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# J: a powerful calculator

Here I shall talk about a powerful calculator called J. In fact, it is so powerful that we should better not call it a calculator. It is a powerful programming language. But it is so concise that it looks more like a calculator than a programming language.

## The very basics

You have all the standard arithmetic operations, + means addition, - is subtraction and * is multiplication. But the symbol for division is %. A bit of a surprise, but not much. The symbol ^ is for exponentiation. You can use ( and ) as usual.

Nothing particularly strange so far. Now comes the first shock. All computations are done right-to-left. Thus 2+3*5 and 3*5+2 will produce different results. Irritating? Well, not quite. Actually, it saves typing a lot of ()'s. In usual notation we have to plan where to put the ()'s, and then have to actually type them. In J we have to plan the order, and there is nothing to type.

exp(x) is ^. Oops, how can that be? Did we not say just a moment ago that ^ is for raising to power? Well, yes. But if you type ^3, then clearly there is no number to the left of ^. So J supplies e there. And since exp(x) is indeed some form of power, it is not much difficult to remember either!

J uses rather short names. They are easy to type (almost like pressing a single button in a calculator), but a bit cryptic. J tries to help by using similar symbols for similar functions. For example, ln(x) is closely related to exp(x), and is denoted in J by ^. The extra period is called an inflection. There is another inflection, the colon. Thus if s is a symbol then typically s. and s: are two related symbols. And each has a monad meaning and a dyad meaning. Thus with each symbol J can associate 6 meanings.

Here is an example of such similar symbols. > means greater than. No surprise here. Closely relate with is greater-than-or-equal which is denoted by >:. If you want to compute maximum of two numbers use >. as a dyad. Ceiling is computed by >.. You have the obvious analogues for <. Of course, it is rather tedious to memorise all the 6 meanings attached to a symbol. It is somewhat like trying to improve your vocabulary by cramming a dictionary. But it is good to know.

If we want to compute absolute value, then use |. Yes, just a single vertical bar! If we want to get remainder, then again the same symbol is used, but as 3 | 5.

You can compute factorial or choose using !.

You may compute $\sin x$, $\cos x$ etc in a rather remarkable way using the same verb o.. For example, 1 o. x is $\sin x$, while 2 o. x is $\cos x$ and 3 o. x is $\tan x$. If these look irritating you may load 'trig' and write more conventionally sin x, cos x etc. If $\sin x$ is 1 o. x then what should be the command for $\sin ^{-1}x$? It is _1 o. x.

## Some goodies

J has some functions that are often not available in a calculator. These are not terribly important, but often comes handy. Let's get acquainted with a few such. % is for reciprocal. If I type 2+%3 that will mean two plus one-third.

%: will compute square root.

4 ^. 56 will compute log of 56 to the base 4. Not many calculators come equipped with a log for arbit bases.

A rather stunning function is #:. Used as a monad it produces binary representations of integers. As a dyad it can fold into hierarchical unit systems. For example, 60 seconds make a minute, 60 minutes an hour, 24 hours a day. You can convert 126547 seconds to day, hr, min, sec as 24 60 60 #: 126547. Of course, you may convert 45678 to base 7 (showing least significant 4 digits) as 7 7 7 7 #: 45678

You can convert it back by using the related symbol #..

J can handle complex numbers. They are written like 1j2. Rational numbers are written like 1r2.

## Assignments

You can use assignments to create variables. =.

## Lists

Most modern languages allow lists. All elements of a list must of the same type (numeric, character, lists or boxes, that we shall talk about in a moment). You cannot mix these. Also, if one elment is list all others must also be lists of the same type and composition. A string is just list of characters. Thus you cannot have a list with one element "emacs" and another "vim", as these differ in size. But this will hardly cause any problem, as J can pad the shorter one with blanks.

Sometimes hwever, you want to have a list with mixed types (or strings with different lengths without having to pad with blanks). Then boxes come handy. A box (whatever it may contain) is always considered a single entity.

It is important to understand that a box may contain a list and list may have a box as its element. But all elements in the same list must be of the same type.

Let us start by understanding how to create lists. The simplest way is to actually write the things one after another. No delimiter of any sort is required.

Of course long lists are not easily created like this. So we have some convenience methods: i. 4 and 4 # 2. These also allow creation of lists of lists, etc.

Two lists may be combined in two different ways to produce a new list. One is by appending using ,. For this both the lists must have elements of identical structure (else padding will be used). The other technique is to create a new doubleton list with the input lists its elements. This is achieved with ,:.