Easy-J 

An Introduction to the World’s most 
Remarkable Programming Language  

by Linda Alvord and Norman Thomson 

October 2002

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Contents 

Introduction  
Section 1.  Getting Started 
Section 2.  Introducing Simulation 
Section 3.  Defining your own verbs   
Section 4.  Fitting equations to data 
Section 5.  Boxes and Records 
Section 6.  Investigating Possibilities  
Section 7.  Editing, System Facilities and Scripts 
Section 8.  Drawing graphs 
Section 9.  Rank 
Index   
Vocabulary 

Introduction 

  1        
  2 
13 
16  
24 
26    
29 
33 
36 
39 
43 
44 

J is both a language and an exceptional programming package which provides a 
highly concise notation for specifying much that is done routinely in the day to day 
business of computing, such as sorting, searching, updating and restructuring data.  Its 
inventor and designer is Dr. K.E. Iverson, who also devised the language APL, out of 
which J developed.  The first J interpreters appeared around 1990, since when the 
language has grown in popularity and application, particularly in the world of finance, 
where its conciseness and power for rapid algorithm development is highly valued.  
Amazingly, this algorithm-rich software is available free by download from  

www.jsoftware.com 

a download which takes only a few minutes, following which installation is easy. 

J also has many enthusiasts in education, where it can be a powerful motivator on 
account of the clarity with which users can express their intentions on a computer. 

The objective of this tutorial is to give a brief introduction which will either 
encourage the user to perform the above download and discover that the claims which 
are made on this page are in no way understated, or, if the download has already been 
performed, encourage him or her to take the first steps up the J learning ladder into a 
world of discovery and delight.  It is impossible in a pamphlet of this size to cover 
anything other than a tiny part of the facilities which J affords.  However, that need be 
no disadvantage, since the J system is fully self-describing through a comprehensive 
Help facility, through Labs (that is prepared sessions) which can be invoked by the 
Studio drop down menu, and through libraries of scripts and packages which 
encapsulate the work of many previous users.  Once the topics in this tutorial have 
been grasped, the user should find little difficulty in extending his or her knowledge 
both by exploiting the help features listed above and by direct exploration on the 
keyboard.   Further, the J language is totally consistent across all the many platforms 
on which it is now available, and a vigorous Internet forum (forum@ jsoftware.com) 
is perhaps the best support mechanism of all.       

 1

 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
SECTION 1 : GETTING  STARTED 

This tutorial assumes that the reader has either 

 (i)  successfully installed J on a computer and is ready to use it; or 
(ii)  is interested in J as a vehicle for reading and writing algorithms and wishes to 
obtain something of its flavour prior to using it on a computer. 

J in its simplest use acts just like a desk calculator.  If you are working at a terminal, 
type the following input, pressing the “enter” key after each line : 

   3+4 
7 
   5*20 
100 

Symbols like + and * for plus and times in the above phrases are called verbs and 
represent functions.  You may have more than one verb in a J phrase, in which case it 
is constructed like a sentence in simple English by reading from left to right, that is 
4+6%2 means 4 added to whatever follows, namely 6 divided by 2 : 

   4+6%2 
7 

Here is another phrase : 

   4%6+2 
0.5 

In a similar way this means 4 divided by everything to the right, namely 6+2 which is 
8.  Hence 4%6+2 evaluates to 4%8 = 0.5. Notice how the verbs themselves are 
executed in left to right order, that is the rightmost verb + is executed before the 
leftmost, %.  The simplicity of these rules for reading phrases on the one hand and 
executing them on the other avoids the need for artificial rules of precedence such as 
apply in ordinary arithmetic to state that for example * has priority over +. 

Pursuing the analogy with English, the sentence “Jack visited the house which he 
built” is read from left to right, but in order to “execute” it you must do the “build” 
first, so that you can properly identify which house was visited. 

The repertoire of basic arithmetic verbs   +  -  *  %    in J is completed with the 
power verb ^ : 

   4^3   
64 

If you wish to change the order of execution of verbs, you may do so in the normal 
way by using parentheses : 

 2

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   1+2%3 
1.66667 
   (1+2)%3 
1 

You might wonder why the first of these answers was represented to 6-digit precision 
rather than any other.  The reason is that print precision is controlled by using a so-
called foreign conjunction – foreign because it “belongs” to the system rather than to 
the J language proper.   The default value is 6 can be confirmed by typing 

   9!:10 '' 
6 

NB. get the current print precision 

and reset to other values, say 4, by : 

   (9!:11)4 
   1+2%3 
1.667 

NB. set print precision 

You should think of  9!:10 and  9!:11 as further verbs which perform system 
functions within the J workspace environment.  The above lines also introduce the 
symbol NB. which indicates that everything to its right on a line should be read as a 
comment. 

The underbar symbol  _ is used to indicate “negative”, and is an inseparable part of a 
negative number.  You may not leave a space between the underbar and the digits of a 
number.  

   3-8 
_5 
   3-_8 
11 

Within the workspace, data is stored by assigning values to variables with names 
chosen by the user, subject to the requirement that the first character in such a name 
must be alphabetic : 

   b=.4 
   value1=._0.3 
   value1=.1.6  

NB. decimals <1 must have leading 0 
NB. a second assignment to value1 

Single-value items such as the above are known as “scalars”.  Entry of a name by 
itself causes the most recently assigned value of the named variable to be output : 

   b 
4 
   value1 
1.6 

 3

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
However, J is much more than just a calculator, since arbitrarily large lists of numbers 
(which are also sometimes called “vectors”)  can be assigned : 

   c=.3 1 4 0.5 _2 
   d=.4 0 3 _1.2 7 

Again the underbar symbol is used to express negative numbers.   Adding two lists 
means that corresponding items in each are added: 

   c+d 
7 1 7 _0.7 5 
   c*d 
12 0 12 _0.6 _14 
   d%c 
1.333 0 0.75 _2.4 _3.5 

Dividing c by d involves a division by 0 in the second position.  The result of this is 
infinity, denoted by a single underbar : 

   d%c 
0.75 _ 1.333 _0.4167 _0.2857 

There are some list operations which are not meaningful, for example adding a list 
with five items to one with only three.   If you attempt to do this the result is as 
follows :   

   c+1 2 3 
|length error 
|   c    +1 2 3 

Three observations should be noted about this error message : 

(a) it is concise; 
(b) the word “length” gives insight into the nature of the error; and 
(c) the added blank spaces indicate the exact point in the phrase where, 

reading from right to left, the error was detected.  

J allows complex numbers, so for some verbs such as square root, denoted by the 
digraph %:, may have results with a number separated without spaces by the letter j.  
The number 0j1.414 below indicates a real component of 0 and an imaginary 
component of  1.414. 

   %:c 
1.732 1 2 0.7071 0j1.414 

NB. square roots of c  

The natural logarithm verb ^. produces logarithms to the base e, and can also 
generate results containing complex numbers :  

   ^.c 
1.099 0 1.386 _0.6931 0.6931j3.142 

NB. natural logarithms 

 4

 
 
  
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
    
Logarithms to base 10 are obtained by : 

   10^.c 
0.4771 0 0.6021 _0.301 0.301j1.364 

and similarly for logarithms to any other number bases, e.g. 2 :   

   2^.1 2 4 8 20 
0 1 2 3 4.322 

If you wish to count or tally  #  the number of items in a list, type : 

   #c 
5 

The verb # is treated like any other verb, so 2 plus the tally of items in c is : 

   2+#c 
7 

The verb from  {   references items in a list. The fourth item in list c is : 

   3{c 
0.5 

You probably expected the answer _2; what you must take into account is that items 
in lists in J are always indexed from 0.  A list can be used to select from a list : 

   0 3 4{c 
3 0.5 _2 

NB. 0 3 4 is a list of indices 

Data need not be numeric – in the next example a list of characters is defined, and the  
characters themselves are tallied: 

   cv=.'J is useful.' 
   cv 
J is useful. 
   #cv 
12 

cv can be indexed by a numeric list containing repeated items : 

   0 7 10 10 2 7 3{cv  
Jellies 

In J a scalar is a single data item which can be either a number, or a literal character, 
which in turn may be a letter of the alphabet, digit or symbol.  A numeric scalar item 
is the value of the number represented by the combination of its characters, sign and 
decimal point.  Literal characters are enclosed in single quotes.  Thus the tally of the 
literal items in the representation of the single number  _3.875 is :   

 5

 
   
 
 
 
    
 
 
 
 
 
 
 
 
  
    
 
 
   #'_3.875' 
6 

A tally of the number as a numeric scalar is :  

   #_3.875 
1 

Some symbols such as # represent two different verbs.  In general, a verb has data on 
both its left and right, called left and right arguments.  Such a verb is called a dyadic 
verb.  A verb with only a right argument is described as monadic, thus square root and 
tally are monadic verbs.  As a dyadic verb # is called copy.  The left argument of 
copy is the number of copies of each item in the right argument : 

   3#c 
3 3 3 1 1 1 4 4 4 0.5 0.5 0.5 _2 _2 _2 

The dyadic verb reshape verb $ can create what appear to be matrices, but which are 
in fact lists of lists.  The left argument provides the structure and the right argument 
gives the data.  The data is reused by “wrapping round” : 

    matrix=.3 4$c 
  3  1 4 0.5 
 _2  3 1   4 
0.5 _2 3   1 

The monadic verb shape of $ gives the structure of the right argument and is always a 
list :     

    $matrix 
3 4 
    #matrix 
3 

Note that tally counts only the first item in the shape, that is  it counts at the topmost 
level only. For simple lists like c and d their tally and shape look identical. 

Arithmetic verbs (the words “verb” and “function” can often be used interchangeably) 
may also have one scalar argument and the other a list : 

   c 
3 1 4 0.5 _2 
   b 
4 
   c+b 
7 5 8 4.5 2 
   c*2 
6 2 8 1 _4 

 6

 
 
  
  
 
 
 
 
 
 
 
 
 
  
Technically what happens is that the scalar is replicated (that is extended) to become a 
list of matching length, and item by item function application takes place as before. 

Use the verb append, represented by a comma, to join lists and scalars : 

   c,b 
3 1 4 0.5 _2 4 

Arithmetic verbs have monadic as well as dyadic forms, for example : 

   +b 
4 
   -b 
_4 

Monadic + (called conjugate) does not change the value of its argument, provided that 
it is numeric and not complex.  However, it can be useful in simultaneously 
calculating and displaying a value : 

   +z=.4+6%2 
7 

Monadic minus is called negate , and monadic * is called signum, which returns 1 for 
a positive value of its argument, _1 for a negative value, and 0 for a zero value : 

   *b 
1 
   *-b 
_1 
   *b-b 
0 

Monadic % is the function reciprocal : 

   %b 
0.25 

The two digraph symbols <. and >. double up as minimum and maximum in their 
dyadic form, and as ceiling and floor (that is, next integer above and below) in their 
monadic form.  Like the arithmetic verbs, they can be applied to lists as well as to 
scalars.  Here are some examples : 

   +value1=.1.6 
1.6 
   >.value1 
2 
   <.value1 
1 
   b>.value1 
4 

NB. round up value1 

NB. round down value1 

NB. maximum of b and value1 

 7

 
 
 
 
 
 
 
 
 
 
 
    
 
 
 
 
   b<.value1 
1.6 
   c  
3 1 4 0.5 _2   
   d  
4 0 3 _1.2 7 
   c>.d   
4 1 4 0.5 7 
   0>.d   
4 0 3 0 7  

NB. minimum of b and value1 

NB. c and d as defined above  

NB. item by item maxima 

NB. d with negative items  
NB.       replaced by zero 

This is a good point at which to interrupt the description of J verbs with some 
elementary arithmetic examples which show how J can be applied to simple problems. 

(a)  Suppose you want to express 1 foot 4½ inches as a percentage of first 2 feet, then 
2 ft. 6 ins., 3 ft., 6 ft., and 10 ft., in each case rounding the answer to the nearest 
percentage above.  Using a list allows all five calculations to be done in parallel : 

   >.100*(1+4.5%12)%2 2.5 3 6 10 
69 55 46 23 14 

(b)  An Indian is said to have sold Manhattan Island to white settlers in 1650 for 12 
dollars.  What would be the dollar value of this sum in the year 2000 if invested at 
compound interest of 3%, 4%, 5%, 10% ? 

   (9!:11)16 

NB. set print precision to 16  

   <.12*(1+0.03 0.04 0.05 0.1)^(2000-1650) 
373430 10986263 312921872 3686557834057256 

(c)  A body falling from rest for y seconds drops a height of ½gt2 cms. where g=981 
cm./sec2.  Find the height fallen after 1,2,4,8,16 secs., then the velocities v at these 
gh2
times (the formula for velocity is v = 

). 

   (9!:11)6 

NB. set print precision to 6 

   +ht=.0.5*981*1 2 4 8 16^2  
490.5 1962 7848 31392 125568 
   +vel=.%:2*981*ht 
981 1962 3924 7848 15696 

Returning to the description of J verbs, the next one to be introduced is | which in its 
monadic form means absolute value, and in its dyadic form the remainder or residue 
when the right argument is divided by the left argument : 

   |c 
3 1 4 0.5 2 
   1.6|b 
0.8 

 8

 
 
 
 
 
 
  
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
Frequently one wants to add, multiply etc. all the items in a list.  This process is called 
insertion, and the notation for doing it is  

   +/c 
6.5 
   */c 
_12 
   -/c 
3.5 

The last example deserves a little more attention.  A more complete description of 
insertion is that the symbol / represents an adverb which when applied to a verb 
modifies its behaviour in some way.  The modification for / consists of placing the 
verb between each item in the list so that +/v is equivalent to 

          3 + 1 + 4 + 0.5 + _2 

and right to left execution (or alternatively left to right reading) takes place as usual. 

Try this with – in the gaps, and it will immediately become clear why -/c  is 3.5.  
You will notice also that 3.5 is the alternating sum of c, that is the sum of the items in 
odd-numbered positions (first, third, etc.) minus the sum of items in even-numbered 
items.  Following a similar argument, %/c is the alternating product of c.  Perhaps 
even more interesting is the effect of putting >. and <. in the gaps, which produces 
the largest and smallest items respectively in the list c. 

J has a full complement of relational verbs, that is 

<    <:    >    >:    =    ~: 

standing for less than, less than or equals, greater than, greater than or equals, 
equals and not equals respectively.  These are verbs which give Boolean results 
representing truth by 1 and falsity by 0.  As with arithmetic verbs, corresponding 
items in lists of the same length are processed in parallel : 

   c<d 
1 0 0 0 1 

Frequently an expression like the above is used as an intermediate step in a further 
operation such as a copy where 1 means “copy the item” and 0 means “don’t copy” :  

      (c<d)#c 
3 _2 
   (c>:d)#c 
1 4 0.5 

Another verb which returns Boolean (that is 0 or 1) values is the membership verb e.: 

   6 e. c 
0 

 9

 
 
 
 
 
 
 
  
 
 
 
 
 
 
Read this as a question “is 6 in c?”  If the left argument is a list, the question is asked 
for each item individually : 

   c e. b  
0 0 1 0 0  
   c e. d  
1 0 1 0 0 

NB. c is  3 1 4 0.5 _2,  b is 4  

NB. recall d is  4 0 3 _1.2 7 

The set of “logical” verbs also give Boolean results.  They are  

*.    +.    -.   “*:    +:  

which represent and, or, not, not-and and not-or respectively.  For example : 

   0 1 1 *.1 1 0 
0 1 0     
   -.c<d 
0 1 1 1 0 
   +./0 1 0 1 0 
1 

The symbol i. called index generator, produces a list of integers starting from 0: 

   i.6 
0 1 2 3 4 5 

If the integer argument is negative the result is a list in descending order : 

      i._6 
5 4 3 2 1 0 

An important special case arises if the argument is 0, when an empty list is generated : 

   i.0 

NB. list with zero items 

   #i.0   
0  
   $i.0   
0 
   $b 

   #b 
1 

NB. tally of empty list is scalar zero 

NB. shape is a one-item list, viz. zero   

NB. shape of a scalar is an empty list 

NB. .. but its tally is 1 

Mathematically inclined readers should also note the following : 

   +/i.0 
0 
   */i.0 
1  

 10

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
These are the “identity values” of the verbs to the left of /, that is, those values id  
for which x verb id is equal to x for all values of x.   Recall that comma means 
append, so +/c,x   and (+/c)++/x must be equal, since the sum of the list c,x is 
equal to the sum of the sums of its parts.  If x is i.0  then +/c,x is just +/c  and 
so +/i.0  must be the identity value for + .   A similar argument applies to * .  

Grids and Tables 

It is easy to generalise i. in order to make a list of any equally spaced values, as you 
might want to do in marking points on the axis of a graph.  For example, the integers 
from –4 to +4 are given by 

   _4+i.9 
_4 _3 _2 _1 0 1 2 3 4 

and those from 3 to 9 at intervals of one half by 

   3+0.5*i.13 
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 

The 256 ASCII characters are collected together in a constant noun a. called 
alphabet, in which the upper case letters of the usual alphabet commence at the item 
indexed as 65 and the lower case ones at 97.  Hence the lower case alphabetic 
characters are   

   (97+i.26){a. 
abcdefghijklmnopqrstuvwxyz 

The dyadic form of / gives verb tables, for example the ordinary addition and 
multiplication tables for the first five natural numbers are, written side by side 

   k=.1 2 3 4 5 
   k+/k   
2 3 4 5  6 
3 4 5 6  7 
4 5 6 7  8 
5 6 7 8  9 
6 7 8 9 10 

   k*/k 
1  2  3  4  5 
2  4  6  8 10 
3  6  9 12 15 
4  8 12 16 20 
5 10 15 20 25 

Tables can be used as indices and so all of the ideas in this subsection can be put 
together in the phrase :  

   (k*/k){(96+i.26){a. 
abcde 
bdfhj 
cfilo 
dhlpt 
ejoty 

 11

 
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
i. also has a dyadic form called index of  which is somewhat like e. Remembering 
that the first item has index 0, compare :  

   c 
3 1 4 0.5 _2 
   c e.b   
0 0 1 0 0 
   c i.b   
2 

NB. is c a member of b(= 4)?  

NB. what is the index of b in c?  

If both arguments are lists  

    d 
4 0 3 _1.2 7 
    d i.c  
2 5 0 5 5  

NB. reminder!  

NB. what are the d-indices of c? 

the three fives (one more than the largest index in c) indicate that the second, fourth 
and fifth items in c  do not occur in d.   

The verbs take  {.  and drop }.  do just this to a specified number of items from a 
list – from the left of the list if the left argument is positive, and from the right if it is 
negative : 

    3{.c 
3 1 4 
   _3{.c 
4 0.5 _2 
   3}.c 
0.5 _2 
   _3}.c 
3 1 

Used monadically, the default left argument is 1.  In addition, the verb  tail  {:  
selects the last item in a list : 

     {:c 
_2 

Take has the further property that “over-taking” is permitted, that is fill items (0 for 
numeric data, and blank for character) are used to pad out when necessary : 

   7{.c 
3 1 4 0.5 _2 0 0 
   _25{.'J is useful' 
              J is useful 

 12

 
 
 
 
 
 
 
 
 
 
  
 
SECTION 2 : INTRODUCING SIMULATION 

You have now seen that using J is in many ways more like handling a very 
sophisticated calculator than a computer.  Used in this way, it is a remarkable tool for 
solving routine problems in many disciplines.  The techniques in this section show 
some ways in which you can take advantage of its capability for taking samples. 

Classifying, tabulating and condensing actual or simulated data are all easy in J.  
Problems involving rolling dice and tossing coins are simple enough to observe what 
actually happens as well as to predict what you would theoretically expect to happen. 

The symbol ? is called roll as in “roll a die”, so that ?6 models rolling a die with six 
faces, or more generally randomly selecting one of the first six non-negative integers : 

   ?6 
0 
   ?6 
4 

NB. first throw 

NB. second throw  

The faces are usually numbered from 1 to 6.  Conveniently, the verb increment  >: 
adds 1 to each item(Likewise decrement  <: subtracts 1 from each item)  : 

NB. b is one greater than a 

   b=.>:a=.?6   
   a 
4 
   b 
5 

Suppose we had ten dice.  J extends roll to a list of integers so that the computer 
simulates the throws of ten dice, or equivalently ten throws of a single die : 

   >:?6 6 6 6 6 6 6 6 6 6 
4 2 1 5 5 6 3 4 5 1 

Suppose that we now simulate a multiple choice test in which there are ten questions 
each with five options : 

   +t=.?10#5 
3 4 4 4 1 2 2 0 2 2 

To convert these into, say, letters of the alphabet use from : 

   t{'abcde' 
deeebccacc 

In a similar way simulate the tossing of six coins where 0 represents a tail and 1 a 
head.  First make six copies of the integer 2 : 

   6#2 
2 2 2 2 2 2 

 13

 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
and then roll (that is) toss  each of them : 

   (?6#2){'th' 
hhthth 

Often what is interesting is aggregates rather than the individual rolls.  For example 
you might want to simulate several times the gender distribution of 1000 births in a 
hospital where 0 represents a girl and 1 a boy : 

   +/?1000#2 
501 
   +/?1000#2 
480 

All the above examples have used roll monadically. The dyadic case is called deal , 
for which the difference is that the left argument gives the number of draws and also 
the values of i.right argument are progressively deleted:,  

   6?6 
4 1 0 3 5 2 

This means that the right argument must be at least as great as the left argument, 
otherwise the left argument is outside the domain, that is, the permitted values   : 

   7?6 
|domain error 
|   7    ?6 

There are no limits to the shape of the structure of the units from which the draws are 
to be made.  For example, suppose the multiple choice test above was taken by each 
of five children, so that the simulation is now one of six ten-item lists, each item 
within which is a random draw from i.5 : 

   u=.?6 10$5 
4 2 3 4 3 4 3 3 0 4 
2 2 0 3 2 3 3 0 4 4 
4 2 0 2 4 1 2 1 2 4 
2 2 4 1 1 4 0 3 3 0 
0 3 3 1 2 3 3 1 4 3 
4 0 0 3 3 1 1 0 1 4 

Now make a draw from each of these, and translate into letters as before : 

   u{'abcde' 
ecdededdae 
ccadcddaee 
ecacebcbce 
ccebbeadda 
addbcddbed 
eaaddbbabe 

 14

 
 
  
 
 
 
 
 
 
   
 
 
A nice application of this technique is to simulate hands of 13 cards from a pack.  
First assume that the cards are ordered in the pack from smallest to highest, and in suit 
order Diamonds, Clubs, Hearts, Spades. (Whether any actual pack is physically 
ordered in this way makes no difference to the quality of the simulation.).  A draw is 
then a choice of 13 integers from 0 to 51 without replacement : 

   +hand=.13?52 
43 50 25 45 40 4 9 11 24 42 17 19 37 

Next calculate the suits by dividing these values by 13 and rounding down (the 
monadic is called ravel and causes the value of its argument, in this case suits , to 
be displayed as a simple list)  : 

   ,suits=.(<.hand%13){'CDHS' 
SSDSSCCCDSDDH 

Now calculate the values by taking remainders on division by 13 : 

   ,values=.(13|hand){'23456789TJQKA' 
6KA836JKK568K 

Finally use a verb laminate (,:) to align the cards nicely in columns : 

   values,:suits 
6KA836JKK568K 
SSDSSCCCDSDDH 

NB. 6 of spades, King of spades etc. 

Simple counts of combinatoric items are obtained by the verbs factorial !  and its 
dyadic form out of,  which gives the number of combinations of r objects out of n : 

NB. factorial 0 thru factorial 5  

NB. e.g.21=no of selections of 2 out of 7  

      !i.6 
1 1 2 6 24 120 
   2!2+i.6 
1 3 6 10 15 21 

Pascal’s Triangle 

The arrangement of integers known as Pascal’s triangle in which the binomial 
coefficients appear in columns is constructed using the table : 

   (i.8)!/i.8 
1 1 1 1 1  1  1  1 
0 1 2 3 4  5  6  7 
0 0 1 3 6 10 15 21 
0 0 0 1 4 10 20 35 
0 0 0 0 1  5 15 35 
0 0 0 0 0  1  6 21 
0 0 0 0 0  0  1  7 
0 0 0 0 0  0  0  1 

 15

 
 
 
 
 
 
 
     
 
 
 
  
 
 
SECTION 3 : DEFINING YOUR OWN VERBS 

The starting point is again the list  

   c 
3 1 4 0.5 _2 

In section 1 we saw how the expressions >./ and <./ supplied the maximum and 
minimum values respectively, and the range of v is simply the difference between 
these two values.  J allows us to express this as 

>./ - <./ 

using a verb structure which is known as a fork : 

   (>./-<./)c 
6 

Instead of constantly having to use parentheses to write this compound verb, it can be 
consolidated by assignment to a user-defined name : 

   range=.>./-<./ 
   range c 
6  

One of the most important features of the verb which has just been defined is that, 
unlike almost every other programming language you are likely to have encountered, 
there is no explicit reference to data (that is, arguments) in the definition.  This 
particular style of programming is known as tacit programming. 

Another example of tacit programming, which also exhibits the fork structure is    

   mean=.+/%# 
   mean c 
1.3 

The central operation, divide, is performed on two functions, namely the sum +/ and 
x∑ , only 
n

the tally # of the data.  This is just what in mathematics might be written 

as noted above, the J definition makes no reference to data such as x and n.  

A composite verb such as +/%# consists of three verbs written one after the other. 
When two rather than three verbs are written together the combination is called a 
hook.  The verbs concerned may be primitive (that is part of the basic J repertoire) or 
user-defined, and so a simple example of this structure is -mean . 

(-mean)c means  c-mean c, that is the single argument is used first as a 
monadic argument to the right verb, and then as the left argument of the left verb.  In 
general, for verbs u and v, the hook (u v)y is equivalent to y u v(y).  So define 

 16

 
 
 
 
 
 
   
  
 
 
 
 
 
 
 
 
 
   mdev=.-mean  
   mdev c 
1.7 _0.3 2.7 _0.8 _3.3 

NB. deviations from the mean 

It is natural to ask what happens if there are four verbs in a row, say u v w x.   The 
answer is that the three rightmost are resolved as a fork leaving two verbs to form a 
hook  u (v w x) : 

   (-+/%#)c 
1.7 _0.3 2.7 _0.8 _3.3 

Similarly five verbs resolve into two forks, and so on. 

Those who have experience of writing programs may protest that “real” programs of 
any magnitude consist of actions in sequence  -  do this, then this, then this …, just 
like main verbs in stories.  However, there is another way in which verbs can be 
combined in English, which becomes  apparent when you think of verbs such as 
‘stirfry’ and ‘sleepwalk’.  If you fried the food completely and then stirred it, there is 
no way in which you could be said to have ‘stirfried’ it!    Rather the two verbs ‘stir’ 
and ‘fry’ are blended together or fused in a manner which says that the primary verb 
‘fry’ is elaborated with an action ‘stir’ which must take place concurrently with it.  
Similarly with ‘sleepwalk’, the primary action is walking, only it is a new kind of 
walking, which happens when sleeping and walking take place simultaneously.   Yet 
another example which emphasises concurrency even more directly is the verb 
‘timestamp’.  Were the event to be timed after the stamping, the two activities of 
timing and stamping would not take place in exact synchronisation, and similarly if 
the stamping were to take place after the timing.  

A generic feature which all these compound verbs have in common is that of ‘joining’ 
simple verbs, which leads, by analogy with usage in English grammar, to the use of 
the term conjunction.  Further there is a generic property involved in the type of 
concurrent fusion which is implicit in all three compound verbs described above;  this 
idea is consolidated in the specific conjunction @ which is called atop.  Thus, in a 
pidgin mixture of English and J :    

       stirfry =. stir @ fry,  sleepwalk =. sleep @ walk,  timestamp =. time @ stamp    

Contrast this with the kind of sequencing which takes place in a childlike narrative : 

“We got dressed, then we ate breakfast, then we cleaned our teeth, then we walked to 
the woods, then we picnicked ….” 

where one thing happens strictly after another in sequence.  Another J conjunction 
called at  @: deals with this method of joining verbs, so that, using the pidgin mixture 
above, and taking account of the fact that “after” in English requires a verb reversal 
compared with “then”, the above story could be related : 

   “We picnicked @: walked @: cleaned teeth @: ate @: got dressed”    

 17

 
 
 
 
 
 
 
 
 
 
 
 
Reflection on ordinary linguistic experience thus shows that in combining verbs there 
is an implicit distinction between fusion with concurrency on the one hand, and strict 
sequencing on the other.  Because J is of necessity a precise language you are required 
to distinguish explicitly between these two cases when you choose to combine verbs, 
which, as we have already seen, is an inevitable activity in writing your own 
programs.  Thus we look next at how the two J conjunctions , @ and @: are used in 
programming practice.    First, here is a new verb square whose symbol is *: 

   *:c 
9 1 16 0.25 4   

Suppose that we wish to total these squared values. The composite operation “total-
atop-square” which is a fusion of total and square means that the combined verb is 
applied to every item in the argument list.   Since “total” can be perfectly reasonably 
applied to a single number x, that is +/x is just x, the effect of “total-atop-square” is 
no different from square by itself :  

   (+/@*:)c 
9 1 16 0.25 4 

NB. total atop square for each item 

The other interpretation, and in practice the more likely one, is that “square” should 
produce an intermediate result before “total” is applied, that is, as in the children’s 
narrative, we seek  “total-after-square” .  In this case @: must be used : 

   (+/@:*:)c 
30.25 

NB. total after square for entire list 

If you like, you can think of @: as providing a weaker linkage between the verbs 
than @ does .   In order to sum the squares of deviations from the mean, say 

   (+/@(*:@mdev))c 
21.8 

which provides the definition of a “sum of squares” verb : 

   ssq=.+/@(*: @mdev) 
   ssq c 
21.8 

Conjunctions are always resolved at the earliest possible point on a left to right 
reading scan, so that if the parentheses were to omitted in the above definition, the 
meaning would be  (+/@(*:)@mdev which, as described above, results in totalling 
being applied to individual items : 

   ssq1=.+/@*: @mdev 
   ssq1 c 
2.89 0.09 7.29 0.64 10.89 

Following the discussion of grids in section 1, here is how a monadic verb grid 
might be developed, given its argument is to be a three-item list consisting of start 

 18

 
 
 
 
 
 
 
 
 
   
 
    
 
point, interval width, number of items.  First use the hook (*i.)to construct the 
intervals, then convert this to a fork using take and drop in order to make  ti 
monadic : 

   2(*i.)5 
0 2 4 6 8 
   ti=.{.(*i.)}. 
   ti 2 5 
0 2 4 6 8 

NB. 2 times 0 1 2 3 4 

NB. ti=times index generator 

Apply this technique a second time to introduce the location parameter, and recognise 
also the fact that ti must be applied after the scale and size parameters have been 
extracted by drop  : 

   grid=.{.+ti@}. 
   grid 97 2 5 
97 99 101 103 105 

It may have struck you in comparing J verb definitions with conventional program 
writing, that although argument data is excluded from the former, some programs  
necessarily involve constants.  For example, to write a verb to round numbers to a 
given number of decimal places, use of the number 10 is unavoidable.  A possible 
starting point is a hook which multiplies x (left argument) by “10 to the power..”  J 
allows you to “bind” the constant 10 to the power verb with a conjunction bond  & .  

    up=.*10&^   
     3.757 4.232 up 2 
375.7 423.2 

NB. x times 10 to the power y 

Simple rounding, that is to the nearest integer, consists of adding 0.5 to a number and 
taking its floor. This calls for another bond to define a verb rnd for simple rounding : 

   rnd=.<.@(0.5&+)   
   rnd 3.4 
3 

NB. add a half and round down 

and the next step is to combine rounding with “upping” in an “after” relationship : 

   3.757 4.232(rnd@up)2 
376 423 

By analogy with up define  

   down=.%10&^   

NB. x divided by 10 to the y 

so that the desired result is  

   (3.757 4.232(rnd@up)2)down 2 
3.76 4.23 

 19

 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
  
Notice that the parameter 2 is used as a right argument twice in the above expression.  
To permit this reuse J has two verbs left and right written as [ and ] respectively which 
extract the right and left arguments.  Using these, the operations embedded in the 
above line are brought together in the form of a fork : 

   round=.rnd@up down ]  
   3.757 4.232 round 2 
3.76 4.23 

Explicit programming 

When you start to write programs which are appreciably larger than those of the 
preceding subsection, joining verbs correctly can begin to make demands on mental 
ingenuity.  Accordingly, a more conventional style of defining programs is allowed 
which allows left and right argument data to be referred explicitly as x. and  y. 
respectively.  Redefining ssq  in this style should make things clear :   

   ssq=.monad define 
+/ *: mdev y. 
) 
   ssq i.5 
10 
   ssq c  
21.8 

Notice that mdev is still defined tacitly showing that the programmer is free to mix 
explicit and tacit styles in whatever way he or she find most comfortable. In the case 
of  round we have a dyadic verb : 

   round=.dyad define 
(rnd x. up y.)down y.  
) 

    3.757 4.232 round 2 
3.76 4.23 

In this case explicit programming avoids the need to use the conjunction @ which 
was necessary in the earlier tacit definition.  If you find conjunctions are tricky to 
master, then the ability to switch between explicit and tacit styles can be invaluable. 
J even provides the capability to “translate” an explicit verb automatically into a tacit 
one as in the following dialogue : 

   9!:3(5) 

NB. set system for linear display of verbs 

   13 : '(rnd x. up y.)down y.'  
([: rnd up) down ] 

 9!:3  is a foreign conjunction like 9!:11 which was used earlier to set the print 
precision.  The available parameters are 2, 4 and 5, which instructs the system to set 
verb display to boxed, tree and linear formats respectively.  

 20

 
 
 
      
 
 
 
 
    
 
    
 
13 : is a code which requests the translation of whatever explicit definition string 
follows in quotes.  The result includes a new symbol  cap  [: which is necessary 
because the translator  resolves everything in terms of forks..  When a monadic 
function like rnd is encountered, a dummy symbol is necessary to fill the place of the 
left tine of the fork.       

Verbs for sorting   

J contains two primitive verbs for sorting, namely grade up /:  and grade down \:. 
These return the permutations which would cause the list given as right argument to 
be sorted in ascending (descending) order : 

   /:c 
4 3 1 0 2 
   4 3 0 1 2{c 
_2 0.5 3 1 4 

   \:c 
2 0 1 3 4 
   2 0 1 3 4{c 
4 3 1 0.5 _2 

NB. upward sorting permutation 

NB. c sorted upwards 

NB. downward sorting permutation 

NB. c sorted downwards 

Each of these pairs of steps can be reduced to a hook.  But first observe that the 
arguments of any dyadic verb can be switched from left to right by applying an adverb 
reflex, so that the immediately preceding result could also have been achieved using 
the verb {~ : 

   c{~2 0 1 3 4 
4 3 1 0.5 _2 

NB. c sorted downwards 

Recall that a hook is a composite verb defined by writing two verbs one after the other 
so that (u v)y is equivalent to y u  v(y).  Thus combining the verbs {~ and  /: gives 

     ({~/:)c 
_2 0.5 1 3 4 

NB. c sorted upwards 

leading to user-defined verbs 

   sortu=.{~/: NB. sort upwards 
   sortu c 
_2 0.5 1 3 4 

and 

   sortd=.{~\: NB. sort downwards 
   sortd c 
4 3 1 0.5 _2 

 21

 
 
       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A further primitive verb nub removes duplicates from a list : 

   ~. 4 6 2 4 4 6 2 5 4 
4 6 2 5 

so that an upwardly sorted list with duplicates removed is given by sortu atop 
nub.  As you have already seen atop is expressed by the conjunction @ 

   (sortu@~.)4 6 2 4 4 6 2 5 4 
2 4 5 6 

This can be made into another user defined verb : 

   unub=.sortu @ ~. NB. upwardly sorted nub   
   unub 4 6 2 4 4 6 2 5 4 
2 4 5 6 

A verb for reversing and shifting   

The verb |.  in its monadic reverses a list, and in its dyadic form performs a shift, to 
the left if the left argument is positive, and to the right if it is negative :   

   |.c 
_2 0.5 4 1 3 
   |.'J is useful.' 
.lufesu si J 
   2|.1 2 3 4 5 
3 4 5 1 2 
   _1|.1 2 3 4 5 
5 1 2 3 4 

NB. reverse c 

NB. shift >:i.5 two places to left 

NB. shift >:i.5 one place to right 

The principle remains the same even if the argument is a list of lists : 

   ]m=.3 5$'rosessmellsweet' NB. list of three words 
roses 
smell 
sweet 
   1 |.m   
smell 
sweet 
roses 

NB. shift words one place up 

Suppose that a verb is wanted which removes duplicate blanks from sentences.  A first  
step is to compare the sentence as a character list with itself shifted one place right, 
and marking where matching items are not equal : 

   test=.~:~1&|. 
   test s=.'how    are    you   today  ?'    
1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 

 22

 
   
 
   
 
 
 
 
 
 
 
 
 
 
 
  
    
Then use copy to retain only the marked items : 

   (#~test) s 
how are you today ? 

A flaw in the above verb is that it will remove duplicates of any character, not just 
space.  This can be adjusted for by making a fork whose centre verb is or +. and 
which does a not equal ~: comparison with space :  

   s1=.'marry   me,   harry   !' 
   (#~test+.~:&' ') s1 
marry me, harry ! 

so the final defined verb for RemoveDuplicateBlanks is rdb=.#~test+.~:&' '. 

Conditional structures in J are catered for using a mechanism called a gerund, in 
which a succession of verbs are separated  by the conjunction  tie `, followed by 
another conjunction called agenda @., followed by a verb whose result is an integer 
index into the tied verb list.  The overall structure is what is known in programming as 
a case statement.  As a simple example applied to an if-then-else situation, consider 
how the median of a list of ordered numbers might be programmed.  If the tally of the 
argument list is odd, then the median is the value of the middle number.  If it is even 
the median is the mean of the values of the two middle numbers. 

A first step is to obtain the index of the median value or values.  Define    

    half=.-:@<:    

NB.  halve one less than rt. argument   

If the argument is odd, the result is a single integer.   However, if the argument is even 
the result is a fraction n.5 and what is wanted is both the floor and ceiling .  The 
condition “is-odd” is the result of  2&| whose result must be 0 or 1 which  are 
appropriate indices for the two case verbs and leads to the following definition : 

  medind=.((<.,>.)@half) ` half @.(2&|) 
  medind # 1 3 4 7 
1 2  

Next apply the index(es) using from and finally use mean defined above : 

   (mean@:{~medind@#) 1 3 4 7  
3.5 

As a final step the verb sortu  can be incorporated, so that the requirement that the 
list be ordered initially can be removed : 

   median=.(mean@:{~medind@#)@sortu 
   median 7 4 1 3 
3.5 

 23

 
 
  
 
 
 
 
 
 
  
            
 
  
 
 
  
SECTION 4 : FITTING EQUATIONS TO DATA 

One of the most powerful mathematical J primitive verbs is matrix inverse, which also 
provides least squares fitting in its dyadic form.    As an example, consider the 
simultaneous equations 

x + 2y –3z = 15 
x +   y +  z = 12 
          2x –   y –  z  =  0 

The coefficient matrix of the left hand side is : 

   +m=.3 3$1 2 _3 1 1 1 2 _1 _1 
1  2 _3 
1  1  1 
2 _1 _1 

Its inverse is 

  %.m 
   0 0.3333   0.3333 
 0.2 0.3333  _0.2667 
_0.2 0.3333 _0.06667 

The solution of the equations is : 

   15 12 0%.m 
4 7 1 

The functionality of  %.does not stop here.  Suppose  

   x,:y   
2.1 2.4 3.6 3.7 4.3  5.1 5.5  5.8 5.9  6.6  7.4  8.2 
4.1 6.0 5.5 8.2 7.5 12.6 8.1 10.8 7.2 13.1 11.3 15.6 

NB. x and y as laminated lists 

are readings from an experiment in which a best-fitting straight line is to be found : 

   y%.x 
1.786 

says that y = 1.786x is the “best-fitting” line (in the least squares sense)  of the form  
y = kx.  It is usually more useful to fit a constant as well, that is either y = kx + c  or   
x = ky +c.  A variant of append called append items adds 1 to every item in a list : 

   y%.1,.x 
1.272 1.563 
   x%.1,.y 
0.8117 0.4624 

so   y = 1.272 + 1.563x  and x = 0.812 + 0.462y  are the two regression lines. 

 24

 
 
 
 
 
 
 
 
 
 
 
 
     
 
 
  
 
  
  
 
The right argument of %.in the first of these two phrases is equivalent to x^/i.2 
since  x^0 is 1 for all values of x .  Thus : 

   y %. x^/i.2  
1.27181 1.56334 

NB. Best fitting straight line 

This idea can be immediately extended by using powers 0,1,2 in order to give the 
best-fitting quadratic : 

   y %. x^/i.3  
2.768 0.8803 0.06781 

NB. Best fitting quadratic 

Now suppose data are the values of a polynomial, say the cubic function  0.5x 3  + 4x2 
+ 5x – 6.  First assign a list of the coefficients in reverse order : 

   coef=._6 5 4 0.5 

Next supply a set of x values for which you wish to evaluate the polynomial : 

   +x=._4+i.9 
_5 _4 _3 _2 _1 0 1 2 3 

Now use a verb p.which evaluates the polynomial at all values of x.:  

   +y=.coef p. x 
38 15 0 _7 _6 3 20 45 78 

An x,y table for the polynomial is : 

   x,:y 
_4  _3 _2   _1  0   1  2    3   4 
 6 1.5 _4 _7.5 _6 3.5 24 58.5 110 

The best fitting straight line and quadratic are 

   y %. x^/i.2 
20.667 10.9 
   y %. x^/i.3 
_6 10.9 4 

that is y = 10.9x + 20.667 and y = 4x2 + 10.9x – 6, whereas the best fitting cubic 
recovers the original coefficient values : 

   y %. x^/i.4 
_6 5 4 0.5 

 25

 
 
 
    
 
    
 
 
 
  
 
     
 
 
      
 
 
SECTION 5 : BOXES AND RECORDS 

The concept of a “box” greatly enhances the variety of data structures that J can 
handle.  The principle is that any list structure, however complex, can be cast into a 
container called a box, which it itself a scalar.  The analogy with records in 
conventional data processing should be clear, as should the application of the verb 
box  <   in the following  example : 

   fname=.'Harry' 
   sname=.'Potter' 
   fpubl=.1998 

   shelfcode=.'childrens' 
   (<fname),(<sname),(<fpubl),<shelfcode 
+-----+------+----+---------+ 
|Harry|Potter|1998|childrens| 
+-----+------+----+---------+ 

Without boxing it is not possible to mix character and numeric data :  

   fname,sname,fpubl 
|domain error 
|   fname,sname    ,fpubl 

Also, boxes are complex scalar objects which are quite different both in character and 
in permissible operations from their contents.  For example, if m=.i.3 4 : 

   (<m),<m 
+---------+---------+ 
|0 1  2  3|0 1  2  3| 
|4 5  6  7|4 5  6  7| 
|8 9 10 11|8 9 10 11| 
+---------+---------+ 

The result is a list of two items in a row, each of which is a container for three lists 
each of four items.  Adding the contents of the boxed lists works fine :   

   m+m 
 0  2  4  6 
 8 10 12 14 
16 18 20 22 

but if you try adding the containers there is a problem : 

   (<m)+<m 
|domain error 
|   (<m)    +<m 

 26

 
 
 
 
 
 
   
            
    
 
 
 
 
 
This error message says clearly that adding in the numerical sense is not appropriate 
for the container domain, and in the same way you may not mix the numeric and 
character domains :    

    fname + fpubl 
|domain error 
|   fname    +fpubl 

An alternative to boxing items individually to construct a boxed list is to use the verb 
link  ; :   

   fname;sname;fpubl;shelfcode  
+-----+------+----+---------+ 
|Harry|Potter|1998|childrens| 
+-----+------+----+---------+ 

A matching verb open undoes boxed structures, but naturally requires type 
consistency for the unboxed contents : 

   >fname;sname;shelfcode  
Harry  
Potter 
childrens 

that is, the result of the above open is three character lists. 

A commercial type data set could be built up with further levels of boxing, for 
example : 

   rec1=.fname;sname;fpubl;shelfcode  
   rec2=.'Jane';'Eyre';1847;'classics'   

   (<rec1),:<rec2 
+-----------------------------+ 
|+-----+------+----+---------+| 
||Harry|Potter|1998|childrens|| 
|+-----+------+----+---------+| 
+-----------------------------+ 
|+----+----+----+--------+    | 
||Jane|Eyre|1847|classics|    | 
|+----+----+----+--------+    | 
+-----------------------------+ 

using laminate  ,:  to align the records vertically.  (n.b.(<rec1),<rec2 would be 
a list of two single items, and so, as with say 2 3, the items would be displayed side by 
side.)    Subsequent additions to the data set are made with simple appends : 

   rec3=.'Peter';'Rabbit';1904;'childrens' 

   ds=.(rec1,:rec2),rec3 

 27

 
 
    
 
 
 
 
 
 
      
     
 
      
   ds 
+-----+------+----+---------+ 
|Harry|Potter|1998|childrens| 
+-----+------+----+---------+ 
|Jane |Eyre  |1847|classics | 
+-----+------+----+---------+ 
|Peter|Pan   |1904|childrens| 
+-----+------+----+---------+ 

This is a list of three four-item lists, and so 

   #(rec1,:rec2),rec3 
3 
   $(rec1,:rec2),rec3 
3 4 

To extract a list of shelfcodes, use transpose and then from :  

   scs=.3{|:ds 
   scs 
+---------+--------+---------+ 
|childrens|classics|childrens| 
+---------+--------+---------+ 

This list can now be used to select the rows which correspond to children’s books : 

   (scs=<'childrens')#ds 
+-----+------+----+---------+ 
|Harry|Potter|1998|childrens| 
+-----+------+----+---------+ 
|Peter|Pan   |1904|childrens| 
+-----+------+----+---------+ 

To sort the records in alphabetical order of surnames use 1  to extract the snames 
fields, then grade-up (introduced in section 3) to find the required permutation : 

   /:1{|:ds 
1 0 2 

Then apply this to make the required selection from the list of records :  

   (/:1{|:ds){ds 
+-----+------+----+---------+ 
|Jane |Eyre  |1847|classics | 
+-----+------+----+---------+ 
|Peter|Pan   |1904|childrens| 
+-----+------+----+---------+ 
|Harry|Potter|1998|childrens| 
+-----+------+----+---------+ 

 28

 
 
 
 
 
  
   
 
 
 
 
   
SECTION 6 : INVESTIGATING POSSIBILITIES 

This section begins with a problem.  Suppose we have a situation with four two-way 
choices, for each of which something must be accepted or rejected.  These could be to 
add (or not add) onions, cheese, sausage or mushrooms to a pizza.  The aim is to make 
a table for the cost for each of the possible selections.  

First two relevant verbs  base #. and antibase #:. are introduced which allow 
numbers to be  represented in all manner of different units.   For example, converting 
4 hrs. 22 mins. 54 secs. to seconds, and then 2 yds. 2 feet 9 inches to inches is 
achieved by : 

   0 60 60 #. 4 22 54 NB. base for mins., secs.,is 60 60 
15774 
   0 3 12 #. 2 2 9    NB. base for ft., ins., is 3 12   
105 

To convert back to the original units use #: : 

   0 60 60 #:15774    NB. secs to hrs. mins. secs. 
4 22 54 
   0 3 12 #:105 
2 2 9 

  NB. inches to yds. ft. ins.  

As another example, antibase generates the individual digits in a decimal integer : 

   10 10 10 #:658    NB. digits in base 10 
6 5 8 

When the right argument is a list, the result is a list of lists : 

   10 10 10 10 10#:4342 8958 4646 243 10 12334  
0 4 3 4 2 
0 8 9 5 8 
0 4 6 4 6 
0 0 2 4 3 
0 0 0 1 0 
1 2 3 3 4 

With no left argument the default number base is 2, which delivers binary numbers : 

NB. 8=*/2 2 2, so it is also #:i.*/2 2 2  

   #:i.8   
0 0 0 
0 0 1 
0 1 0 
0 1 1 
1 0 0 
1 0 1 
1 1 0 
1 1 1 

 29

 
 
 
 
 
 
 
 
 
 
 
 
 
For the purposes of display, it is often convenient to use a verb transpose |: which 
reorganises a list of lists into a new set of lists in which the first list is a list of all first 
items, the second list is a list of all second items and so on  : 

   |:#:i.8 
0 0 0 0 1 1 1 1 
0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 

Thus, returning to the pizza problem, given 0 means absent and 1 means present  

   |:#:i.16 
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 

represents all the possible choices of toppings. 

There is no requirement that all the items in the left argument of dyadic #: are the 
same.  If we wished to allow three possibilities for the second row, say because two 
types of cheese become available making a total of 24 possible pizzas in total, we can 
supply 2 3 2 2 as the left argument : 

   |:2 3 2 2#:i.24 
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 
0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 1 1 1 1 2 2 2 2 
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 

Now the 24 is just the product of 2 3 2 and 2, or in J terminology */2 3 2 2, and 
so there is some redundancy in the above expression.  J allows us to avoid this by 
using a hook with #: as the left verb, and i.@(*/) as the right verb :  

   poss=.#:i.@(*/)  
   |: poss 2 3 2 2  
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 
0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 1 1 1 1 2 2 2 2 
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 

Now make and display a list of the ingredients, using open and link described in 
section 5 : 

   ]tops=.>'onions';'cheese';'sausage';'mushrooms ' 
onions    
cheese    
sausage   
mushrooms  

 30

 
     
  
 
   
 
 
 
 
  
  
 
The leftmost verb right ], by virtue of its definition, causes a display of everything to 
its right.  Earlier monadic + and , were used for this purpose;  in practice right is much 
the commonest way of doing simultaneous assignment and display because it is 
independent of type (character or numeric).  Now define    

   tab=.|:poss 2 3 2 2 
   tops,tab 
|domain error 
|   tops    ,tab 

The problem here is that J does not allow mixed types (character and numeric) to be 
appended.   However, J does provide a very convenient verb format ":  which 
transforms any numeric object into its character equivalent which looks identical on 
display : 

   tops,":tab 
onions                                          
cheese                                          
sausage                                         
mushrooms                                       
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 
0 0 0 0 1 1 1 1 2 2 2 2 0 0 0 0 1 1 1 1 2 2 2 2 
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 

Not quite what we were thinking of  - what is needed is append items ,. which was 
first met in section 4, and is now used to append the two lists on an item by item basis 
as opposed to one list after the other.  At the same time we have incorporated the 
dyadic form of format which allows a field size to be specified, in this case 1  : 

   tops,.1 ":tab 
onions    000000000000111111111111 
cheese    000011112222000011112222 
sausage   001100110011001100110011 
mushrooms 010101010101010101010101 

Now suppose that we want to compute the costs of the various types of pizza, given 
the costs of the various toppings : 

   cost=.0.80 1.80 2.20 1.50 

The required calculation is to multiply each of the four lists (rows) in tab by the 
matching element in cost and then add “down the columns”, that is a +/ applied to 
each column.  This combination of addition and multiplication, sometimes known as 
an “inner product” is expressed by the dot conjunction, and so 

   cost +/ .* tab 
0 1.5 2.2 3.7 1.8 3.3 4 5.5 3.6 5.1 5.8 7.3 0.8 2.3 3 4.5 
2.6 4.1 4.8 6.3 4.4 5.9 6.6 8.1 

 31

 
 
 
 
 
 
 
 
 
  
 
gives a list of the 24 possible pizzas. 

To display the combinations in the form of a price list use format once again, only 
now applying it dyadically to specify a field width of 1 with no decimals for the first 
four columns, and 6 with 2 decimal places for the price column : 

   ((4#1j0),6j2)":|:tab,cost +/ .* tab 
0000  0.00 
0001  1.50 
0010  2.20 
0011  3.70 
0100  1.80 
0101  3.30 
0110  4.00 
0111  5.50 
 ……………..    .. 
 … 
and so on.  The leftmost column is not very informative, so use copy to make it more 
meaningful, having first used append items monadically to change it from a list of 24 
items to a list of 24 single-item lists : 

   $prices=.cost+/ .*tab 
24    
   $,.prices 
24 1      
   ((|:tab)#'OCSM'),.6j2":,.prices 
       0.00 
M      1.50 
S      2.20 
SM     3.70 
C      1.80 
CM     3.30 
CS     4.00 
CSM    5.50 
CC     3.60 
CCM    5.10 
CCS    5.80 
CCSM   7.30 
O      0.80 
OM     2.30 
OS     3.00 
OSM    4.50 
OC     2.60 
OCM    4.10 
OCS    4.80 
OCSM   6.30 
OCC    4.40 
OCCM   5.90 
OCCS   6.60 
OCCSM  8.10 

 32

 
 
 
 
SECTION 7 : EDITING, SYSTEM FACILITIES and SCRIPTS 

Sooner or later you will want to write “programs” which you cannot express on a 
single line.  While J contains many sophisticated features which make it possible to 
express a great deal in a single phrase, it can be comforting to know that a more 
conventional style of conditional and looping programming is also available.  We look 
at three ways in which a user-defined verb for Fibonacci numbers can be constructed.  
Fibonacci numbers are sequences which are started with two arbitrary numbers (most 
commonly 0 and 1), and thereafter each succeeding number is the sum of the previous 
two.  Clearly such a series could go on indefinitely, and so for practical purposes one 
of the parameters of a Fibonacci verb must provide a stopping condition, for example, 
either a total number of numbers is given, or the series may stop after a given value 
has been exceeded. 

The English of the previous sentence can be rendered in J as _2&{.  (take the last 
two) and then +/@(_2&{.) (sum the last two).  Finally we want to join this to what 
we started with, which is the verb structure previously recognised as a hook    
,+/@(_2&{.) .  Hence a user-defined verb for a single Fibonacci step is  

   fib=.,+/@(_2&{.) 
   fib 2 3 
2 3 5 

Suppose that the stopping condition is that this step has to be performed a given 
number of times, say 10.  J provides a conjunction power  ^:  which allows us to say 
just this.  Its symbol demonstrates the analogy with the power verb which prescribes 
how often a number is to be multiplied by itself.  

   fib^:10(0 1) 
0 1 1 2 3 5 8 13 21 34 55 89 

We can even write this series all in a single line without any named verb, although the 
expression could be criticised as becoming a little bit hard to disentangle : 

   (,+/@(_2&{.))^:10(0 1) 
0 1 1 2 3 5 8 13 21 34 55 89 

Next, here, side by side, are two alternative ways in which we could have written and 
executed this program, the second shows incidentally that J supports recursion : 

Fib=.dyad define 
r=.y. [ i=.0 
while. i<x. do.     
  i=.i+1   
  r=.r,+/_2{.r  
end.  
) 

Fib=.dyad define 
if.x.>0 do. 
  r=.(x.-1)Fib y. 
  r=.r,+/_2{.r 
else. r=.y.   
end. 
) 

   10 Fib 0 1  
0 1 1 2 3 5 8 13 21 34 55 89 

 33

 
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
     
In the body of the code x. and y. are used to reference the left and right arguments 
as in Section 3.  and the control words (while. do. end. if. else.) must 
be terminated with dots. The code itself is terminated with a right parenthesis at which 
point the session reverts to execution mode, as opposed to object definition mode.  
The first code line in the left hand program could have been written as two separate 
lines – the left bracket acts as a statement separator, although you might have 
recognised it as the verb left whose value is what lies to its left, ignoring any right 
argument.  Its role as a statement separator is a happy consequence of this definition. 

The opening line of the above definition could also have been written Fib=.4 :0, 
in which case Fib would be a strictly dyadic verb.  Another possibility which allows 
both a monadic and a dyadic definition is :  

Fib=.3 : 0 
10 Fib y.  
: 
r=.y. [ i=.0 
… ….. etc. 

The first line states that it is a dyadic verb (that is an object of class 3) which is to be 
defined.  The 0 means that the subsequent input lines are to made from the keyboard.  
The colon on the second line separates the monadic and dyadic definitions, and in the 
present case establishes a default left argument of 10.  

More System Facilities 

You have already seen how the foreign conjunction is used to get and set print 
precision.  With different integer arguments it has many other uses for bridging the 
gap between programs and the underlying operating system.  For example, a left 
argument of 1 is associated with reading and writing.  1!:1(1) means “read from 
the keyboard”  : 

   g=.1!:1(1)   
I am now typing a message ...  
   g 
I am now typing a message ...  

NB. On the keyboard 

NB. From the computer  

Thus  

   ask=.monad define 
1!:1(1) 
) 
   h=.ask 'what''s the score?' 
round about 20  
   h 
round about 20  

NB. From the keyboard 

NB. From the computer 

You can ask how many verbs there are presently in the workspace by  

 34

 
 
 
 
 
 
 
 
 
 
 
 
 
     
 
   4!:1(3) 
 +---+---+---+ 
|Fib|ask|fib| 
+---+---+---+ 

For nouns (that is constants and variables), adverbs and conjunctions replace 3 above   
with 0, 1 and 2 respectively. 

You can ask the time of day with 6!:0’’(1) , the elapsed time since start of session 
in microseconds by 6!:1’’(1),together with much, much more which is fully 
documented within the J system help facilities. 

Scripts 

Naturally you do not want to repeat the typing of input every time you want to use a 
sequence of user-defined verbs, so J provides for scripts, which are text files, (usually 
with qualifier .ijs although any qualifier except .ijx is acceptable) into which you can 
save objects for later use.   The extension .ijx is reserved for executable J sessions 
which run under the control of software known as the “session manager”.  It is this 
software which, for example, allows you to run an arrow up the screen and re-execute 
a previously submitted line.  Assume that you have saved work from executing 
sessions into a script files.  When you want to reuse the script file you can either  

(a) load the file explicitly by    load'c:\j406\temp\fib.ijs' 

 (b) Use File/Open to open the script file, which results in the file appearing in 

a new window.  Once the script has been opened, use Run/Window or Run/Line 
from a drop-down menu; 

(c) with the script file open, press Ctl-W which is equivalent to (a). 

You will find that there are already many existing scripts in your J system, and by 
loading these you are able to take advantage of a great deal of other people’s past 
work and experience.  An example is the statistics package which consists of three 
separate scripts obtainable by Open/System/stats.ijs.  This script contains three lines   

script_z_ <'system\packages\stats\random.ijs' 
script_z_ <'system\packages\stats\statfns.ijs' 
script_z_ <'system\packages\stats\statdist.ijs'  

Do Run/Line on the second of these and you can now execute all the verbs in the 
script, for example median which we laboured to program in section 3  is just one 
of several statistics immediately available : 

   median 6 7 9 0 _2 1 5 7   
5.5 

 35

 
 
 
 
 
 
 
 
 
 
 
 
 
  
SECTION 8 : DRAWING GRAPHS 

One feature of the J package which you are bound to welcome is the ease with which 
you can draw graphs.  First do the following 

   load 'plot' 
   plot c=.3 1 4 0.5 _2 

and you should observe a new window containing a plot of these five values plotted 
against i.5 on the x axis.  Any one-dimensional sequence can be plotted in this way; 
you will find that there are verbs in another script which help you draw, for example, 
trig series. To draw y= 2sin(5x) + 3cos(12x) from 0 to 2π: 

      load 'numeric trig' 
   x=.steps 0,(o.1),100   
NB. o. means“’ 'pi times' 
   cos=.2&o.  [   sin=.1&o.  NB. dyadic o. supplies the

NB. trig ratios sin,cos,etc 

   plot x;(2*sin 5*x)+3*cos(12*x) 

To print this graph and simultaneously save it as a file, create the following verb : 

pg=.dyad define 
'res fn'=.x.;y.   NB. resolution / filename 
require'opengl' 
glfile fn 
glsavebmp res 
printbmp_jzopengl_ fn  
) 

Then, with the graph displayed in the plot window, issue the following : 

 36

 
 
 
 
     
 
 
 
 
 
 
 
 
 
 
 
 
 
 
   500 300 pg 'c:\j406\temp\plot1.bmp' 

Graphs can be divided into two categories, those which are essentially algebraic, and 
those which are inherently geometrical.  One example of each is given. 

To start with, the polynomial y = 4x2 + 5x – 16 (parabola) is plotted from –4 to 4 by   

  x=.steps _4 4 100 
    plot x;_16 5 4 p.x 

In order to make the graph a little more interesting the second item in the link is 
changed into a list of three lists, corresponding to a straight line, a parabola and a 
cubic respectively : 

     plot x;((6 5 p.x),:_16 5 4 p.x),_3 _4 2 1 p.x 

The example to illustrate geometric graphs involves one possible parametrisation of 
the curves known as epicycloids and hypocycloids.  The first line below illustrates 
incidentally a technique for making multiple assignments on a single line : 

'r R a b'=.2.6 2.36 _40.1 50  
x=.(0.01*o.2)*i.101  
p=.(r*cos(a*x))+R*cos(b*x)        
q=.(r*sin(a*x))+R*sin(b*x) 

NB. x from 0 to 2pi 

The command to make the plot is 

     'labels 0'plot p;q 

The left argument of plot is a character string which describes just one of the many 
possible drawing options, in this case it says “do not display axis labels”. 

 37

 
 
 
 
    
  
  
 
 
  
 
 
 
   
 
Facilities are also provided which allow the construction of a wide range of drawings.  
A selection of the basic drawing tools is given below.  The graphics window is 
assumed to be 2 units high and 2 units deep with the origin (0,0) at the centre. 

   load 'graph' 
   gdopen'' 
   gdrect _0.4 _0.4 0.8 0.8     NB. x y width height 
   gdellipse 0.5 _0.5 0.2 0.4   NB. centre, axis lens.  
   gdpolygon t=.0.1*7 1 _7 1 7 _4 _2 6 _1 _6 7 1  

    NB. t is six coordinate pairs in succession 

   gdshow''   NB. window displayed at this point  

The scope for drawing graphics and plots is endless, and ample guidance to all the 
facilities can be found in the Graph Utilities Lab which is part of the J system. 

 38

 
  
 
     
 
 
 
  
 
 
SECTION 9 : RANK 

If you have successfully followed the preceding sections, you are now ready to 
encounter what is one of the most important concepts in J, namely rank..   In section 1 
an object called matrix was defined which, although it looked like a matrix, was in 
fact a list of three four-lists. 

   matrix  
  3  1 4 0.5 
 _2  3 1   4 
0.5 _2 3   1 

The fact that it is a list of three lists is made explicit by tally : 

   #matrix 
3 

and the fact that each of these three lists is a four-list by shape : 

   $matrix 
3 4 

Tallying the shape 

    #$matrix 
2 

indicates that matrix is a list of lists, that is after penetrating two list levels 
primitive objects, in this case numbers, are reached.   This can be expressed more 
concisely by saying that “matrix is an object of rank 2”, and leads to the definition 
of a verb  

   rank=.#@$ 

It is very important to get the order of verbs right in rank since $@#  means tally 
first which always results in a scalar, whose shape is an empty list.  An alternative and 
equivalent definition of rank uses cap which was encountered earlier in section 3 :     

   rank=.[:#$ 

The structure involved here is called a capped fork, although a capped hook might be  
a more appropriate name because the effect is to make  the verb pair u v return u v(y) 
rather than y u v(y) which it would do under the hook rule.  In either case :  

   rank matrix 
2 
   rank c=.3 1 4 0.5 _2 
1 
   rank 7 
0 

 39

 
 
   
 
 
 
 
 
 
 
 
 
 
 
 
What makes rank a particularly important concept is that verbs, both user-defined and 
primitive, can be made to operate at different rank levels.  For example we have seen 
how box < can make any object into a scalar.  Thus : 

   rank <matrix 
0 

However, if  box operates at rank 1, that is 1 level in, then it is only the lowest level 
rank 1 objects which are boxed : 

   <"1 matrix 
+---------+--------+----------+ 
|3 1 4 0.5|_2 3 1 4|0.5 _2 3 1| 
+---------+--------+----------+ 
   rank <"1 matrix 
1  

Because it joins a verb and a noun, the symbol " is a conjunction, and because of 
what it does it is called the rank conjunction.  When we come to insert verbs like +/ 
rank becomes important and useful in enabling different types of subtotalling to be 
performed.  To “add-insert” two lists without any reference to rank, means, in a 
reasonably obvious sense, add their corresponding items : 

   +/matrix 
1.5 2 8 5.5 

that is, in matrix terms, add “down the columns”.  To “add-insert” at rank 1 means to 
add at the lowest list level, that is to add the items in each of the boxes above : 

    +/"1 m 
8.5 6 2.5 

in matrix terms, this is adding “along the rows” : 

To find the grand total of all the items in matrix we have two options using atop :  

   gt1=.+/@”(+/)  or    gt2=.+/@, 
   gt1 matrix   
17 

   gt2 matrix 
17 

NB. sum atop ravel 

and two using cap : 

   gt3=.[:+/+/   
   gt3 matrix   
17 

   gt4=.[:+/,  NB. sum after ravel  
   gt4 matrix 
17 

Which verb you choose is entirely a matter of personal taste.  If using the atop form, 
notice that parentheses are essential in the definition of gt1 .  Without them +/@+/  

 40

 
 
 
    
  
 
   
 
 
    
 
 
 
 
 
 
 
 
 
 
means “insert the verb +/@”+”” which, following the discussion in section 3, is just 
the same as inserting the verb + between each of the three top level lists, which in 
turn, as we saw above, is the same as adding down the columns : 

   (+/@+)/ matrix 
1.5 2 8 5.5 

To attach column totals to a matrix use a hook : 

  ct=.,+/  
  ct matrix 
  3  1 4 0.5 
 _2  3 1   4 
0.5 _2 3   1 
1.5  2 8 5.5 

and to attach both row and column totals : 

   ct ct"1 matrix 
  3  1 4 0.5 8.5 
 _2  3 1   4   6 
0.5 _2 3   1 2.5 
1.5  2 8 5.5  17 

To make this into a user-defined verb there are, as with  gt , two options, you may 
use either a conjunction with parentheses or cap :  

   totals1=.ct@:(ct"1) 
   totals1 m 
  3  1 4 0.5 8.5 
 _2  3 1   4   6 
0.5 _2 3   1 2.5 
1.5  2 8 5.5  17 

or   totals2=.[:ct ct"1 
   totals2 m 
  3  1 4 0.5 8.5 
 _2  3 1   4   6 
0.5 _2 3   1 2.5 
1.5  2 8 5.5  17 

In  totals1 note again the importance of both the parentheses and the need for at 
@: as opposed to atop @ .  at is vital in this case since the grand total is itself the 
sum of totals, namely the row totals, and so it is necessary that column totalling does 
not commence until the row totalling and appending is complete, a form of 
sequencing which is implicit in cap.  

Another example of a verb operating at different rank levels is taken from section 3 
and involves the verb reverse : 

   m=.3 5$'rosessmellsweet' 
   w 
roses 
smell 
sweet 

 41

 
    
 
 
 
 
 
     
 
 
 
 
 
     
 
 
 
    
 
 
 
 
  |."1 w 
sesor 
llems 
teews 

   |.w    
sweet 
smell 
roses 

Similarly 

  sortu m 
 _2  3 1   4 
0.5 _2 3   1 
  3  1 4 0.5 

sorts the rows in ascending order of the size of the first item, whereas 

   sortu"1 m 
0.5   1 3 4 
 _2   1 3 4 
 _2 0.5 1 3 

sorts each rows into ascending order individually. 

Here are the effects of rank with a deeper (that is, rank 3) list :  

   i.2 2 3 
0  1  2   
3  4  5   

6  7  8   
9 10 11   

   +/"1 i.2 2 3 
 3 12 
21 30 
   +/"2 i.2 2 3 
 3  5  7 
15 17 19 

i.2 2 3 is a two-list, each item of which is a two-list of three lists.  Summing at 
rank 1, the result is the sums of each of the inner two-lists of which there are four 
altogether.  Summing at rank 2, the result is the sums of each of the two pairs of three-
lists.   Without the rank conjunction the result is the sum of the two outer two-lists 
item by item, exactly as with matrix : 

   +/i.2 2 3 
 6  8 10 
12 14 16 

Armed with the rank conjunction the potential of J for expressing data manipulations 
is truly formidable.  

Here the present investigation of J must cease.  As hinted in the introduction there is 
much, much more left for you to discover.  If you have found what you have met so 
far has stimulated you, then you will now proceed at a very fast rate under your own 
direction.        

 42

 
 
 
 
 
 
 
 
 
 
 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
Index 

Absolute value, 8 
Adverb, 9, 21, 35 
Agenda, 23 
Alphabet, 11 
Alternating sum, 9 
Anti-base, 29 
Append, 7, 27, 29, 41 
Append items, 24, 31 
Arguments, 6 
Arithmetic functions, 2 
ASCII characters, 11 
Assignment, 3 
At, 17, 41 
Atop, 17, 40  

Base, 29 
Best fitting 
      polynomials,  25 
Bond, 19 
Box, 26, 40 

Cap, 21, 39, 41 
Capped fork, 39 
Cards, 15 
Case statement, 23  
Ceiling, 7, 23 
Character list, 5, 12 
Combinations, 15 
Comments, 4 
Complex numbers, 4 
Concurrency, 17 
Conjunctions, 17, 35 
Control words, 34 
Copy, 6, 9  

Deal, 13 
Decrement, 13 
Dice, 13 
Domain errors, 14, 26, 31 
Dot conjunction, 31 
Drawing, 36 
Drop, 12, 19 
Dyadic, 6 

Empty list, 10 
Epicycloids, 37 
Explicit Programming,           
             20 

Factorial, 15 
Fibonacci numbers, 33 
Fill items, 12 
Floor, 7, 23 
Foreign conjunction, 
          3, 20, 34 

Fork, 16, 19 
Format, 31 
From, 5, 13, 23 
Gerund, 23 
Grade down, 21 
Grade up,  21, 28 
Graphs, 36 
Grid, 11,18 

Halve, 23 
Hook, 16, 19 
Hypocycloids, 37 

Identity values, 10 
Increment, 13 
Index generator,10, 11, 
  29, 41 

Index of, 12 
Inner product, 31   
Insert, 9, 40 
Keyboard input, 34 

Laminate, 15, 24, 27 
Least squares, 24 
Left, 20, 34 
Link, 27 
Lists, 4 
Logarithms, 4 
Logical verbs, 10 

Matrix, 6, 24, 39 
Matrix inversion, 24 
Maximum, 7 
Mean, 16 
Median, 23, 35 
Membership, 9 
Minimum, 7 
Monadic, 6 
Multiple choice, 14 

Negate, 7 
Not equal, 23 
Noun, 11, 33 
Nub, 22 

Open, 27 
Or, 23 
Out of, 15 
Overtaking, 12  

Pascal’s Triangle, 15 
Permutations, 21, 28 
Pi, 36 
Polynomials, 25, 37 

 43

Power, 2, 33 
Primitives, 16 
Print precision, 3 
Quadratics, 25 

Range, 16 
Rank, 39 
Ravel, 15 
Reciprocal, 7 
Recursion, 33 
Reflex, 21 
Regression lines, 25 
Relational verbs, 9 
Remove blanks, 22  
Reshape, 6 
Residue, 8 
Reverse,  22, 41 
Right, 20, 31 
Roll, 13 
Rounding, 19 

Scalars, 3 
Scripts, 35  
Sequential 
      programming, 17 
Session Manager, 35 
Shape of, 6, 39 
Shift, 22 
Signum, 7 
Sorting, 21, 41 
Simultaneous 
      equations, 24 
Square Root, 4 
Square, 18 
System facilities, 
                3, 20,34 
Tables, 11,15 
Tacit programming, 16 
Tail, 12 
Take, 12 
Tally, 5, 39 
Tie, 23  
Time, 35 
Tossing coins, 13 
Transpose, 30 
Trigonometry, 36 

Underbar, 3  

Vectors, 4 
Verbs, 2 

Workspace, 3 
Wrap around, 6 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Vocabulary 

The table below gives the J symbols which are used in this booklet.  A few other 
symbols have also been included there are obvious analogies with those which are 
used in the booklet.  Where two meanings are given, these are monadic/dyadic. 

+.  or (dyadic) 
-.  not (monadic) 
*.  and (dyadic) 
%.   matrix inverse (monadic)  %:  square root 
^.  natural logarithm/logarithm 

+:  double/not-or 
-:  halve 
*:  square/not-and 

|.  reverse/shift 

|:  transpose 

#.  base 

#:  antibase 

{.  take 
,.  append items  
<.  floor/minimum 
>.  ceiling/maximum 

{:  drop 
,:  laminate 
<:  decrement/less than or equal 
>:  increment/greater than or equal 
~:  not equals 

[:  cap 

/:  grade-up 
\:  grade-down 
 ”": format 

e.  membership 
i.    index generator/index of 
o.    multiply by pi 
p.    polynomial 

Verbs: 

+  conjugate/plus 
-  negate/minus 
*  signum/times 
%  reciprocal/divide 
^  power 
!  factorial/out of   
|  absolute value/residue 
=  equals 
?  roll/deal 
#  tally/copy 
$  shape of/reshape   
{  from 
, 
ravel/append 
<  box/less than 
>  open/greater than 

;  link  
[  left   
]  right   

Adverbs : 

/  insert/table 
~  reflex  

Conjunctions : 

@  atop (strong linkage)       @. agenda 
&  bond 
.  dot 
`  tie 

     @: after (weak linkage)      

  ^:  power 
  !:  foreign conjunction 

Noun :  

     a.    alphabet 

Special symbol : 

     =.  assignment                                 

Control words :   if. do. else. while. end.     

 44