# BigDecimal(Variable Precision Floating Library for Ruby)

Japanese
BigDecimal is an extension library for the Ruby interpreter. Using BigDecimal class, you can obtain any number of significant digits in computation. For the details about Ruby see:

- http://www.ruby-lang.org/en/:Official Ruby page(English).
- http://kahori.com/ruby/ring/:Mutually linked pages relating to Ruby(Japanese).

NOTE:
This software is provided "AS IS" and without any express or implied warranties,including,without limitation,the implied warranties of merchantability and fitness for a particular purpose. For the details,see COPYING and README included in this distribution.

## Contents

- Introduction
- Usage and methods
- Infinity,NaN,Zero
- Internal structure
- Binary or decimal number representation
- Resulting number of significant digits

## Introduction

Ruby already has builtin (variable length integer number) class Bignum. Using Bignum class,you can obtain any integer value in magnitude. But, variable length decimal number class is not yet built in. This is why I made variable length floating class BigDecimal. Feel free to send any comments or bug reports to me. shigeo@tinyforest.gr.jp I will try(but can't promise) to fix bugs reported.

## Installation

The Ruby latest version can be downloaded from Official Ruby page. Once decompress the downloaded Ruby archive,follow the normal installation procedures according to the documents included.

## Usage and methods

Suppose you already know Ruby programming, to create BigDecimal objects,the program would like:
`

       require 'bigdecimal'
       a=BigDecimal::new("0.123456789123456789")
       b=BigDecimal("123456.78912345678",40)
       c=a+b

`

### List of methods

In 32 bits integer system,every 4 digits(in decimal) are computed simultaneously. This means the number of significant digits in BigDecimal is always a multiple of 4.

Some more methods are available in Ruby code (not C code). Functions such as sin,cos ...,are in math.rb in bigdecimal directory. To use them,require math.rb as: ``

    require "bigdecimal/math.rb"

For details,see the math.rb code and comments. Other utility methods are in util.rb. To use util.rb, require it as: `

    require "bigdecimal/util.rb"

` For details,see the util.rb code.

#### Class methods

- new

> "new" method creates a new BigDecimal object.
> a=BigDecimal::new(s[,n]) or
> a=BigDecimal(s[,n]) or
> where:
> s: Initial value string. Spaces will be ignored. Any unrecognizable character for representing initial value terminates the string.
> n: Maximum number of significant digits of a. n must be a Fixnum object. If n is omitted or is equal to 0,then the maximum number of significant digits of a is determined from the length of s. Actual number of digits handled in computations are usually greater than n.
> n is useful when performing divisions like `

>     BigDecimal("1")    / BigDecimal("3")    # => 0.3333333333 33E0
>     BigDecimal("1",10) / BigDecimal("3",10) # => 0.3333333333 3333333333 33333333E0

> ` but the resulting digits obtained may differ in future version.

- mode

> f = BigDecimal.mode(s[,v])
> mode method controls BigDecimal computation. If the second argument is not given or is nil,then the value of current setting is returned. Following usage are defined.

> [EXCEPTION control]

> Actions when computation results NaN or Infinity can be defined as follows.

> > f = BigDecimal::mode(BigDecimal::EXCEPTION_NaN,flag)
> > f = BigDecimal::mode(BigDecimal::EXCEPTION_INFINITY,flag)
> > f = BigDecimal::mode(BigDecimal::EXCEPTION_UNDERFLOW,flag)
> > f = BigDecimal::mode(BigDecimal::EXCEPTION_OVERFLOW,flag)
> > f = BigDecimal::mode(BigDecimal::EXCEPTION_ZERODIVIDE,flag)
> > f = BigDecimal::mode(BigDecimal::EXCEPTION_ALL,flag)

> EXCEPTION_NaN controls the execution when computation results to NaN.
> EXCEPTION_INFINITY controls the execution when computation results to Infinity.
> EXCEPTION_UNDERFLOW controls the execution when computation underflows.
> EXCEPTION_OVERFLOW controls the execution when computation overflows.
> EXCEPTION_ZERODIVIDE controls the execution when zero-division occurs.
> EXCEPTION_ALL controls the execution when any defined exception occurs.
> If the flag is true,then the relating exception is thrown.
> No exception is thrown when the flag is false(default) and computation continues with the result:

> > EXCEPTION_NaN results to NaN
> > EXCEPTION_INFINITY results to +Infinity or -Infinity
> > EXCEPTION_UNDERFLOW results to 0.
> > EXCEPTION_OVERFLOW results to +Infinity or -Infinity
> > EXCEPTION_ZERODIVIDE results to +Infinity or -Infinity

> EXCEPTION_INFINITY,EXCEPTION_OVERFLOW, and EXCEPTION_ZERODIVIDE are currently the same.
> The return value of mode method is the value set.
> If nil is specified for the second argument,then current setting is returned.
> Suppose the return value of the mode method is f,then f & BigDecimal::EXCEPTION_NaN !=0 means EXCEPTION_NaN is set to on.

> [ROUND error control]

> Rounding operation can be controlled as:

> > f = BigDecimal::mode(BigDecimal::ROUND_MODE,flag)

> where flag must be one of:

> ROUND_UP | round away from zero.
> ROUND_DOWN | round towards zero(truncate).
> ROUND_HALF_UP | round up if the digit >= 5 otherwise truncated(default).
> ROUND_HALF_DOWN | round up if the digit >= 6 otherwise truncated.
> ROUND_HALF_EVEN | round towards the even neighbor(Banker's rounding).
> ROUND_CEILING | round towards positive infinity(ceil).
> ROUND_FLOOR | round towards negative infinity(floor).

> New rounding mode is returned. If nil is specified for the second argument,then current setting is returned.
> The digit location for rounding operation can not be specified by this mode method, use truncate/round/ceil/floor/add/sub/mult/div methods for each instance instead.

- limit[(n)]

> Limits the maximum digits that the newly created BigDecimal objects can hold never exceed n. This means the rounding operation specified by BigDecimal.mode is performed if necessary. limit returns the value before set if n is nil or is not specified. Zero,the default value,means no upper limit.
> The limit has no more priority than instance methods such as truncate,round,ceil,floor,add,sub,mult,and div.
> mf = BigDecimal::limit(n)

- double_fig

> double_fig is a class method which returns the number of digits the Float class can have. `

>       p BigDecimal::double_fig  # ==> 20 (depends on the CPU etc.)

> ` The equivalent C programs which calculates the value of double_fig is: `

>      double v          = 1.0;
>      int    double_fig = 0;
>      while(v + 1.0 > 1.0) {
>         ++double_fig;
>         v /= 10;
>      }

> `

- BASE

> Base value used in the BigDecimal calculation. On 32 bits integer system,the value of BASE is 10000.
> b = BigDecimal::BASE

#### Instance methods

- +

> addition(c = a + b)
> For the resulting number of significant digits of c,see Resulting number of significant digits.

- -

> subtraction (c = a - b) or negation (c = -a)
> For the resulting number of significant digits of c,see Resulting number of significant digits.

- *

> multiplication(c = a * b)
> For the resulting number of significant digits of c,see Resulting number of significant digits.

- /

> division(c = a / b)
> For the resulting number of significant digits of c,see Resulting number of significant digits.

- add(b,n)

> c = a.add(b,n)
> c = a.add(b,n) performs c = a + b.
> If n is less than the actual significant digits of a + b, then c is rounded properly according to the BigDecimal.limit.
> If n is zero,then the result is the same as +'s.

- sub(b,n)

> c = a.sub(b,n)
> c = a.sub(b,n) performs c = a - b.
> If n is less than the actual significant digits of a - b, then c is rounded properly according to the BigDecimal.limit.
> If n is zero,then the result is the same as -'s.

- mult(b,n)

> c = a.mult(b,n)
> c = a.mult(b,n) performs c = a * b.
> If n is less than the actual significant digits of a * b, then c is rounded properly according to the BigDecimal.limit.
> If n is zero,then the result is the same as *'s.

- div(b[,n])

> c = a.div(b,n)
> c = a.div(b,n) performs c = a / b.
> If n is less than the actual significant digits of a / b, then c is rounded properly according to the BigDecimal.limit.
> If n is zero,then the result is the same as /'s. If n is not given,then the result will be an integer(BigDecimal) like Float#div.

- fix

> c = a.fix
> returns integer part of a.

- frac

> c = a.frac
> returns fraction part of a.

- floor[(n)]

> c = a.floor
> returns the maximum integer value (in BigDecimal) which is less than or equal to a. `

>      c = BigDecimal("1.23456").floor  #  ==> 1
>      c = BigDecimal("-1.23456").floor #  ==> -2

> ` As shown in the following example,an optional integer argument (n) specifying the position of the target digit can be given.
> If n> 0,then the (n+1)th digit counted from the decimal point in fraction part is processed(resulting number of fraction part digits is less than or equal to n).
> If n<0,then the n-th digit counted from the decimal point in integer part is processed(at least n 0's are placed from the decimal point to left). `

>      c = BigDecimal("1.23456").floor(4)   #  ==> 1.2345
>      c = BigDecimal("15.23456").floor(-1) #  ==> 10.0

> `

- ceil[(n)]

> c = a.ceil
> returns the minimum integer value (in BigDecimal) which is greater than or equal to a. `

>      c = BigDecimal("1.23456").ceil  #  ==> 2
>      c = BigDecimal("-1.23456").ceil #  ==> -1

> ` As shown in the following example,an optional integer argument (n) specifying the position of the target digit can be given.
> If n>0,then the (n+1)th digit counted from the decimal point in fraction part is processed(resulting number of fraction part digits is less than or equal to n).
> If n<0,then the n-th digit counted from the decimal point in integer part is processed(at least n 0's are placed from the decimal point to left). `

>      c = BigDecimal("1.23456").ceil(4)   # ==> 1.2346
>      c = BigDecimal("15.23456").ceil(-1) # ==> 20.0

> `

- round[(n[,b])]

> c = a.round
> round a to the nearest 1(default)ÅD
> `

>      c = BigDecimal("1.23456").round  #  ==> 1
>      c = BigDecimal("-1.23456").round #  ==> -1

> ` The rounding operation changes according to BigDecimal::mode(BigDecimal::ROUND_MODE,flag) if specified. As shown in the following example,an optional integer argument (n) specifying the position of the target digit can be given.
> If n>0,then the (n+1)th digit counted from the decimal point in fraction part is processed(resulting number of fraction part digits is less than or equal to n).
> If n<0,then the n-th digit counted from the decimal point in integer part is processed(at least n 0's are placed from the decimal point to left). `

>     c = BigDecimal::new("1.23456").round(4)   #  ==> 1.2346
>     c = BigDecimal::new("15.23456").round(-1) #  ==> 20.0

> ` Rounding operation can be specified by setting the second optional argument b with the valid ROUND_MODE.
> `

>     c = BigDecimal::new("1.23456").round(3,BigDecimal::ROUND_HALF_EVEN)   #  ==> 1.234
>     c = BigDecimal::new("1.23356").round(3,BigDecimal::ROUND_HALF_EVEN)   #  ==> 1.234

> `

- truncate[(n)]

> c = a.truncate
> truncate a to the nearest 1ÅD
> As shown in the following example,an optional integer argument (n) specifying the position of the target digit can be given.
> If n>0,then the (n+1)th digit counted from the decimal point in fraction part is processed(resulting number of fraction part digits is less than or equal to n).
> If n<0,then the n-th digit counted from the decimal point in integer part is processed(at least n 0's are placed from the decimal point to left). `

>     c = BigDecimal::new("1.23456").truncate(4)   #  ==> 1.2345
>     c = BigDecimal::new("15.23456").truncate(-1) #  ==> 10.0

> `

- abs

> c = a.abs
> returns an absolute value of a.

- to_i

> changes a to an integer.
> i = a.to_i
> i becomes to Fixnum or Bignum. If a is Infinity or NaN,then i becomes to nil.

- to_s[(n)]

> converts to string(default results look like "0.xxxxxEn"). `

>     BigDecimal("1.23456").to_s  #  ==> "0.123456E1"

> ` If n(>0) is given,then a space is inserted to each of two parts divided by the decimal point after every n digits for readability. `

>     BigDecimal("0.1234567890123456789").to_s(10)   #  ==> "0.1234567890 123456789E0"

> ` n can be a string representing a positive integer number. `

>     BigDecimal("0.1234567890123456789").to_s("10") #  ==> "0.1234567890 123456789E0"

> ` If the first character is '+'(or ' '),then '+'(or ' ') will be set before value string when the value is positive. `

>     BigDecimal("0.1234567890123456789").to_s(" 10") #  ==> " 0.1234567890 123456789E0"
>     BigDecimal("0.1234567890123456789").to_s("+10") #  ==> "+0.1234567890 123456789E0"
>     BigDecimal("-0.1234567890123456789").to_s("10") #  ==> "-0.1234567890 123456789E0"

> ` At the end of the string,'E'(or 'e') or 'F'(or 'f') can be specified to change number representation. `

>     BigDecimal("1234567890.123456789").to_s("E")  #  ==> "0.1234567890123456789E10"
>     BigDecimal("1234567890.123456789").to_s("F")  #  ==> "1234567890.123456789"
>     BigDecimal("1234567890.123456789").to_s("5E") #  ==> "0.12345 67890 12345 6789E10"
>     BigDecimal("1234567890.123456789").to_s("5F") #  ==> "12345 67890.12345 6789"

> `

- exponent

> returns an integer holding exponent value of a.
> n = a.exponent
> means a = 0.xxxxxxx*10**n.

- precs

> n,m = a.precs
> precs returns number of significant digits (n) and maximum number of significant digits (m) of a.

- to_f

> Creates a new Float object having (nearly) the same value. Use split method if you want to convert by yourself.

- sign

> n = a.sign
> returns positive value if a > 0,negative value if a < 0, otherwise zero if a == 0.
> where the value of n means that a is:
> n = BigDecimal::SIGN_NaN(0) : a is NaN
> n = BigDecimal::SIGN_POSITIVE_ZERO(1) : a is +0
> n = BigDecimal::SIGN_NEGATIVE_ZERO(-1) : a is -0
> n = BigDecimal::SIGN_POSITIVE_FINITE(2) : a is positive
> n = BigDecimal::SIGN_NEGATIVE_FINITE(-2) : a is negative
> n = BigDecimal::SIGN_POSITIVE_INFINITE(3) : a is +Infinity
> n = BigDecimal::SIGN_NEGATIVE_INFINITE(-3) : a is -Infinity
> The value in () is the actual value,see (Internal structure.

- nan?

> a.nan? returns True when a is NaN.

- infinite?

> a.infinite? returns 1 when a is Infinity, -1 when a is -Infinity, nil otherwise.

- finite?

> a.finite? returns true when a is neither Infinity nor NaN.

- zero?

> c = a.zero?
> returns true if a is equal to 0,otherwise returns false

- nonzero?

> c = a.nonzero?
> returns nil if a is 0,otherwise returns a itself.

- split

> decomposes a BigDecimal value to 4 parts. All 4 parts are returned as an array.
> Parts consist of a sign(0 when the value is NaN,+1 for positive and -1 for negative value), a string representing fraction part,base value(always 10 currently),and an integer(Fixnum) for exponent respectively. a=BigDecimal::new("3.14159265")
> f,x,y,z = a.split
> where f=+1,x="314159265",y=10 and z=1
> therefore,you can translate BigDecimal value to Float as:
> s = "0."+x
> b = f*(s.to_f)*(y**z)

- inspect

> is used for debugging output.
> p a=BigDecimal::new("3.14",10)
> should produce output like "#<0x112344:'0.314E1',4(12)%gt;". where "0x112344" is the address, '0.314E1' is the value,4 is the number of the significant digits, and 12 is the maximum number of the significant digits the object can hold.

- sqrt

> c = a.sqrt(n)
> computes square root value of a with significant digit number n at least.

- **

> c = a ** n
> returns the value of a powered by n. n must be an integer.

- power

> The same as ** method.
> c = a.power(n)
> returns the value of a powered by n(c=a**n). n must be an integer.

- divmod,quo,modulo,%,remainder

> See,corresponding methods in Float class.

- <=>

> c = a <=> b
> returns 0 if a==b,1 if a > b,and returns -1 if a < b.

Following methods need no explanation.

- ==
- ===
same as ==,used in case statement.
- !=
- <
- <=
- >
- >=

### About 'coerce'

For the binary operation like A op B:

1.Both A and B are BigDecimal objects
  A op B is normally performed.
2.A is the BigDecimal object but B is other than BigDecimal object
  Operation is performed,after B is translated to corresponding BigDecimal object(because BigDecimal supports coerce method).
3.A is not the BigDecimal object but B is BigDecimal object
  If A has coerce method,then B will translate A to corresponding BigDecimal object and the operation is performed,otherwise an error occures.

String is not translated to BigDecimal in default. Uncomment /* #define ENABLE_NUMERIC_STRING */ in bigdecimal.c, compile and install again if you want to enable string to BigDecimal conversion. Translation stops without error at the character representing non digit. For instance,"10XX" is translated to 10,"XXXX" is translated to 0.
String representing zero or infinity such as "Infinity","+Infinity","-Infinity",and "NaN" can also be translated to BigDecimal unless false is specified by mode method.
BigDecimal class supports coerce method(for the details about coerce method,see Ruby documentations). This means the most binary operation can be performed if the BigDecimal object is at the left hand side of the operation.
For example: `

      a = BigDecimal.E(20)
      c = a * "0.123456789123456789123456789" # A String is changed to BigDecimal object.

` is performed normally.
But,because String does not have coerce method,the following example can not be performed.
`

      a = BigDecimal.E(20)
      c = "0.123456789123456789123456789" * a # ERROR

` If you actually have any inconvenience about the error above. You can define a new class derived from String class, and define coerce method within the new class.

## Infinity,Not a Number(NaN),Zero

Infinite numbers and NaN can be represented by string writing "+Infinity"(or "Infinity"),"-Infinity",and "NaN" respectively in your program. Infinite numbers can be obtained by 1.0/0.0(=Infinity) or -1.0/0.0(=-Infinity).
NaN(Not a number) can be obtained by undefined computation like 0.0/0.0 or Infinity-Infinity. Any computation including NaN results to NaN. Comparisons with NaN never become true,including comparison with NaN itself.
Zero has two different variations as +0.0 and -0.0. But,still, +0.0==-0.0 is true.
Computation results including Infinity,NaN,+0.0 or -0.0 become complicated. Run following program and confirm the results. Send me any incorrect result if you find. `

     require "bigdecimal"
     aa  = %w(1 -1 +0.0 -0.0 +Infinity -Infinity NaN)
     ba  = %w(1 -1 +0.0 -0.0 +Infinity -Infinity NaN)
     opa = %w(+ - * / <=> > >=  < == != <=)
     for a in aa
      for b in ba
        for op in opa
          x = BigDecimal::new(a)
          y = BigDecimal::new(b)
          eval("ans= x #{op} y;print a,' ',op,' ',b,' ==> ',ans.to_s,\"\n\"")
        end
      end
     end

`

## Internal structure

BigDecimal number is defined by the structure Real in BigDecimal.h. Digits representing a float number are kept in the array frac[] defined in the structure. In the program,any floating number(BigDecimal number) is represented as:
= 0.xxxxxxxxx*BASE**n
where 'x' is any digit representing mantissa(kept in the array frac[]), BASE is base value(=10000 in 32 bit integer system), and n is the exponent value.
Larger BASE value enables smaller size of the array frac[],and increases computation speed. The value of BASE is defined ind VpInit(). In 32 bit integer system, this value is 10000. In 64 bit integer system, the value is 1000000000. When BASE is 10000,an element of the array frac[] can have value of from 0 to 9999. (up to 4 digits).
The structure Real is defined in bigdecimal.h as:
`

      typedef struct {
         VALUE  obj;     /* Back pointer(VALUE) for Ruby object.         */
         unsigned long MaxPrec; /* The size of the array frac[]          */
         unsigned long Prec;    /* Current size of frac[] actually used. */
         short    sign;         /* Attribute of the value.  */
                                /*  ==0 : NaN               */
                                /*    1 : +0                */
                                /*   -1 : -0                */
                                /*    2 : Positive number   */
                                /*   -2 : Negative number   */
                                /*    3 : +Infinity         */
                                /*   -3 : -Infinity         */
         unsigned short flag;   /* Control flag             */
         int      exponent;     /* Exponent value(0.xxxx*BASE**exponent) */
         unsigned long frac[1]; /* An array holding mantissa(Variable)   */
      } Real;

` The decimal value 1234.56784321 is represented as(BASE=10000):

        0.1234 5678 4321*(10000)**1

where frac[0]=1234,frac[1]=5678,frac[2]=4321, Prec=3,sign=2,exponent=1. MaxPrec can be any value greater than or equal to Prec.

## Binary or decimal number representation

I adopted decimal number representation for BigDecimal implementation. Of cource,binary number representation is common on the most computers.

### Advantages using decimal representation

The reason why I adopted decimal number representation for BigDecimal is:

Easy for debugging
  The floating number 1234.56784321 can be easily represented as:
  frac[0]=1234,frac[1]=5678,frac[2]=4321,exponent=1,and sign=2.
Exact representation
  Following program can add all numbers(in decimal) in a file without any error(no round operation).
  `

         file = File::open(....,"r")
         s = BigDecimal::new("0")
         while line = file.gets
            s = s + line
         end

  ` If the internal representation is binary,translation from decimal to binary is required and the translation error is inevitable. For example, 0.1 can not exactly be represented in binary.
  0.1 => b1*2**(-1)+b1*2**(-2)+b3*2**(-3)+b4*2**(-4)....
  where b1=0,b2=0,b3=0,b4=1...
  bn(n=1,2,3,...) is infinite series of digit with value of 0 or 1, and rounding operation is necessary but where we should round the series ? Of course, exact "0.1" is printed if the rounding operation is properly done,
Significant digit we can have is automatically determined
  In binary representation,0.1 can not be represented in finite series of digit. But we only need one element(frac[0]=1) in decimal representation. This means that we can always determine the size of the array frac[] in Real structure.

### Disadvantage of decimal representation

Because most computers have no internal decimal representation. Once you use BigDecimal,you need to keep using it without considering computation cost if exact computation is required.

#### Which is the first input?

Because most people uses decimal notation for numeric data representation, BigDecimal can handle numeric data without loss of translation error.

## Resulting number of significant digits

For the fundamental arithmetics such as addition,subtraction, multiplication,and division,I prepared 2 group of methods

### 1. +,-,*,/

For the operation + - * /,you can not specify the resulting number of significant digits.
Resulting number of significant digits are defined as:
1.1 For *,resulting number of significant digits is the sum of the significant digits of both side of the operator. For / ,resulting number of significant digits is the sum of the maximum significant digits of both side of the operator.
1.2 For + and -,resulting number of significant digits is determined so that no round operation is needed.
For example, c has more than 100 significant digits if c is computed as:
c = 0.1+0.1*10**(-100)
As +,-,and * are always exact(no round operation is performed unless BigDecimal.limit is specified), which means more memory is required to keep computation results. But,the division such as c=1.0/3.0 will always be rounded.

### 2. add,sub,mult,div

The length of the significant digits obtained from +,-,*,/ is always defined by that of right and left side of the operator. To specify the length of the significant digits by your self, use methos add,sub,mult,div. `

     BigDecimal("2").div(3,12) # 2.0/3.0 => 0.6666666666 67E0

`

### 3. truncate,round,ceil,floor

Using these methods,you can specify rounding location relatively from decimal point. `

     BigDecimal("6.66666666666666").round(12) # => 0.6666666666 667E1

`

### 4. Example

Following example compute the ratio of the circumference of a circle to its diameter(pi=3.14159265358979....) using J.Machin's formula.
`

    #!/usr/local/bin/ruby

    require "bigdecimal"
    #
    # Calculates 3.1415.... (the number of times that a circle's diameter
    # will fit around the circle) using J. Machin's formula.
    #
    def big_pi(sig) # sig: Number of significant figures
      exp    = -sig
      pi     = BigDecimal::new("0")
      two    = BigDecimal::new("2")
      m25    = BigDecimal::new("-0.04")
      m57121 = BigDecimal::new("-57121")

      u = BigDecimal::new("1")
      k = BigDecimal::new("1")
      w = BigDecimal::new("1")
      t = BigDecimal::new("-80")
      while (u.nonzero? && u.exponent >= exp)
        t   = t*m25
        u   = t.div(k,sig)
        pi  = pi + u
        k   = k+two
      end

      u = BigDecimal::new("1")
      k = BigDecimal::new("1")
      w = BigDecimal::new("1")
      t = BigDecimal::new("956")
      while (u.nonzero? && u.exponent >= exp )
        t   = t.div(m57121,sig)
        u   = t.div(k,sig)
        pi  = pi + u
        k   = k+two
      end
      pi
    end

    if $0 == __FILE__
      if ARGV.size == 1
        print "PI("+ARGV[0]+"):\n"
        p big_pi(ARGV[0].to_i)
      else
        print "TRY: ruby pi.rb 1000 \n"
      end
    end

`

Shigeo Kobayashi (E-Mail:<shigeo@tinyforest.gr.jp>)