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
-- Reading numbers in program text or using readitem, 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
(See also REF * ITEMISE)
-- 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
16 -> pop_pr_radix;
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.
radix_apply(15,pr,16);
F
radix_apply(15.9,pr,16);
F.E66666
radix_apply(15,8,4,prnum,16);
F.000
There are several different global variables, with names starting
'pop_pr' that control printing formats.
For more information on printing numbers (and other things) see
HELP * POP_PR_EXPONENT
HELP * POP_PR_PLACES
HELP * POP_PR_QUOTES
HELP * POP_PR_RADIX
HELP * POP_PR_RATIOS
HELP * PRINT
HELP *PRNUM
REF * PRINT-- Reading numbers in program text or using readitem, etc. ------------
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 *POPRADIANS - specifies whether using radians or degrees
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 -------------------------------------------------
--- Copyright University of Sussex 1991. All rights reserved. ----------