Pascal Programming/Arrays
An array is a structure concept for custom data types. It groups elements of the same data type. You will use arrays a lot if you are dealing with lots of data of the same data type.
ListsEdit
NotionEdit
In general, an array is a limited and arranged aggregation of objects, all of which having the same data type called base type or component type.^{[1]} An array has at least one discrete, bounded dimension, continuously enumerating all its objects. Each object can be uniquely identified by one or more scalar values, called indices, along those dimensions.
DeclarationEdit
In Pascal an array data type is declared using the reserved word array
in combination with the auxiliary reserved word of
, followed by the array’s base type:
program arrayDemo(output);
type
dimension = 1..5;
integerList = array[dimension] of integer;
Behind the word array
follows a nonempty commaseparated list of dimensions surrounded by brackets.
All array’s dimensions have to be ordinal data types, yet the base type type can be of any kind.
If an array has just one dimension, like the one above, we may also call it a list.
AccessEdit
A variable of the data type integerList
as declared above, holds five independent integer
values.
Accessing them follows a specific scheme:
var
powerN: integerList;
begin
powerN[1] := 5;
powerN[2] := 25;
powerN[3] := 125;
powerN[4] := 625;
powerN[5] := 3125;
writeLn(powerN[3]);
end.
This program
will print 125
, since it is the value of powerN
that has the index value 3
.
Arrays are like a series of “buckets” each holding one of the base data type’s values.
Every bucket can be identified by a value according to the dimension specifications.
When referring to one of the buckets, we have to name the group, that is the variable name (in this case powerN
), and a proper index surrounded by brackets.
Character arrayEdit
Lists of characters frequently have and had special support with respect to I/O and manipulation.
This section is primarily about understanding, as in the next chapter we will get to know a more sophisticated data type called string
.
Direct assignmentEdit
String literals can be assigned to array[…] of char
variables using an assignment statement, thus instead of writing:
program stringAssignmentDemo;
type
fileReferenceDimension = 1..4;
fileReference = array[fileReferenceDimension] of char;
var
currentFileReference: fileReference;
begin
currentFileReference[1] := 'A';
currentFileReference[2] := 'Z';
currentFileReference[3] := '0';
currentFileReference[4] := '9';
You can simply write:
currentFileReference := 'AZ09';
end.
Note, that you do not need to specify any index anymore.
You are referring to the entire array variable on the LHS of the assignment symbol (:=
).
This only works for overwriting the values of the whole array.
There are extensions allowing you to overwrite merely parts of a char
array, but more on that in the next chapter.
Most implementations of string literal to char
array assignments will pad the given string literal with insignificant char
values if it is shorter than the variable’s capacity.
Padding a string means to fill the remaining positions with other characters in order meet a certain size.
The GPC uses space characters (' '
), whereas the FPC uses char
values whose ord
inal value is zero (#0
).
Reading and printingEdit
Although not standardized, read
/readLn
and write
/writeLn
usually support writing to and reading from array[…] of char
variables.
program interactiveGreeting(input, output);
type
nameCharacterIndex = 1..20;
name = array[nameCharacterIndex] of char;
var
host, user: name;
begin
host := 'linuxnotebook'; // in lieu of getHostName()
writeLn('Hello! What is your name?');
readLn(user);
write('Hello, ', user, '. ');
writeLn('My name is ', host, '.');
end.
Hello! What is your name?
Aïssata
Hello, Aïssata. My name is linuxnotebook.
user
. The user input has been highlighted, and the program
was compiled with the FPC.This works because text
files, like the standard files input
and output
, are understood to be infinite sequences of char
values.^{[fn 1]}
Since our array[…] of char
is also, although finite sequence of char
values, individual values can be copied pretty much directly to and from text
files, not requiring any kind of conversion.
Primitive comparisonEdit
Unlike other arrays, array[…] of char
variables can be compared using =
and <>
.
if user <> host then
begin
write('Hello, ', user, '. ');
writeLn('My name is ', host, '.');
end
else
begin
writeLn('No. That is my name.');
end;
This kind of comparison only works as expected for identical data types.
It is a primitive characterbycharacter comparison.
If either array is longer or shorter, an =
comparison will always fail, because not all characters can be compared to each other.
Correspondingly, an <>
comparison will always succeed for array[…] of char
values that differ in length.
Note, the EP standard also defines the EQ
and NE
functions, beside many more.
The difference to =
and <>
is that blank padding (i. e. #0
in FPC or ' '
in GPC) has no significance in EQ
and NE
.
In consequence, that means using these functions you can compare strings and char
arrays regardless of their respective maximum capacity and still get the naturally expected result.
The =
and <>
comparisons on the other hand look at the memory’s internal representation.
MatricesEdit
An array’s base type can be any data type,^{[fn 2]} even another array. If we want to declare an “array of arrays” there is a short syntax for that:
program matrixDemo(output);
const
columnMinimum = 30;
columnMaximum = 30;
rowMaximum = 10;
rowMinimum = 10;
type
columnIndex = columnMinimum..columnMaximum;
rowIndex = rowMinimum..rowMaximum;
plot = array[rowIndex, columnIndex] of char;
This has already been described above. The last line is identical to:
plot = array[rowIndex] of array[columnIndex] of char;
It can be expanded to two separate declarations allowing us to “reuse” the “inner” array data type:
row = array[columnIndex] of char;
plot = array[rowIndex] of row;
Note that in the latter case plot
uses row
as the base type which is an array by itself.
Yet in the short notation we specify char
as the base type, not a row
but its base type.
When an array was declared to contain another array, there is a short notation for accessing individual array items, too:
var
curve: plot;
x: columnIndex;
y: rowIndex;
v: integer;
begin
// initialize
for y := rowMinimum to rowMaximum do
begin
for x := columnMinimum to columnMaximum do
begin
curve[y, x] := ' ';
end;
end;
// graph something
for x := columnMinimum to columnMaximum do
begin
v := abs(x)  rowMaximum;
if (v >= rowMinimum) and (v <= rowMaximum) then
begin
curve[v, x] := '*';
end;
end;
This corresponds to the array’s declaration.
It is vital to ensure the indices you are specifying are indeed valid.
In the latter loop the the if
branch checks for that.
Attempting to access nonexistent array values, i. e. by supplying illegal indices, may crash the program, or worse remain undetected thus causing difficult to find mistakes.
A program compiled with the GPC will terminate with./a.out: value out of range (error #300 at 402a76)
‑Cr commandline switch, or placing the compiler directive comment{$rangeChecks on}
Confer also chapter “Enumerations”, subsection “Restriction”. 
Note, the “unusual” order of x
and y
has been chosen to facilitate drawing an upright graph:
// print graph, note reverse `downto` direction
for y := rowMaximum downto rowMinimum do
begin
writeLn(curve[y]);
end;
end.
That means, it is still possible to refer to entire “sub”arrays as a whole. You are not forced to write all dimension an array value has, given it makes sense.
Array data types that have exactly two dimensions are also called matrices, singular matrix.
In mathematics a matrix does not necessarily have to be homogenous, but could contain different “data types”. 
Real valuesEdit
As introduced in one of the first chapters the data type real
is part of the Pascal programming language.
It is used to store integer values in combination with an optional fractional part.
Real literalsEdit
In order to distinguish integer
literals from real
literals, specifying real
values in your source code (and also for read
/readLn
) differs slightly.
The following source code excerpt shows some real
literals:
program realDemo;
var
x: real;
begin
x := 1.61803398;
x := 1E9;
x := 500e12
x := 1.7320508;
x := +0.0;
end.
To summarize the rules:
 A
real
value literal always contains either a.
as a radix mark, or anE
/e
to separate an exponent (the in ), or both.  There is at least one WesternArabic decimal digit before and one after the
.
(if there is any).  The entire number and exponent are preceded by signs, yet a positive sign is optional.
0.0 accepts a sign, but is in fact (in mathematics) a signless number. 0.0 and +0.0 denote the same value.

As it has always been, all number values cannot contain spaces.
LimitationsEdit
The real
data type has many limitations you have to be aware of in order to effectively use it.
First of all, we want to reemphasize an issue that was mentioned when data types were introduced:
In computing real
variables can only store a subset of rational numbers (ℚ).^{[2]}
That means, for example, you cannot store the (mathematically speaking) real number (ℝ) .
This number is an irrational number (i. e. not a rational number).
If you cannot write a number as a finite real
literal, it is impossible to store it in a system using a finite amount of memory, such as computer systems do.
Fortunately, in EP three constants aide your usage of real
values.
minReal
is the smallest positive real
value.
It conjunction with the constant maxReal
, it is guaranteed that all arithmetic operations in produce, quote, reasonable approximations.^{[3]}
It is not specified what exactly constitutes a “reasonable” approximation, thus it can, for example, mean that maxReal  1
yields as “an approximation” maxReal
.^{[fn 3]}
Also, it is quite possible that real
variables can store larger values than maxReal
.
epsReal
is short for “epsilon real
”.
The small Greek letter ε (epsilon) frequently denotes in mathematics an infinitely small (positive) value, yet not zero.
According to the ISO standard 10206 (“Extended Pascal”), epsReal
is the result of subtracting 1.0
from the smallest value that is greater than 1.0
.^{[3]}
No other value can be represented between this value and 1.0
, thus epsReal
represents the highest precision available, but just at that point.^{[fn 4]}
Most implementations of the real
data type will show a significantly varying degree of precision.
Most notable, the precision of real
data type implementations complying with the IEEE standard 754 format, decays toward the extremes, when approaching (and going beyond) maxReal
and maxReal
.
Therefore you usually use, if at all, a reasonable multiple of epsReal
that fits the given situation.
Transfer functionsEdit
Pascal’s strong typing system prevents you from assigning real
values to integer
variables.
The real
value may contain a fractional part that an integer
variable cannot store.
Pascal defines, as part of the language, two standard functions addressing this issue in a welldefined manner.
 The function
trunc
, short for “truncation”, simply discards any fractional part and returns, as aninteger
, the integer part. As a consequence this is effectively rounding a number toward zero.trunc(1.999)
will return the value1
.  If this feels “unnatural”, the
round
function rounds areal
number to its closestinteger
value.round(x)
is (regarding the resulting value) equivalent totrunc(x + 0.5)
for nonnegative values, and equivalent totrunc(x  0.5)
for negativex
values.^{[fn 5]} In other words, this is what you were probably taught in grade school, or the first rounding method you learned in homeschooling. It is commonly referred to as “commercial rounding”.
Both functions will fail if there is no such integer
value fulfilling the function’s respective definition.
There is no function if you want to (explicitly) “transfer” an integer
value to a real
value.
Instead, one uses arithmetically neutral operations:
integerValue * 1.0
(using multiplicative neutral element), orintegerValue + 0.0
(using summing neutral element)
These expressions make use of the fact, as it was mentioned earlier as a passing remark in the chapter on expressions, that the expression’s overall data type will become real
as soon as one real
value is involved.^{[4]}
Printing real
valuesEdit
By default write
/writeLn
print real
values using “scientific notation”.
real
value constant pi
. It was adopted by many compilers.
program printPi(output);
begin
writeLn(pi);
end.
3.141592653589793e+00
 sign, where a positive sign is replaced with a blank
 one digit
 a dot
 a positive number of postdecimal digits
 an
E
(uppercase or lowercase)  the sign of the exponent, but this time a positive sign is always written and a zero exponent is preceded by a plus sign, too
 the exponent value, with a fixed minimum width (here 2) and leading zeros
While this style is very universal, it may also be unusual for some readers.
Particularly the E
notation is something now rather archaic, usually only seen on pocket calculators, i. e. devices lacking of enough display space.
Luckily, write
/writeLn
allow us to customize the displayed style.
real
parameters, too, but this also shows more digits:
program printPiDigits(output);
begin
writeLn(pi:40);
end.
3.141592653589793238512808959406186e+00
40
refers to the entire width including a sign, the radix mark, the e
and the exponent representation.The procedures write
/writeLn
accept for real
type values (and only for real
values) another colonseparated format specifier.
This second number determines the (exact) number of postdecimal digits, the “fraction part”.
Supplying two format specifiers also disables scientific notation.
All real
values are printed using regular positional notation.
That may mean for “large” numbers such as 1e100
printing a one followed by a hundred zeros (just for the integer part).
program realFormat(output);
var
x: real;
begin
x := 248e9 + 500e6;
writeLn(x:32:8);
end.
248000000000.00048828
.
and in the case of negative numbers 
, is 32 characters. After the .
follow 8 digits. Bear in mind the precise number, especially the fractional part, may vary.In some regions and languages it is customary to use a ,
(comma) or other character instead of a dot as a radix mark.
Pascal’s onboard write
/writeLn
procedures will, on the other hand, always print a dot, and for that matter – read
/readLn
will always accept dots as radix marks only.
Nevertheless, all current Pascal programming suites, Delphi, FPC and GPC provide appropriate utilities to overcome this restriction.
For further details we refer to their manuals.
This issue should not keep us from continuing learning Pascal.
write
/writeLn
(and in EP also writeStr
) generate will be rounded with respect to the last printed digit.
program roundedWriteDemo(output);
begin
writeLn(3.75:4:1);
end.
3.8
real
’s limitations. It was verified that the computer used for this demonstration could indeed store precisely the specified value. The rounding you see in this particular case is not due to any technical circumstances.ComparisonsEdit
First of all, all (arithmetic) comparison operators do work with real
values.
The operators =
and <>
, though, are particularly tricky to handle.
In most applications you do not compare real
values to each other when checking for equality or inequality.
The problem is that numbers such as ⅓ cannot be stored exactly with a finite series of binary digits, only approximated, yet there is not one valid approximation for ⅓ but many legit ones.
The =
and <>
operators, however, compare – so to speak — for specific bit patterns.
This is usually not desired (for values that cannot be represented exactly).
Instead, you want to ensure the values you are comparing are within a certain range, like:
function equal(x, y, delta: real): Boolean;
begin
equal := abs(x  y) <= delta;
end;
Delphi and the FPC’s standard RTS provide the function sameValue
as part of the math
unit.
You do not want to program something other people already have programmed for you, i. e. use the resources.
DivisionEdit
Now that we know the data type used for storing (a subset of) rational numbers, in Pascal known as real
, we can perform and use the result of another arithmetic operation:
The division.
FlavorsEdit
Pascal defines two different division operators:
 The
div
operator performs an integer division and discards, if applicable, any remainder. The expression’s resulting data type is alwaysinteger
. Thediv
operator only works if both operands areinteger
expressions.  The, probably more familiar, operator
/
(a forward slash), divides the LHS number (the dividend) by the RHS number (the divisor), too, but a/
division always yields areal
type value.^{[4]} This is also the case if the fractional part is zero. A “remainder” does not exist for the/
operation.
The div
operation can be put in terms of /
:
divisor div dividend = trunc(divisor / dividend)
However, this is only a semantic equivalent,^{[fn 7]} it is not how it is actually calculated.
The reason is, the /
operator would first convert both operands to real
values and since, as explained above, not all integer
values can necessarily be represented exactly as real
values, this would produce results potentially suffering from rounding imprecisions.
The div
operator on the other hand is a pure integer
operator and works with “integer precision”, that means no rounding is involved in actually calculating the div
result.
Off limits divisorEdit
Note, since there is no generally accepted definition for division by zero, a zero divisor is illegal and will result in your program to abort. If your divisor is not a constant and depends on runtime data (such as a variable read from user input), you should check that it is nonzero before doing a division:
if divisor <> 0 then
begin
x := dividend / divisor;
// ...
end
else
begin
writeLn('Error: zero divisor encountered when calculating rainbow');
end;
Alternatively, you can declare data types excluding zero, so any assignment of a zero value will be detected:
type
natural = 1..maxInt;
var
divisor: natural;
begin
write('Enter a positive integer: ');
readLn(divisor); // entering a nonpositive value will fail
Some Pascal dialects introduce the concept of “exceptions” that can be used to identify such problems. Exceptions may be mentioned again in the “Extensions” part of this Wikibook.
Arrays as parametersEdit
Arrays can be copied with one simple assignment dataOut := dataIn;
.
This requires, however, as it is customary with Pascal’s strong type safety, that both arrays are assignmentcompatible:
That means their base type and dimension specifications are the same.^{[fn 8]}
Because calling a routine involves invisible assignments, writing generalpurpose code dealing with lots of different situations would be virtually impossible if the entire program had to use one array type only. In order to mitigate this situation, conformant array type parameters allow writing routines accepting differing array dimension lengths. Array dimension data types still have to match.
Let’s look at an example program using this feature:
program tableDemo(output);
procedure printTableRows(data: array[minimum..maximum: integer] of integer);
var
i: integer; // or in EP `i: type of minimum;` [preferred alternative]
begin
for i := minimum to maximum do
begin
writeLn(i:20, '  ', data[i]:20);
end;
end;
A conformantarray parameter looks pretty similar to a regular array variable declaration, but the dimensions are specified differently.
Usually, when declaring new arrays you provide constant values as dimension limits.
Since we do not want constants, though, we name placeholder identifiers for the actual dimension limits of any array printTableRows
will receive.
In this case they are named minimum
and maximum
, joined by ..
inbetween indicating a range.
minimum
and maximum
become variables inside the definition of printTableRows
, but you are not allowed to assign any values to them.^{[fn 9]}
You are not allowed to declare new identifiers bearing the same name as the array boundary variables.
In Pascal all constants implicitly have an unambiguously determinable data type.
Since our array limit identifiers are in fact variables they require a data type.
The : integer
indicates both minimum
and maximum
have the data type integer
.
In a conformant array paramter, the short notation for nested arrays uses a ; to separate multiple dimensions, thus resembling a regular parameter list.

Once we have declared and defined printTableRows
we can use it with lots of differently sized arrays:
var
table: array[0..3] of integer;
nein: array[9..9] of integer;
begin
table[0] := 1;
table[1] := 2;
table[2] := 4;
table[3] := 8;
printTableRows(table);
nein[9] := 9;
printTableRows(nein);
end.
Delphi and the FPC (as of version 3.2.0 released in 2020) do not support conformantarray parameters, but the GPC does.
LogarithmsEdit
Special supportEdit
Prior the 21^{st} century logarithm tables and slide rules were heavily utilized tools in manual calculations, so much so it led to the inclusion of two basic functions in Pascal.
 The function
exp
exponentiates a number to the base , Euler’s constant. The value ofexp(x)
is equivalent to the mathematical term .  The function
ln
, short for Latin “logaritmus naturalis”, takes the natural logarithm of a positive number. “Natural” refers to again.
Both functions always return real
values.
IntroductionEdit
Since the use of logarithms is not necessarily taught in all curricula, or you might just need a refresher, here is a short primer: The basic idea of logarithms is that all operations are lifted one level.
logarithm level  

real level 
On the lifted level many operations become simpler, especially if the numbers are large. For instance, you can perform a rather easy addition if you actually mean to take the product of two numbers. For this you have to “lift” all operands one level up: This is done by taking the logarithm. Pay particular attention to : On the logarithm level is a non“logarithmized” factor, you only take the logarithm of
Once you are done, you have descend one level again to get the actual “real” result of the intended operation (on the underlying level).
The reverse operation of ln
is exp
.^{[fn 10]}
To put this principle in Pascal terms:
x * y = exp(ln(x) + ln(y))
Remember, x
and y
have to be positive in order to be valid parameters to ln
.
ApplicationEdit
Taking the logarithm and then exponentiating values again are considered “slow” operations and introduce a certain overhead. In programming, overhead means taking steps that are not directly related to the actual underlying problem, but only facilitate solving it. In general, overhead is avoided, since (at first) it takes us even farther away from the solution.
real
data type if intermediate results are out of its range, but it is known that the final result will definitely be within the range of real
again.
program logarithmApplicationDemo(output);
const
operand = maxReal;
var
result: real;
begin
// for comparison
writeLn(maxReal:40);
result := sqr(operand);
result := sqrt(result);
writeLn(result:40);
// “lift” one level
result := ln(operand);
result := 2.0 * result; // corresponds to sqr(…)
result := 0.5 * result; // corresponds to sqrt(…)
// reverse `ln`: bring `result` “back down”
result := exp(result);
writeLn(result:40);
end.
1.7976930000000000495189307440746532950903200318892038e+308
Inf
1.7976929999999315921963138504476453672053533033977331e+308
sqr
in line 10 exceeds the range of real
rendering any subsequent results invalid. In this particular implementation this situation is displayed as Inf
(infinity). Since we know that reversing this operation by taking the principal square root results in a storable result again, we can perform the same operation with logarithms instead.
The shown output was generated by the program
being compiled with the GPC. The program was executed on a 64bit platform with an FPU using 80bit numbers.As you can see, this goes to the detriment of precision. It is a compromise between fast operations, and “accurate enough” results.
The best solution is, of course, finding a better algorithm.
The above demonstration is effectively , that is abs(x)
(remember, squaring a number always yields a nonnegative number).
This operation’s result will stay in the range of real
.
TasksEdit
All tasks, including those in the following chapters, can be solved without conformantarray parameters. This takes account of the fact that not all major compilers support them.^{[fn 11]} Nonetheless, using routines with conformantarray parameters are often the most elegant solution.
program
that prints the following multiplication table:
1 2 3 4 5 6 7 8 9 10
2 4 6 8 10 12 14 16 18 20
3 6 9 12 15 18 21 24 27 30
4 8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100
writeLn
). The generation of data and printing data shall be implemented by separate routines (these routine may not call each other).program multiplicationTable(output);
const
xMinimum = abs(1);
xMaximum = abs(10);
yMinimum = abs(1);
yMaximum = abs(10);
// NB: Only Extended Pascal allows constant definitions
// based on expressions. The following two definitions
// are illegal in Standard Pascal (ISO 7185).
zMinimum = xMinimum * yMinimum;
zMaximum = xMaximum * yMaximum;
Calculating the maximum and minimum expected value now (as constants) has the advantage that the compiler will emit a warning during compilation if any value exceeds maxInt
.
The abs
were inserted as a means of documentation:
The program only works properly for nonnegative values.
type
x = xMinimum..xMaximum;
y = yMinimum..yMaximum;
z = zMinimum..zMaximum;
table = array[x, y] of z;
Using z
as the table
array’s base type (and not just integer
) has the advantage that if we accidentally implement the multiplication incorrectly, assigning outofrange values will fail.
For such a trivial task like this one it is of course irrelevant, but for more difficult situations deliberately constricting the allowed range can thwart programming mistakes.
Do not worry if you just used a plain integer
.
var
product: table;
Note, the product
variable has to be declared outside and before populateTable
and printTable
are defined.
This way both routines refer to the same product
variable.^{[fn 12]}
procedure populateTable;
var
factorX: x;
factorY: y;
begin
for factorX := xMinimum to xMaximum do
begin
for factorY := yMinimum to yMaximum do
begin
product[factorX, factorY] := factorX * factorY;
end;
end;
end;
It is also possible to reuse previously calculated values, make use of the fact that the table can be mirrored along the diagonal axis, or do other “optimization stunts”.
The important thing for this task, though, is to correctly nest the for
loops.
procedure printTable;
var
factorX: x;
factorY: y;
begin
for factorY := yMinimum to yMaximum do
begin
for factorX := xMinimum to xMaximum do
begin
write(product[factorX, factorY]:5);
end;
writeLn;
end;
end;
An advanced implementation would, of course, first determine the maximum expected length and store it as a variable, instead of using the hardcoded format specifier :5
.
This, though, is out of this task’s scope.^{[fn 13]}
It just should be mentioned hardcoded values like this one are considered bad style.
begin
populateTable;
printTable;
end.
real
value literals shorter than five characters, all denoting the value “positive one”. What are they? The user will terminate his input with an empty line. Print this instruction beforehand.
 When done, print the message.
 When printing, a line may be at most 80 characters long (or whatever is reasonable for you). You are allowed to presume the user’s input lines are at most 80 characters long.
 Ensure you only wrap lines at space characters (unless there are no space characters in a line).
More tasks you can solve can be found on the following Wikibook pages:
 Alevel Computing 2009/AQA/Problem Solving, Programming, Data Representation and Practical Exercise/Fundamentals of Programming/OneDimensional Arrays
 Alevel Computing 2009/AQA/Problem Solving, Programming, Data Representation and Practical Exercise/Fundamentals of Programming/TwoDimensional Arrays
Sources:
 ↑ Jensen, Kathleen; Wirth, Niklaus. Pascal – user manual and report (4th revised ed.). p. 56. doi:10.1007/9781461244509. ISBN 9780387976495. "An array type consists of a fixed number of components (defined when the array type is introduced) all having the same type, called the component type."
 ↑ This limitation comes from Pascal: ISO 7185:1990 (Report). ISO/IEC. 1991. p. 16. "realtype. The required typeidentifier real shall denote the realtype. […] The values shall be an implementationdefined subset of the real numbers, denoted as specified in 6.1.5 by signedreal." The end of the last sentence implies only writable, those you can specify in your source code, are legal
real
values. For example, the value is different from , the decimial representation of which would be infinitely long (a. k. a. irrational number), thus the actual, correct value of cannot appear in source code as a “real
” value.  ↑ ^{a} ^{b} Joslin, David A. (19890601). "Extended Pascal – Numerical Features". Sigplan Notices 24 (6): 77–80. doi:10.1145/71052.71063. "The programmer can obtain some idea of the real range and precision from the (positive) implementationdefined constants
MINREAL
,MAXREAL
andEPSREAL
. Arithmetic in the ranges[maxreal,minreal]
,0
, and[minreal,maxreal]
“can be expected to work with reasonable approximations”, whereas outside those ranges it cannot. As what constitutes a “reasonable approximation” is a matter of opinion, however, and is not defined in the standard, this statement may be less useful than it appears at first sight. The measure of precision is on somewhat firmer ground:EPSREAL
is the commonly employed measure of (typically floatingpoint) accuracy, i.e the smallest value such that1.0 + epsreal > 1.0
.".  ↑ ^{a} ^{b} Jensen, Kathleen; Wirth, Niklaus. Pascal – user manual and report (4th revised ed.). p. 20. doi:10.1007/9781461244509. ISBN 9780387976495. "As long as at least one of the operands is of type
Real
(the other possibly being of typeInteger
) the following operators yield a real value:
"*
multiply /
divide (both operands may be integers, but the result is always real) +
add 
subtract
Notes:
 ↑ More precisely,
text
files are (possibly empty) sequences of lines, each line consisting of a (possibly empty) sequence ofchar
values.  ↑ Some compilers (such as the FPC) allow zerosized data types [not allowed in any ISO standard]. If that is the case, an array that has a zerosize base type will be rendered ineffective, virtually forfeiting all characteristics of arrays.
 ↑ Modern
real
arithmetic processors can indicate a precision loss, i. e. when the result of an arithmetic operation had to be “approximated”. However, there is no standardized way to access this kind of information from your Pascal source code, and usually this kind of signaling is also not favorable, since the tiniest precision loss will set off the alarm, thus slowing down your program. Instead, if it matters, one uses software that allows arbitrary precision arithmetics, like for example the GNU Multiple Precision Arithmetic Library.  ↑ This number is not completely arbitrary. The most prevalent
real
number implementation IEEE 754 uses a “hidden bit”, making the value1.0
special.  ↑ Not all compilers comply with this definition of the standard. The FPC’s standard
round
implementation will round in the case of equidistance toward even numbers. Knowing this is relevant for statistical applications. The GPC uses for itsround
implementation functionality provided by the hardware. As such, the implementation is hardwaredependent, on its specific configuration, and may deviate from the ISO 7185 standard definition.  ↑ Given the most prevalent implementations Two’s complement for
integer
values and IEEE 754 forreal
values, you have to consider the fact that (virtually) all bits in aninteger
contribute to its (mathematical) value, whereas areal
number stores values for the expressionmantissa * base pow exponent
. In very simple terms, themantissa
stores theinteger
part of a value, but the problem is that it occupies fewer bits than aninteger
would use, thus there is (for values that require more bits) a loss in information (i. e. a loss in precision).  ↑ The exact technical definition reads like: The value of
dividend div divisor
shall be such thatwhere the value shall be zero ifabs(dividend)  abs(divisor) < abs((dividend div divisor) * divisor) <= abs(dividend)
abs(dividend) < abs(divisor)
; otherwise, […]  ↑ Furthermore, both arrays have to be either
packed
or “unpacked”.  ↑ The EP standard calls this characteristic
protected
.  ↑ It is important that both functions use one common base, in this case it is .
 ↑ The ISO standard 7185 (“Standard Pascal”) calls this, lack of conformantarray parameters, “Level 0 compliance”. “Level 1 compliance” includes support for conformant array parameters. Of the compilers presented in Getting started only the GPC is a “Level 1”compliant compiler.
 ↑ This style of programming is slightly disfavored, keyword “global variables”, but as for now we do not know appropriate syntax (
var
parameters) to not do that.  ↑ For extra credit: You can make use of the fact that (assuming that
zMaximum
is positive) . You can use this value to find out the minimum number of digits required.