 in 0.31415e1) An exponent (e.g., the 1 in 0.31415e1)

    The floating point number representation will be discussed in detail in the CS255 course.

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• Floating point variables
  • There are 2 kinds of floating point variables in Java

    We will discuss the double precision kind first.

    (We discuss double precision floating point variable first to avoid using the casting operation - casting will be explained later)

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  • A double precision floating point variable is a variable that:

    uses 8 consecutive bytes of memory as a single 64 bit memory cell uses a modified binary number system as encoding scheme For more details on the floating point encoding method, please take CS255 A double precision floating point variable can represent a floating point number: in range of from −10308 to 10308 and with about 15 decimal digits accuracy

    This kind of variable is commonly used in scientific computations.

    A calculator uses double precision floating point variables.

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  • Important fact:

    A floating point variable behaves like a piece of paper where you can record (exactly) one floating point number:

    BTW, that's how you perceive a double typed variable.

    Inside the computer, the number is represented with bits.

    A double typed variable looks something like this in reality:

    The 64 bits encodes a floating point number

    The encoding method used is called the IEEE 754 Standard - See: click here

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  Note to lecturer:

  Teach "Defining floating variables" with 2 terminal windows window 1: edit a Java program window 2: compile and run Lecture while editing the Java program (from scratch)

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• Defining floating point variables

  • Every variable in Java must be defined
    This rule applies to a floating point variable.

    Also, you must use the correct syntax construct to write a variable definition

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  • Syntax to define an floating point variable:

    double NameOfVariable ;

    Notes:

    The keyword double announces the variable definition clause The NameOfVariable is an identifier which is the name of the variable. The variable definition clause is must be ended with a semi-colon ";"

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  • Example:

    public class Var01 { public static void main(String[] args) { double x; // Define floating point variable with name "x" System.out.println("Hello Class"); System.out.println(" The variable x contains this number:"); System.out.println(x); // Print variable "x" } }

    Notes:

    The name of the Java program is Var01 Therefore, the UNIX filename that contain this Java program must be: Var01.java The statement "System.out.println(x);" will print the content of the variable x to the terminal Notice that the statement "System.out.println(x);" does not uses quotes ".." around the variable x The statement "System.out.println("x");" (with quotes ".." around x) will print the text x Remember that a double variable is like a piece of paper (where you can write a one floating point number) The variable name is like writing that name in a corner of the paper (for identification purpose)

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• Effect of a variable definition clause      (use black board to illustrate variable definition)
  • When a Java program starts to run inside the computer, it will use up a portion of the RAM memory to store various things

    A (large) portion of the RAM memory will remain unused

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  • The effect of the definition:

    double x;

    is the following:

    The computer will find 8 consecutive bytes of RAM memory that is unused The computer will then reserve these memory bytes It also associate the name x with the (8 bytes of) reserved memory Whenever the Java program uses the name x, the computer will translate that into read/write operations to these (8 bytes of) reserved memory The computer can store (exactly) one floating point number in the reserved memory bytes. That is why: A floating point variable behaves like a piece of paper where you write down (exactly) one floating point number

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  • Example: (draw these figures on black board - show the progression)

    When the Java program is run, it starts with the main() method: A portion of the RAM memory is used. But a large portion will be unused When the variable definition "double x" is processed, it causes the computer to reserve consecutive 8 bytes of RAM memory: These 8 bytes of reserved memory can now be referred to by the name x !!! (How this is done exactly will be explained in CS255)

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

    How to run the program:

    Right click on link and save in a scratch directory To compile:   javac Var01.java To run:          it will not run due to an error...

    Output: (Compile error)

    Var01.java:11: variable x might not have been initialized System.out.println(x); // Print the content in variable "x" ^ 1 error

    Meaning:

    The variable x is "blank" (i.e., has nothing "written" on it) (Note: blank is not the same as contains the value 0)

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• Terminology: Creating variables
  • Computer Science Jargon:

    Creating a variable = reserve memory space for a variable

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• Update the value stored in a variable
  • Assignment statement:

    The assignment statement in the Java programming language instructs the computer to update the value stored in a variable

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  • Syntax of an assignment statement:

    VariableName = Expression ;

    Notes:

    The symbol "=" is called the assignment operator in Java The variable on left-hand-side of the "=" operator is the receiving variable The expression on right-hand-side of the "=" operator is the value that is assigned to the receiving variable The assignment statement must be ended with a semi-colon ";" Meaning of the assignment statement: The expression on the RHS of the "=" operator is first computed The computed value is then assigned to the receiving variable

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  • Example:

    public class Var02 { public static void main(String[] args) { double x; // Define floating point variable with name "x" x = 0.31415e1; // Assign 3.1415 to x System.out.println("Hello Class"); System.out.println(" The variable x contains this number:"); System.out.println(x); // Print variable "x" } }

    Notes:

    The name of the Java program is now Var02 Therefore, the UNIX filename that contain this Java program must be: Var02.java (I will not make this comment anymore from this point on...)

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  • Example:

    When the variable definition "double x" is processed, it causes the computer to reserve consecutive 8 bytes of RAM memory: These 8 bytes of reserved memory can now be referred to by the name x !!! The assignment statement x=0.31415e1 will store the value 3.1415 into the memory cell identified by the name x: The Print statement System.out.println(x) will read the value stored at the memory cell identified by the name x and print it to the terminal:

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

    How to run the program:

    Right click on link and save in a scratch directory To compile:   javac Var02.java To run:          java Var02

    Output:

    Hello Class The variable x contains this number: 3.1415

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• Defining and initializing a variable at the same time
  • When a variable is defined, it does not contain a legitimate value

  • You can give an initial value to a variable when you define the variable

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  • Syntax to define an initialized double precision floating point variable:

    double varName = initialValue ;

    Example:

    public class Var03 { public static void main(String[] args) { double x = 0.31415e1; // Define an initialized variable System.out.println("Hello Class"); System.out.println(" The variable x contains this number:"); System.out.println(x); // Print variable "x" } }

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• Defining multiple variables of the same type
  • You can define multiple variables of the same type with multiple clauses

    Example: defining 3 variables

    public class Var04 { public static void main(String[] args) { double x = 0.31415e1; // <----- Focus here double y; double z = 2.71828; y = 1.0; System.out.println(x); // Print variable "x" System.out.println(y); // Print variable "y" System.out.println(z); // Print variable "z" } }

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  • You can also define multiple variables of the same type with one clauses by separating the different variable names with a comma.

    Example: defines that same 3 variables with 1 clause

    public class Var04 { public static void main(String[] args) { double x = 0.31415e1, y, z = 2.71828; // <----- Focus here y = 1.0; System.out.println(x); // Print variable "x" System.out.println(y); // Print variable "y" System.out.println(z); // Print variable "z" } }

    Notice the comma (,) separating the variable names.