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```HELP NUMBERS                                          Nic Ford, Jul 1987
A.Sloman March 1989

Keywords: integer, biginteger, number, decimal, ddecimal, ratio,
complex, printing.

CONTENTS - (Use <ENTER> g to access required sections)

-- Introduction
-- Integers and Bigintegers
-- Ratios
-- Decimals and ddecimals
-- Complex Numbers
-- Printing numbers in binary, octal, hexadecimal, etc
-- Associated documentation

-- Introduction --------------------------------------------------------

There  are  six  different  types  of  number  within  POP-11:  integer,
biginteger, ratio, decimal, ddecimal and complex number. This help  file
briefly describes what each type is and the role it plays within POP-11,
and then details the lexical syntax used to represent it within POP-11 -
that is, the sequences of characters which can be used to build  numbers
of that type.

-- Integers and Bigintegers --------------------------------------------

An integer  in POP-11  is a  rational number  with no  fractional  part.
Although decimals can also have no fractional part (or, more accurately,
a fractional  part of  zero) integers  are useful  because they  can  be
stored  more  compactly  and   accessed  more  quickly  than   decimals.
Bigintegers are integers  which cannot  be stored  compactly because  of
their magnitude.

An integer can be represented in a program file as:

a)  A sequence  of digits  (characters  in the  range `0`  to  `9`),
optionally preceded by a minus sign.

Examples: 463, -67

b)  An integer INT1  directly  preceded by an  integer of type  (a),
INT2, and a colon - INT2 must be in the range 2-36, and is taken
as the  base of  the  integer (for  digits  greater than  9  the
characters A-Z are  used). No  digit in  INT1 may  have a  value
equal to  or  greater than  INT2.  If  either INT1  or  INT2  is
negative (not both), then INT1 will be forced negative.

Examples: 2:11, -8:5641, 16:-FFA5E

c)  A character  constant (the  character "`",  followed by  another
character, optionally followed by  another "`"). This yields  an
integer in the range 0-255.

Examples: `A`, `B, `\^H`

Integers of type (a)  and (b) may optionally  be followed the  character
`e`, and  a signed  or unsigned  class (a)  integer -  this defines  the
exponent of  the  integer. Thus  the  sequence "R:NeI"  will  build  the
integer N *  (R ** I)  WARNING: Specifying an  exponenent may force  the
building of a ratio instead of an integer.

Examples: 3e2, 12:B98Ae7

A biginteger is  created whenever  a character  sequence evaluates  to a
number too large to be a simple integer.

Examples: 3e67, 45e10

-- Ratios --------------------------------------------------------------

A ratio  is a  decimal  number which  has  a non-zero  fractional  part,
described as the ratio of two integers. A ratio can be used to represent
an irrational  number (that  is,  a decimal  with an  "infinitely  long"
fractional part).

A ratio is built from an integer  of type (a) or (b), INT1, followed  by
the character sequence "_/", followed by  an integer of type (a),  INT2,
which may  not be  negative. If  INT1 is  of type  (b) (that  is, it  is
preceded by a radix) then that radix is also used for INT2. A ratio will
be translated to its simplest possible representation (i.e. "3_/6"  will
be translated  into the  ratio "1_/2"),  and if  the denominator  (INT2)
evaluates to 1, then the resulting number will be an integer.

Examples: 3_/10, -1_/2, 18:GF_/45

The variable * POP_PR_RATIOS (default true) can be made false to
make ratios print like decimals.

false -> pop_pr_ratios;
3/4 =>
** 0.75

true -> pop_pr_ratios;
3/4 =>
** 3_/4

When it is false it is only PRINTING that is altered. To prevent
the creation of ratios ensure that division is by a decimal not
an integer, e.g.

true -> pop_pr_ratios;
3/4.0 =>
** 0.75

Since ratios are records that require temporary storage in the heap,
repeated division of integers by integers can lead to a lot of store
use, and trigger unwanted garbage collections. See HELP * EFFICIENCY

-- Decimals and ddecimals ----------------------------------------------

A decimal is a  floating point number -  a rational number,  represented
with a fractional part. Decimals with  a fractional part of zero may  be
represented as integers. Irrational decimals are coerced into ratios  by
POP-11.

A floating point number is represented by a sequence of characters,  all
of which are digits, except for one,  which is a period (the period  may
not appear at the start  or end of the  sequence). As with integers,  if
the number is preceded by an integer and a colon, then that integer will
be taken as the base.

Examples: 0.6557, 45.56, 9:67.0

Also as with integers, an exponent may be specified, using the character
`e` in the same way. In addition, the characters `d` and `s` may be used
in place of  `e` - both  `d` and `e`  force the resultant  number to  be
double precision (a DDECIMAL), `s` forces  it to be single precision  (a
DECIMAL). A sequence of characters denoting a floating point number will
always be coerced into a DDECIMAL by POP-11, unless the `s` exponent  is
used in this way.

Examples: 232.056e-2, 5:131.43d5, 345.2s+7

-- Complex Numbers -----------------------------------------------------

A complex  number is  made  up of  two parts:  its  real part,  and  its
imaginary. Both  parts (confusingly)  are real  numbers -  however,  the
imaginary part is multiplied  (or, considered to  be multiplied -  ask a
mathematician) by the  square root  of minus  one before  any and  every
mathematical function is carried out.  Complex numbers are described  in
two parts because the square  root of minus one  cannot exist as a  real
number, and so must be kept separate from the mathematical "real" world.

A complex number in POP-11 is represented by any two of the above  types
of number  (the  first  representing  the  real  part,  the  second  the
imaginary) separated  by the  character sequences  "_+:" or  "_-:".  The
imaginary part of a complex number may  not be preceded by a minus  sign
or base specification - if the real part has a base then it will be used
by the imaginary also, and if the  imaginary part has to be negative  it
is made such by use of the "_-:" sequence. Unlike integers, if the  real
part both is negative, and  has a base, then  the unary minus sign  MUST
succeed the radix and preceed the number (i.e. "16:-A_+:B" must be  used
in preference to "-16:A_+:B").

Real and  imaginary parts may  be of different  types - however, if  the
are both floating point then  they will be coerced  to the same type  of
floating point (i.e. DECIMALS or DDECIMALS). If the imaginary part  of a
complex number is zero, and both parts are rational, then the  resultant
number will be not complex, but one of the other number types.

Examples: 6_+:7, 1.2_-:5.6, 3:21.1_+:1_/2

-- Printing numbers in binary, octal, hexadecimal, etc ----------------

By default, numbers are printed in decimal format. However the variable
*POP_PR_RADIX can be given an integer value between 2 and 36 to alter
the base. E.g. to print in hexadecimal

2, 8,9, 15, 16, 17, 31, 32, 33 =>
** 2 8 9 F 10 11 1F 20 21

The procedure RADIX_APPLY, described in REF * PRINT, can be used
to apply an arbitrary printing function with a given base. E.g.

F

F.E66666

F.000

There are several different global variables, with names starting
'pop_pr' that control printing formats.

HELP * POP_PR_EXPONENT
HELP * POP_PR_PLACES
HELP * POP_PR_QUOTES
HELP * POP_PR_RATIOS
HELP * PRINT
HELP *PRNUM
REF * PRINT

For full information on the formats for reading in different sorts of
numbers, see

REF * ITEMISE

-- Associated documentation -------------------------------------------

HELP files concerned with numbers and mathematical procedures:

HELP *MATH           - mathematical procedures in POP-11
HELP *ABS            - returns the absolute value of a number
HELP *BIGINTEGERS    - on BIGINTEGERS
HELP *COS            - returns the cosine of a number
HELP *DIV            - divide with remainder and quotient
HELP *DECIMALS       - on DECIMALS and DDECIMALS
HELP *EXP            - returns the exponential of a number
HELP *FRACOF         - returns the fractional part of a decimal
HELP *INTOF          - returns the integral part of a decimal
HELP *ISBIGINTEGER   - check an item is a BIGINTEGER
HELP *ISDDECIMAL     - check an item is a DDECIMAL
HELP *ISDECIMAL      - check an item is a DDECIMAL or DECIMAL
HELP *ISINTEGER      - check an item is an INTEGER or BIGINTEGER
HELP *LOG            - returns the log to base 2 of a number
HELP *PI             - variable holding the value of PI
HELP *RANDOM         - the random number generator
HELP *RANSEED        - the seed for the random number generator
HELP *REALOF         - returns the decimal version of a number
HELP *SIGN           - returns the sign of a number
HELP *SIN            - returns the sine of a number

TEACH files concerned with numbers:

TEACH *ARITH          - arithmetic in POP-11
TEACH *DECIMALS       - tutorial on decimals and ratios in POP-11

REF files concerned with numbers:

REF *NUMBERS        - on numbers & procedures using numbers in POP-11

--- C.all/help/numbers -------------------------------------------------