Help / Learning / Ch 19: Numbers
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Chapter 19: NumbersThe topics covered in this chapter are:
19.1 Numbers of Six Different KindsJ supports computation with numbers of these kinds:
Each kind of number has its own internal representation in memory. For example, an array containing only the truth-values 0 and 1
is stored in a compact internal form, called "boolean",
rather
than in the floating-point form.
Similarly an array containing only (relatively small)
whole numbers is stored in a compact form
called "integer". types =: 'bool';'int';'float';'complex';'ext int';'rational'
type =: > @: ({ & types) @: (1 4 8 16 64 128 & i.) @: (3 !: 0)
19.1.1 BooleansThere are built-in functions for logical computation with boolean values. Giving conventional names to these functions: and =: *. or =: +. not =: -. notand =: *: notor =: +: we can show their truth-tables: p =: 4 1 $ 0 0 1 1 q =: 4 1 $ 0 1 0 1
Further logical functions can be defined in the usual way. For example, logical implication, with the scheme p implies q means not (p and not q) is defined by not composed with the hook and not implies =: not @ (and not)
Notice that in the truth-table above the rows are given in an order
such that p,q = successively 00 01 10 11 in binary or 0 1 2 3.
Call this the standard order. to define implication as 13 b. We see:
We regard the booleans as numbers because they can
be interpreted as having arithmetic values. To illustrate,
implication has the same truth-table as
less-than-or-equal:
For another example of booleans as numbers,
the sum of the positive numbers in a list is shown by:
19.1.2 IntegersOn a 32-bit machine integers range between
_2147483648 and 2147483647.
19.1.3 Bitwise Logical Functions on IntegersJ provides all the expected functions on integers, so not much need be said here. However, this might be a good place to mention that bitwise logical functions on integers are available through the built-in adverb b. we met above. To begin, here is a utility function to show the first and last few bits of an integer (assumed 32-bit). bits =: (3 : ' (8{.y), ''...'', (24 }.y) ') @: ({&'01')@:((32#2) & #:)
Recall that logical function k is given by k b. where k is in the range 0-15.
The function (k+16) b. is logically the same, but applies, not to booleans, but to integers bitwise, that is, on machine words. bits 15 (17 b.) _15 00000000...00000001 The verb (32 b.) rotates the bits of y leftward by x places, or rightward for negative x. Similarly (33 b.) shifts and (34 b.) performs a "signed shift" that is, propagating the sign bit on a rightward move. For example: ] bits a =: 1 00000000...00000001 ] bits b =: _1 (32 b.) a NB. rotating rightwards 10000000...00000000 ] bits c =: _1 (34 b.) b NB. shifting right, propagating sign-bit 11000000...00000000 ] bits d =: 2 (33 b.) c NB. shifting left, removing sign-bits 00000000...00000000 For one more example, recall the Collatz function from Chapter 10 : halve if even, otherwise triple and add one. Here is a bitwise version. odd =: (17 b.) & 1 halve =: _1 & (33 b.) NB. OK for an even number ! triple =: + (1 & (33 b.)) collatz =: halve ` (1 + triple) @. odd collatz ^: (i. 10) 5 5 16 8 4 2 1 4 2 1 4
19.1.4 Floating-Point NumbersA floating-point number is a number represented in the computer in such a way that:
(1) there may be a a fractional part as well as a whole-number part.
(2) a fixed amount of computer storage is occupied by the number, whatever the value of the number.
and therefore (3) the precision with which the number is represented is limited to
at most about 17 significant decimal digits (on a PC). 19.1.5 Scientific NotationWhat is sometimes called "scientific notation" is a convenient way of writing very large or very small numbers. For example, 1500000 may be written as 1.5e6, meaning 1.5 * 10^6. The general scheme is that a number written in the form XeY, where Y is a
(positive or negative) integer means (X * 10^Y).
Note that in 3e2 the letter e is not any kind of function;
it is part of the notation for writing numbers,
just as a decimal point is part of the notation. 19.1.6 Comparison of Floating-Point NumbersTwo numbers are regarded as equal if their difference is relatively small.
For example, we see that a and b have a non-zero difference,
but even so the expression a = b produces "true".
If we say that the "relative difference" of two numbers is the magnitude of the difference divided by the larger of the magnitudes: RD =: (| @: -) % (>. &: |) then for a=b to be true, the relative difference
(a RD b) must not exceed a small value
called the "comparison tolerance" which is by default 2^_44
Thus to compare two numbers we need to compare relative difference with tolerance.
The latter comparison is itself strict, that is, does not involve any tolerance.
However, 1+tiny is tolerantly equal to 1.
The value of the comparison tolerance currently in effect
is given by the built-in verb
9!:18 applied to a null argument. It is currently 2^_44.
Applying the built-in
verb 9!:19 to an argument y sets the tolerance to y
subsequently.
The following example shows that when the
tolerance is 2^_44, then
a = b but when the tolerance is set to zero
it is no longer the case that a = b.
The tolerance queried by 9!:18 and set by 9!:19 is a global parameter, influencing the outcome of computations with =. A verb to apply a specified tolerance t, regardless of the global parameter, can be written as = !. t. For example, strict (zero-tolerance) equality can be defined by: streq =: = !. 0 Resetting the global tolerance to the default value, we see:
Comparison with = is tolerant,
and so are comparisons with <, <:, >, >:, ~: and -:.
For example, the difference a-b is positive but too small to make it true that
a>b
Permissible tolerances range between 0 and 2^_35.
That is, an attempt to set the tolerance larger
than 2^_35 is an error:
The effect of disallowing large tolerances is that no two different integers compare equal when converted to floating-point. 19.1.7 Complex NumbersThe square root of -1 is the imaginary number conventionally
called "i".
A complex number which is conventionally written as,
for example, 3+i4 is in J written as 3j4.
A complex number can be built from two separate real numbers by arithmetic
in the ordinary way,
or more conveniently with the built-in function j. (lowercase j dot, called
"Complex").
Some more examples of arithmetic with complex numbers:
A complex number such as 3j4 is a single number, a scalar. To extract
its real part and imaginary part separately we can use the built-in verb
+.(plus dot, called "Real/Imaginary").
To extract separately the magnitude and angle (in radians) we can use the
built-in verb *. (asterisk dot, called "Length/Angle").
Given a magnitude and angle, we can build a complex number by taking sine and cosine, or more conveniently with the built-in function r. (lowercase r dot, called "Polar"). sin =: 1 & o. cos =: 2 & o. mag =: 5 ang =: 0.92729522 NB. radians
A complex constant with magnitude X and angle (in radians) Y
can be written in the form
XarY, meaning X r. Y. Similarly, if the angle is given in degrees,
we can write XadY.
19.1.8 Extended IntegersA floating-point number, having a limited storage space in the computer's memory, can represent an integer exactly only up to about 17 digits. For exact computations with longer numbers, "extended integers" are available.
An "extended integer" is a number which exactly represents an integer no matter how many digits are needed.
An extended integer is written with the digits followed with the letter 'x'. Compare the following:
Here a is an approximation while b is an exact result.
We can see that adding 1 to a makes no difference, while adding 1 to b does make a difference:
19.1.9 Rational NumbersA "rational number" is a single number which represents exactly the ratio of two
integers, for example, two-thirds is the ratio of 2 to 3. Two-thirds
can be written as a rational number with the notation 2r3.
Rationals can be constructed by dividing extended integers.
Compare the following:
A rational can be constructed from a given floating-point
number with the verb x:
A rational number can be converted to a floating-point approximation with the inverse ofx: , that is, verb x: ^: _1
Given a rational number, its numerator and denominator can
be recovered with the verb 2 & x:, which gives a list of length 2.
19.1.10 Type ConversionWe have numbers of six different types:
boolean, integer, extended integer, rational, floating-point and complex.
The general scheme is that the six types form a progression: from boolean to integer to extended to rational to floating-point to complex.
We say that boolean is the simplest or "lowest" type
and complex as the most general or "highest" type
19.2 Special Numbers
19.2.1 "Infinity"A floating-point number can (on a PC) be no larger than about 1e308,
because of the way it is stored in the computer's memory.
Any arithmetic which attempts to produce a larger result
will in fact produce a special number called "infinity"
and written _ (underscore). For example:
There is also "negative infinity" written as __ (underscore underscore). Infinity is a floating-point number: type _ float
19.2.2 "Indeterminate" NumbersA floating-point number is a 64-bit value, but not all 64-bit values are
valid as floating-point numbers.
Any which is not valid is said to be "Not a Number", or a "NaN".
Such a value might occur, for example, in data imported into a
J program from an unreliable external source.
19.3 Number BasesThe number 5 is represented in binary as 1 0 1. There is a built- in function, monadic #: ( hash colon, called "Antibase Two" ) to compute the binary digits. Note that the result is a list. #: 5 1 0 1 We say that the binary digits are the base-2 representation. More generally, a base-n representation can be produced. The left argument of dyadic #: (called "Antibase") specifies the both the base and the number of digits. To get four binary digits we can write: 2 2 2 2 #: 5 0 1 0 1 63 as two base-8 (octal) digits: 8 8 #: 63 7 7 A mixed-base representation is possible. How many hours, minutes and seconds are there in 7265 seconds? 24 60 60 #: 7265 2 1 5 The inverse functions produce numbers from lists of digits in specified bases. Monadic #. is called "Base Two". Binary 1 0 1 is 5 #. 1 0 1 5 Dyadic #. is called "Base". Its left argument specifies a number-base for the digits of the right argument. 2 #. 1 0 1 5 Equivalently: 2 2 2 #. 1 0 1 5 There must be a base specified on the left for each digit on the right, otherwise an error is signalled 2 2 #. 1 0 1 |length error | 2 2 #.1 0 1 |[-535] c:\users\homer\14\js\19.ijs Again, mixed bases are possible: 2 hours 1 minute 5 seconds is 7265 seconds 24 60 60 #. 2 1 5 7265
19.4 Notations for NumeralsWe have seen above numerals formed with the letters e, r and j, for example:
1e3, 2r3, and 3j4. Here we look at more letters for forming numerals.
Similarly, a numeral written with letter x, of the form XxY means X * e ^ Y
where e is the familiar value 2.718281828....
These p and x forms of numeral provide a convenient way of writing constants accurately without writing out many digits. Finally, we can write numerals with a base other than 10. For example the binary or base-2 number with binary digits 101 has the value 5 and can be written as 2b101. 2b101 5 The general scheme is that NbDDD.DDD is a numeral in number-base N with digits DDD.DDD .
With bases larger than 10, we will need
digits larger than 9, so we take letter 'a' as a digit with value 10, 'b' with value 11,
and so on up to 'z' with value 35.
One more example. 10b0.9 is evidently a base-10 number meaning "nine-tenths" and so, in base 20, 20b0.f means "fifteen twentieths" 10b0.9 20b0.f 0.9 0.75 19.4.1 Combining the NotationsThe notation-letters e, r, j ar ad p x and b may be used in combination. For example we can write 1r2p1 to mean "pi over two". Here are some further examples of possible combinations. A numeral in the form XrY denotes the number
X%Y. A numeral in the form XeYrZ denotes
the number (XeY) % Z because e is considered before r.
A numeral in the form XjY denotes the complex number
(X j. Y) (that is, (X + (%: _1) * Y).
A numeral in the form XrYjZ denotes the number
(XrY) j. Z because r is considered before j
A numeral in the form XpY denotes the number
X*pi^Y. A numeral in the form XjYpZ denotes
(XjY) *pi^Z because j is considered before p.
A numeral in the form XbY denotes the number
Y-in-base-X. A numeral in the form XpYbZ
denotes the number Z-in-base-(XpY) because
p is considered before b.
19.5 How Numbers are DisplayedA number is displayed by J with, by default, up to 6 or 7
significant digits.
This means that, commonly, small integers are shown exactly,
while large numbers, or numbers with many
significant digits, are shown approximately.
The number of significant digits used for display is determined by a global variable called the "print-precision". If we define the two functions: ppq =: 9 !: 10 NB. print-precision query pps =: 9 !: 11 NB. print-precision set then the expression ppq gives the value
of print-precision currently in effect, while
pps n will set the print-precision to n.
19.5.1 The "Format" VerbThere is a built-in verb ": (doublequote colon, called "Format"). Monadic Format converts a number into a string representing the number with the print-precision currently in effect. In the following example,
note that a is a scalar,
while the formatted representation of a is a list of characters.
The argument can be a list of numbers and the result is
a single string.
Dyadic Format allows more control over the
representation.
The left argument is complex: a value of
say, 8j4 will format the numbers in a width of
8 characters and with 4 decimal places.
If the width is specified as zero (as in say 0j3)
then sufficient width is allowed. If the number of decimal places
is negative (as in 10j_3) then numbers are shown in "scientific notation"
Note that monadic format shows a complex number
in the usual way, but dyadic format shows only the real part of a complex number.
More formatting options are
provided by the built-in verbs 8!:n.
Here is a small example to show a few of the many options described in the
J dictionary. ] N =: 3 2 $ 3.8 _2000 0 123.45 _3.14 15000
3.8 _2000
0 123.45
_3.14 15000
We can format each column of N separately. Suppose numbers in the first column are to be presented as blank when zero, 6 characters wide with 0 decimal places. We write a "formatting phrase" like this fp1 =: 'b6.0' Here the 'b' means blank when zero. 2 decimal places. A suitable formatting phrase is like this: fp2 =: 'cn{CR}q{ }12.2'
Here the 'c' means commas, n{CR} means CR after a negative number
and q{ } means 2 spaces after a non-negative.
The formatted result is a character table of dimensions 3 by 18, because N has 3 rows, and we specified widths of 6 and 12 for first and second columns. $ (fp1;fp2) (8!:2) N 3 18
19.6 Random Numbers19.6.1 RollThere are built-in functions for generating random numbers. Monadic ? is called "Roll", because ? n gives the result of rolling a die with n faces marked 0 to n-1. ? 6 4 That is, ? n is selected from the items of i. n
randomly with equal probability. ? 6 6 6 6 3 4 5 2 or more conveniently by: ? 4 $ 6 2 3 5 3 19.6.2 Uniform DistributionWith an argument of zero, monadic ? generates random reals uniformly distributed, greater than 0 and less than 1. ? 0 0 0 0 0.14112615 0.083891464 0.41388488 0.055053198 19.6.3 Other DistributionsThe built-in verb ? generates equiprobable numbers. Various other distributions are provided by the J Application Library stats/distribs Addon 19.6.4 DealDyadic ? is called "Deal". x ? y is a list of x integers randomly chosen
from i. y without replacement, that is, the x integers are all different. 13 ? 52 44 2 36 30 0 6 43 26 28 1 34 48 41 A shuffle of the whole deck is given by 52 ? 52. To shuffle and then deal four hands: 4 13 $ 52 ? 52 15 18 3 8 11 25 27 51 20 31 50 48 35 45 39 21 29 10 33 32 41 36 0 34 16 22 19 14 37 2 24 42 6 7 30 46 47 28 26 38 23 40 13 4 9 5 12 49 1 44 43 17 This brings us to the end of Chapter 19. |
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The examples in this chapter were executed using J version 701.
This chapter last updated 02 Jul 2013
Copyright © Roger Stokes 2012.
This material may be freely reproduced,
provided that this copyright notice is also reproduced.
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