root/docs/yaht.kwid

# This is a transcription of Hal Daume III's "Yet Another Haskell Tutorial"
# The original is 198 PDF pages which is hard to read. 
# This copy is currently incomplete.
# Original is available at http://www.isi.edu/~hdaume/htut/

= About This Report 

The goal of the /Yet Another Haskell Tutorial/ is to provide a complete
intoduction to the Haskell programming language. It assumes no knowledge
of the Haskell language or familiarity with functional programming in
general. However, general familiarity with programming concepts (such as
algorithms) will be helpful. This is not intended to be an introduction
to programming in general; rather, to programming in Haskell. Sufficient
familiarity with your operating system and a text editor is also
necessary (this report only discusses installation on configuration on
Windows and *Nix system; other operating systems may be supported --
consult the documentation of your chosen compiler for more information
on installing on other platforms).

== What is Haskell? 

Haskell is called a lazy, pure functional programming language. It is
called /lazy/ because expressions which are not needed to determine the
answer to a problem are not evaluated. The opposize of lazy is /strict/,
which is the evaluation strategry of most common programming languages
(C, C++, Java, even ML). A strict language is one in which every
expression is evaluated, whether the result of its computation is
important or not. (This is probably not entirely true as optimizing
compilers for strict languages often do what's called "dead code
elmination" -- this removes unused expressions from the program.) It
is called /pure/ because it does not allow side effects (A side effect is
something that affects the "state" of the world. For instance, a
function that prints something to the screen is said to be side-
effecting, as is a function which affects the value of a global
variable.) -- of course, a programming language without side effects
would be horribly useless; Haskell uses a system of /monads/ to isolate
all impure computations from the rest of the program and perform them in
the safe way (see Chapter 9 for a discussion of monads proper or Chapter
5 for how to do input/output in a pure language). 

Haskell is called a /functional/ language because the evaluation of a
program is equivalent to evaluating a function in the pure mathematical
sense. This also differs from standard languages (like C and Java) which
evaluate a sequence of statements, one after the other (this is termed
an /imperative/ langauge).

The History of Haskell The history of Haskell is best described using the
words of the authors. The following text is quoted from the published version
of the Haskell 98 Report: 

.indent
In September of 1987 a meeting was held at the conference on Functional
Programming Languages and Computer Architecture (FPCA '87) in Portland,
Oregon, to discuss an unfortunate situation in the functional
programming community: there had come into being more than a dozen
nonstrict, purely functional programming languages, all similar in
expressive power and semantic underpinnings. There was a strong
consensus at this meeting that more widespread use of this class of
functional languages was being hampered by the lack of a common
language. It was decided that a committee should be formed to design
such a language, providing faster communication of new ideas, a stable
foundation for real applications development, and a vehicle through
which others would be encouraged to use functional languages. This
document describes the result of that committee's efforts: a purely
functional programming language called Haskell, named after the logician
Haskell B. Curry whose work provides the logical basis for much of ours.

The committee's primary goal was to design a language that satisfied
these constraints: 
+ It should be suitable for teaching, research, and applications,
  including building large systems.
+ It should be completely described via the publication of a formal
  syntax and semantics.
+ It should be freely available. Anyone should be permitted to implement
  the language and distribute it to whomever they please.
+ It should be based on ideas that enjoy a wide consensus. 
+ It should reduce unnecessary diversity in functional programming
  languages.

The committee intended that Haskell would serve as a basis for future
research in language design, and hoped that extensions or variants of
the language would appear, incorporating experimental features.

Haskell has indeed evolved continuously since its original publication.
By the middle of 1997, there had been four iterations of the language
design (the latest at that point being Haskell 1.4). At the 1997 Haskell
Workshop in Amsterdam, it was decided that a stable variant of Haskell
was needed; this stable language is the subject of this Report, and is
called "Haskell 98". 

Haskell 98 was conceived as a relatively minor tidy-up of Haskell 1.4,
making some simplifications, and removing some pitfalls for the unwary.

It is intended to be a "stable" language in sense the /implementors are
committed to supporting Haskell 98 exactly as specified, for the
foreseeable future/. 

The original Haskell Report covered only the language, together with a
standard library called the Prelude. By the time Haskell 98 was
stabilised, it had become clear that many programs need access to a
larger set of library functions (notably concerning input/output and
simple interaction with the operating system). If these program were to
be portable, a set of libraries would have to be standardised too. A
separate effort was therefore begun by a distinct (but overlapping)
committee to fix the Haskell 98 Libraries.
.indent.

== Why Use Haskell? 

Clearly you're interested in Haskell since you're reading this tutorial.
There are many motivations for using Haskell. My personal reason for
using Haskell is that I have found that I write more bug-free code in
less time using Haskell than any other language. I also find it very
readable and extensible. 

Perhaps most importantly, however, I have consistently found the Haskell
community to be incredibly helpful. The language is constantly evolving
(that's not to say it's instable; rather that there are numerous
extensions that have been added to some compilers which I find very
useful) and user suggestions are often heeded when new extensions are to
be implemented. 

== Why Not Use Haskell? 

My two biggest complaints, and the complaints of most Haskellers I know,
are: (1) the generated code tends to be slower than equivalent programs
written in a language like C; and (2) it tends to be difficult to debug.

The second problem tends not be to a very big issue: most of the code
I've written is not buggy, as most of the common sources of bugs in
other languages simply don't exist in Haskell. The first issue certainly
has come up a few times in my experience; however, CPU time is almost
always cheaper than programmer time and if I have to wait a little
longer for my results after having saved a few days programming and
debugging.

Of course, this isn't the case of all applications. Some people may find
that the speed hit taken for using Haskell is unbearable. However,
Haskell has a standardized /foreign-function interface/ which allow you
to link in code written in other languages, for when you need to get the
most speed out of your code. If you don't find this sufficient, I would
suggest taking a look at the language O'Caml, which often /outperforms/
even C++, yet also has many of the benefits of Haskell.

== Target Audience 

There have been many books and tutorials written about Haskell; for a
(nearly) complete list, visit the http://haskell.org/bookshelf (Haskell
Bookshelf) at the Haskell homepage. A brief survey of the tutorials
available yields: 

* /A Gentle Introduction to Haskell/ is an introduction to
  Haskell, given that the reader is familiar with functional
  programming en large.
* /Haskell Companion/ is a short reference of common concepts and
  definitions.
* /Online Haskell Course/ is a short course (in German) for beginning
  with Haskell.
* /Two Dozen Short Lessons in Haskell/ is the draft of an excellent
  textbook that emphasizes user involvement.
* /Haskell Tutorial/ is based on a course given at the 3rd International
  Summer School on Advanced Functional Programming.
* /Haskell for Miranda Programmers/ assumes knowledge of the
  language Miranda.
* /PLEAC-Haskell/ is a tutorial in the style of the Perl Cookbook.

Though all of these tutorials is excellent, they are on their own
incomplete: The "Gentle Introduction" is far too advanced for beginning
Haskellers and the others tend to end too early, or not cover
everything. Haskell is full of pitfalls for new programmers and
experienced non-functional programmers alike, as can be witnessed by
reading through the archives of the Haskell mailing list.

It became clear that there is a strong need for a tutorial which is
introductory in the sense that it does not assume knowledge of
functional programming, but which is advanced in the sense that it
/does/ assume some background in programming. Moreover, none of the
known tutorials introduce input/output and iteractivity soon enough (not
even until the 248th page, as in the case of the Hudak book). This
tutorial is not for beginning programmers; some experience and knowledge
of programming and computers is assumed (though the appendix does
contain some background information).

The Haskell language underwent a standardization process and the result
is called Haskell 98. The majority of this book will cover the Haskell
98 standard. Any deviations from the standard will be noted (for
instance, many compilers offer certain extensions to the standard which
are useful; some of these may be discussed). The goals of this tutorial
are: 

* to be /practical/ above all else
* to provide a comprehensive, free introduction to the Haskell language
* to point out common pitfalls and their solutions
* to provide a good sense of how Haskell can be used in the real world

== Acknowledgements 

It would be inappropriate not to give credit also to the
original designers of Haskell. Those are: Arvind, Lennart Augustsson, Dave
Barton, Brian Boutel, Warren Burton, Jon Fairbairn, Joseph Fasel, Andy Gordon,
Maria Guzman, Kevin Hammond, Ralf Hinze, Paul Hudak, John Hughes, Thomas
Johnsson, Mark Jones, Dick Kieburtz, John Launchbury, Erik Meijer, Rishiyur
Nikhil, John Peterson, Simon Peyton Jones, Mike Reeve, Alastair Reid, Colin
Runciman, Philip Wadler, David Wise, Jonathan Young. 

Finally, I would like to specifically thank Simon Peyton Jones, Simon
Marlow, John Hughes, Alastair Reid, Koen Classen, Manuel Chakravarty,
Sigbjorn Finne and Sven Panne, all of whom have made my life learning
Haskell all the more enjoyable by always being supportive. There were
doubtless others who helped and are not listed, but these are those who
come to mind. 

\- Hal Daumé III

= Chapter 1 -- Introduction 

This tutorial contains a whole host of example code, all of which should
have been included in its distribution. If not, please refer to the
links off of the Haskell web site (`haskell.org`) to get it. This book
is formatted to make example code stand out from the rest of the text.

    Code will look like this. 

Occasionally, we will refer to interaction betwen you and the operating
system and/or the interactive shell (more on this in Section 2).

    Interaction will look like this. 

Strewn throughout the tutorial, we will often make additional notes to
something written. These are often for making comparisons to other
programming languages or adding helpful information. 

    *NOTE* Notes will appear like this. 

If we're covering a difficult or confusing topic and there is something
you should watch out for, we will place a warning. 

    *WARNING* Warnings will appear like this. 

Finally, we will sometimes make reference to built-in functions (so-
called Preludefunctions). This will look something like this: 

.math
map :: (a->b)->[a]->[b]
.math

Within the body text, Haskell keywords will appear like this: *where*,
identifiers as `map`, types as /*String*/ and classes as */Eq/*.

= Chapter 2 -- Getting Started 

There are three well known Haskell system: Hugs, GHC and NHC. Hugs is
exclusively an interpreter, meaning that you cannot compile stand-alone
programs with it, but can test and debug programs in an interactive
environment. GHC is both an interpreter (like Hugs) and a compiler which
will produce stand-alone programs. NHC is exclusively a compiler. Which
you use is entirely up to you. I've tried to make a list of some of the
differences in the following list but of course this is far from
exhaustive:

- Hugs
very fast; implements almost all of Haskell 98 (the standard) and most
extensions; built-in support for module browsing; cannot create stand-
alones; written in C; works on almost every platform; build in graphics
library. 
- GHC
interactive environment is slower than Hugs, but allows
function definitions in the environment (in Hugs you have to put them in
a file); implements all of Haskell 98 and extensions; good support for
interfacing with other languages; in a sense the "de facto"
standard. 
- NHC
less used and no interactive environment, but produces smaller and often
faster executables than does GHC; supports Haskell 98 and some
extensions. 

I, personally, have all of them installed and use them for different
purposes. I tend to use GHC to compile (primarily because I'm most
familiar with it) and the Hugs interactive environment, since it is much
faster. As such, this is what I would suggest. However, that is a fair
amount to download an install, so if you had to go with just one, I'd
get GHC, since it contains both a compiler and interactive environment.

Following is a descrition of how to download and install each of this as
of the time this tutorial was written. It may have changed -- see
http://haskell.org (the Haskell website) for up-to-date information. 

== 2.1 Hugs 

Hugs supports almost all of the Haskell 98 standard (it lacks some
of the libraries), as well as a number of advanced/experimental
extensions, including: multi-parameter type classes, extensible records,
rank-2 polymorphism, existentials, scoped type variables, and restricted
type synonyms. 

=== 2.1.1 Where to get it 

The official Hugs web page is at: http://haskell.org/hugs. 

If you go there, there is a link titled "downloading" which will
send you to the download page. From that page, you can download the
appropriate version of Hugs for your computer.

=== 2.1.2 Installation procedures 

Once you've downloaded Hugs, installation differs depending on your
platform, however, installation for Hugs is more of less identical to
installation for any program on your platform. 

- For Windows 
when you click on the "msi" file to download, simply choose "Run This
Program" and the installation will begin automatically. From there, just
follow the on-screen instructions. 
- For RPMs 
use whatever RPM installation program you know best. 
- For source 
first gunzip the file, then untar it. Presumably if you're using a
system which isn't otherwise supported, you know enough about your
system to be able to run configure scripts and make things by hand.

== 2.1.3 How to run it 

On Unix machines, the Hugs interpreter is usually started with a command
line of the form: hugs `[option - file]` ... 

On Windows , Hugs may be started by selecting it from the start menu or
by double clicking on a file with the .hs or .lhs extension. (This
manual assumes that Hugs has already been successfully installed on your
system.) 

Hugs uses options to set system parameters. These options are
distinguished by a leading + or - and are used to customize the
behaviour of the interpreter. When Hugs starts, the interpreter performs
the following tasks: 

* Options in the environment are processed. The variable HUGSFLAGS holds
  these options. On Windows 95/NT, the registry is also queried for Hugs
  option settings. 
* Command line options are processed. 
* Internal data structures are initialized. In particular, the heap is
  initialized, and its size is fixed at this point; if you want to run
  the interpreter with a heap size other than the default, then this
  must be specified using options on the command line, in the
  environment or in the registry.
* The Prelude file is loaded. The interpreter will look for the Prelude
  file on the path specified by the -P option. If the Prelude, located
  in the file Prelude.hs, cannot be found in one of the path
  directories or in the current directory, then Hugs will terminate;
  Hugs will not run without the Prelude file. 
* Program files specified on the command line are loaded. The effect of
  a command hugs `f1 ... fn` is the same as starting up Hugs with the
  hugs command and then typing `:load f1 ... fn`. In particular, the
  interpreter will not terminate if a problem occurs while it is trying
  to load one of the specified files, but it will abort the attempted
  load command. The environment variables and command line options used
  by Hugs are described in the following sections.

=== 2.1.4 Program options 

To list all of the options would take too much space. The most important
option at this point is "+98" or "-98". When you start hugs with "+98"
it is in Haskell 98 mode, which turns off all extensions. When you start
in "-98", you are in Hugs mode and all extensions are turned on. If
you've downloaded someone elses code and you're having trouble loading
it, first make sure you have the "98" flag set properly. 

Further information on the Hugs options is in the manual:
http://cvs.haskell.org/Hugs/pages/hugsman/started.html. 

=== 2.1.5 How to get help 

To get Hugs specific help, go to the Hugs web page. To get general
Haskell help, go to the Haskell web page. 

== 2.2 Glasgow Haskell Compiler

The Glasgow Haskell Compiler (GHC) is a robust, fully-featured,
optimising compiler and interactive environment for Haskell 98; GHC
compiles Haskell to either native code or C. It implements numerous
experimental language extensions to Haskell 98; for example:
concurrency, a foreign language interface, multi-parameter type classes,
scoped type variables, existential and universal quantification, unboxed
types, exceptions, weak pointers, and so on. GHC comes with a
generational garbage collector, and a space and time profiler.

=== 2.2.1 Where to get it 

Go to the official GHC web page http://haskell.org/ghc (GHC) to download
the latest release. The current version as of the writing of this
tutorial is 5.04.2 and can be downloaded off of the GHC download page
(follow the "Download" link). From that page, you can download the
appropriate version of GHC for your computer.

=== 2.2.2 Installation procedures 

Once you've downloaded GHC, installation differs depending on your
platform; however, installation for GHC is more of less identical to
installation for any program on your platform.

- For Windows 
when you click on the "msi" file to download, simply choose "Run This
Program" and the installation will begin automatically. From there,
just follow the on-screen instructions.

- For RPMs 
use whatever RPM installation program you know best. 

- For source 
first gunzip the file, then untar it. Presumably if you're using a
system which isn't otherwise supported, you know enough about your
system to be able to run configure scripts and make things by hand.

For a more detailed description of the installation procedure, look at
the GHC users manual under "Installing GHC".

=== 2.2.3 How to run the compiler 

Running the compiler is fairly easy. Assuming that you have a program
written with a mainfunction in a file called Main.hs, you can compile it
simply by writing:

    % ghc --make Main.hs -o main

The "make" option tells GHC that this is a program and not just a
library and you want to build it and all modules it depends on.
"Main.hs" stipulates the name of the file to compile; and the "-o main"
means that you want to put the output in a file called "main".


    *NOTE* In Windows, you should say "-o main.exe" to tell Windows
    that this is an executable file. 

You can then run the program by simply typing "main" at the prompt.

=== 2.2.4 How to run the interpreter 

GHCi is invoked with the command "ghci" or "ghc -interactive".
One or more modules or filenames can also be specified on the command
line; this instructs GHCi to load the specified modules or filenames
(and all the modules they depend on), just as if you had said :load
modules at the GHCi prompt.

=== 2.2.5 Program options 

To list all of the options would take too much space. The most important
option at this point is "-fglasgow-exts". When you start GHCi without
"-fglasgow-exts" it is in Haskell 98 mode, which turns off all
extensions. When you start with "-fglasgowexts", all extensions are
turned on. If you've downloaded someone elses code and you're having
trouble loading it, first make sure you have this flag set properly.

Further information on the GHC and GHCi options are in the manual off of the 
GHC web page. 

=== 2.2.6 How to get help 

To get GHC(i) specific help, go to the GHC web page. To get general
Haskell help, go to the Haskell web page.

== 2.3 NHC 

About NHC 3 ...

=== 2.3.1 Where to get it 
=== 2.3.2 Installation procedures 
=== 2.3.3 How to run it 
=== 2.3.4 Program options 
=== 2.3.5 How to get help 

== 2.4 Editors 

With good text editor, programming is fun. Of course, you can get along
with simplistic editor capable of just cut-n-paste, but good editor is
capable of doing most of the chores for you, letting you concentrate on
what you are writing. With respect to programming in Haskell, good text
editor should have as much as possible of the following features:

* Syntax highlighting for source files 
* Indentation of source files 
* Interaction with Haskell interpreter (be it Hugs or GHCi) 
* Computer-aided code navigation 
* Code completion 

At the time of writing, several options were available: Emacs/XEmacs
support Haskell via haskell-mode and accompanying Elist code (available
from http://www.haskell.org/haskellmode), and 3 ... .

What's else available? ...

(X)Emacs seem to do the best job, having all the features listed above.
Indentation is aware about Haskell's 2-dimensional layout rules (see
Section 7.11, very smart and have to be seen in action to be believed.
You can quickly jump to the definition of chosen function with the help
of "Definitions" menu, and name of the currently edited function is
always displayed in the modeline.

= Chapter 3 -- Language Basics 

In this chapter we present the basic concepts of Haskell. In addition
to familiarizing you with the interactive environments and showing you
how to compile a basic program, we introduce the basic syntax of
Haskell, which will probably be quite alien if you are used to
languages like C and Java.

However, before we talk about specifics of the language, we need to
establish some general properties of Haskell. Most importantly,
Haskell is a /lazy/ language, which lazy means that no computation
takes place unless it is forced to take place when the result of that
computation is used.

This means, for instance, that you can define infinitely large data
structures, provided that you never use the entire structure. For
instance, using imperative-esque psuedo-code, we could create an
infinite list containing the number 4in each position by doing
something like:

    List makeList()
    {
      List current = new List();
      current.value = 1;
      current.next = makeList();
      return current;
    }

By looking at this code, we can see what it's trying to do: it creates a
new list, sets its value to 4and then recursively calls itself to make
the rest of the list. Of course, if you actually wrote this code and
called it, the program would never terminate, because makeListwould keep
calling itself ad infinitum.

This is because we assume this imperative-esque language is /strict/,
the opposite of strict lazy. Strict languages are often referred to as
"call by value," while lazy languages are referred to as "call by name."
In the above psuedo-code, when we "run" makeList on the fifth line, we
attempt to get a /value/ out of it. This leads to an infinite loop.

The equivalent code in Haskell is: 

    makeList = 1 : makeList

This program reads: we're defining something called makeList(this is
what goes on the left-hand side of the equals sign). On the right-hand
side, we give the definition of makeList. In Haskell, the colon operator
is used to create lists (we'll talk more about this soon). This right-
hand side says that the value of makeListis the element 1stuck on to the
beginning of the value of makeList.

However, since Haskell is lazy (or "call by name"), we do not actually
attempt to evaluate what makeListis at this point: we simply remember
that if ever in the future we need the second element of makeList, we
need to just look at makeList.

Now, if you attempt to write makeListto a file, print it to the screen,
or calculate the sum of its elements, the operation won't terminate
because it would have to evaluate an infinitely long list. However, if
you simply use a finite portion of the list (say the first 45 elements),
the fact that the list is infinitely long doesn't matter. If you only
use the first 4/5elements, only the first 4/5elements are ever
calculated. This is laziness.

Second, Haskell is case-sensitive. Many languages are, but Haskell
actually uses case to give meaning. Haskell distinguishes between /values/
(for instance, numbers: 1, 2, 3, ...); strings: "abc", "hello",
... ; characters: 'a', 'b', ' ', ... ; even functions: for instance, the
function squares a value, or the square-root function); and /types/ (the
categories to which values belong). 

By itself, this is not unusual. Most languages have some system of
types. What is unusual is that Haskell /requires/ that the names given to
functions and values begin with a lower- case letter and that the names
given to types begin with an upper-case letter. The moral is: if your
otherwise correct program won't compile, be sure you haven't named your
function Foo, or something else beginning with a capital letter.

Being a functional language, Haskell eschews side effects. A side effect
is essentially something that happens in the course of executing a
function that is not related to the output produced by that function.

For instance, in a language like C or Java, you are able to modify
"global" variables from within a function. This is a side effect because
the modification of this global variable is not related to the output
produced by the function. Furthermore, modifying the state of the real
world is considered a side effect: printing something to the screen,
reading a file, etc., are all side effecting operations.

Functions that do not have side effects are called /pure/. An easy test
for whether or not a function is pure is to ask yourself a simple
question: "Given the same arguments, will this function always produce
the same result?". 

All of this means that if you're used to writing code in an imperative
language (like C or Java), you're going to have to start thinking
differently. Most importantly, if you have a value x, you must /not/
think of x as a register, a memory location or anything else of that
nature. xis simply a name, just as "Hal" is my name. You cannot
arbitrarily decide to store a different person in my name any more than
you can arbitrarily decide to store a different value in x. This means
that code that might look like the following C code is invalid (and has
no counterpart) in Haskell:

    int x = 5;
    x = x + 1;

A call like x = x + 1 is called /destructive update/ because we are
destroying whatever was in x before and replacing it with a new value.
Destructive update does not exist in Haskell. 

By not allowing destructive updates (or any other such side effecting
operations), Haskell code is very easy to comprehend. That is, when we
define a function `f`, and call that function with a particular argument
`a` in the beginning of a program, and then, at the end of the program,
again call `f` with the same argument `a`, we know we will get out the
same result. This is because we know that `a` cannot have changed and
because we know that `f` only depends on `a` (for instance, it didn't
increment a global counter). This property is called /referential
transparency/ and basically states that if two functions `f` and `g`
produce the same values for the same arguments, then we may replace `f`
with `g` (and vice-versa).


    *NOTE* There is no agreed-upon exact definition of referential
    transparency. The definition given above is the one I like best.
    They all carry the same interpretation; the differences lie in how
    they are formalized.

== 3.1 Arithmetic 

Let's begin our foray into Haskell with simple arithmetic. Start up your
favorite interactive shell (Hugs or GHCi; see Chapter 2 for installation
instructions). The shell will output to the screen a few lines talking
about itself and what it's doing and then should finish with the cursor
on a line reading:

    Prelude>

From here, you can begin to evaluate /expressions/. An expression is
basically something that has a value. For instance, the number `5` is an
expression (its value is 5). Val ues can be built up from other values;
for instance, `5 + 6` is an expression (its value is 11). In fact, most
simple arithmetic operations are supported by Haskell, including plus
(+), minus (-), times (*), divided-by (/), exponentiation (^) and square-
root (sqrt). You can experiment with these by asking the interactive
shell to evaluate expressions and to give you their value. In this way,
a Haskell shell can be used as a powerful calculator. Try some of the
following:

    Prelude> 5*4+3
    23
    Prelude> 5^5-2
    3123
    Prelude> sqrt 2
    1.4142135623730951
    Prelude> 5*(4+3)
    35

We can see that, in addition to the standard arithmetic operations,
Haskell also allows grouping by parentheses, hence the difference
between the values of *5*4+3* and *5*(4+3)*. The reason for this is that
the "understood" grouping of the first expression is operator precedence
*(5*4)+3*, due to /operator precedence/.

Also note that parentheses aren't required around function arguments.
For instance, we simply wrote `sqrt 2`, not `sqrt(2)`, as would be
required in most other languages. You could write it with the
parentheses, but in Haskell, since function application is so common,
parentheses aren't required.

.warning
Even though parentheses are not always needed, sometimes it is better to
leave them in anyway; other people will probably have to read your code,
and if extra parentheses make the intent of the code clearer, use them.
.warning.

Now try entering *2.5000*. Does it work? 

.note
If you're familiar with programming in other languages, you may find
it odd that *sqrt 2* comes back with a decimal point (i.e., is a
floating point number) even though the argument to the function seems
to be an integer. This interchangability of numeric types is due to
Haskell's system of /type classes/ and will be discussed in detail in
Section 4.3). 
.note.

.exercises
- Exercise 3.1 
We've seen that multiplication binds more tightly than division. Can you
think of a way to determine whether function application binds more or
less tightly than multiplication?
.exercises.

== 3.2 Pairs, Triples and More 

In addition to single values, we should also address multiple values.
For instance, we may want to refer to a position by its @/Acoordinate,
which would be a pair of integers. To make a pair of integers is simple:
you enclose the pair in parenthesis and separate them with a comma. Try
the following:

    Prelude> (5,3)
    (5,3)