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Fractions: what changed

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Title: Fractions

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1+## What is Fractions?
2+
3+Fractions is a [[Unit|unit]] for doing calculations with fractions.\
4+The unit was inspired by a [discussion on the Lazarus forum](http://forum.lazarus.freepascal.org/index.php/topic,27805.0.html).
5+
6+## Subversion
7+
8+You can get a copy of fractions on
9+
10+`svn `[`https://lazarus-ccr.svn.sourceforge.net/svnroot/lazarus-ccr/components/fractions`](https://lazarus-ccr.svn.sourceforge.net/svnroot/lazarus-ccr/components/fractions)
11+
12+Or you can use viewcv to preview the files:
13+
14+[<http://sourceforge.net/p/lazarus-ccr/svn/HEAD/tree/components/fractions/>](http://sourceforge.net/p/lazarus-ccr/svn/HEAD/tree/components/fractions/)
15+
16+## Features
17+
18+### The TFraction type
19+
20+TFraction is an "advanced record": a [[Record|record]] with [[Method|methods]] and [[Property|properties]].
21+
22+#### Porperties of TFraction
23+
24+- property Numerator: [[Int64]];
25+ - Gets and sets the Numerator part of the fraction
26+- property Denominator: Int64;
27+ - Gets and sets the Denominator part of the fraction.
28+ - Trying to assign 0 (zero) to TFraction.Denomiator will raise an EZeroDivide [[Exceptions|exception]]
29+
30+#### Methods of TFraction
31+
32+- [[Procedure|procedure]] Normalize;
33+ - Normalizes the fraction by dividing the Numerator and Dominatr by their [[Greatest common divisor|greatest common divisor]], e.g. 10/5 becomes 2/1
34+- [[Function|function]] ToString: [[String]];
35+ - returns the string representation of the fraction (no normalizaltion is done), e.g. when Numerator = 10 and Denominator = 5, the function returns '10/5'
36+- function Resolve: String;
37+ - retruns the string representation of the fraction <b>after normalization</b>, e.g. when Numerator = 4 and Denominator = 3, the function returns '1 1/3'
38+- function ToFloat: [[Double]];
39+ - returns the floating point representation of the fraction, e.g. when Numerator = 1 and Denominator = 3, the function returns 0.3333333333333
40+
41+### Assigning and creating Fractions
42+
43+- function Fraction(ANumerator, ADenominator: Int64): TFraction;
44+ - creates the fraction ANumerator/ADenominator, e.g. Fraction(1,2) gives 1/2 (one half).
45+- function Fraction(AIntPart, ANumerator, ADenominator: Int64): TFraction;
46+ - creates the fraction ANumerator/ADenominator, e.g. Fraction(1,1,2) gives 1 1/2 (one and one half).
47+
48+### Supported calculations
49+
50+#### [[Operator overloading|Overloaded]] [[Operator|operators]]
51+
52+- = (equals)
53+- [[Less than|&lt;]] (less than)
54+- [[Greater than|&gt;]] (greater than)
55+- \<= (less than or equal)
56+- \>= (greater than or equal)
57+- [[Becomes|:=]] (assign): this operator allows an Int64 or a String on the right hand side
58+- [[Plus|+]] (add)
59+- [[Minus|-]] (subtract)
60+- [[Asterisk|*]] (multiply): overloaded for fractions and Int64
61+- [[Slash|/]] (divide): overloaded for fractions and Int64
62+- ** (power): only allowed for Integers (on right hand side)
63+
64+#### Mathematical functions
65+
66+- function Min(a, b: TFraction): TFraction; [[Inline|inline]]; [[Overload|overload]];
67+ - returns a if (a \<= b), otherwise retruns b
68+- function Max(a, b: TFraction): TFraction; inline; overload;
69+ - returns a if (a \>= b), otherwise returns b
70+- function InRange(const AValue, AMin, AMax: TFraction): [[Boolean]]; inline; overload;
71+ - returns [[True]] if (AValue \>= AMin) and (AValue \<= AMax), otherwise returns [[False]]
72+- function EnsureRange(const AValue, AMin, AMax: TFraction): TFraction; inline; overload;
73+ - returns AMin if (AValue \< AMin), otherwise returns AMax if (AValue \> AMax), otherwise returns AValue
74+- function Sign(const AValue: TFraction): TValueSign; inline; overload;
75+ - retruns NegativeValue if (AValue \< 0), otherwise returns PositiveValue if (AValue \>0), otherwise returns ZeroValue
76+- function IsZero(const AValue: TFraction): Boolean; overload;
77+ - returns True if (AValue = 0), otherwise returns False
78+- function Abs(const AValue: TFraction): TFraction; overload;
79+ - return AValue if (AValue \>= 0), otherwise returns -AValue;
80+
81+### General purpose functions
82+
83+- function GreatestCommonDivisor(a, b: Int64): Int64;
84+ - returns the greates common divisor of a and b
85+- function Floor(D: Double): Int64; overload;
86+ - returns the integer part of D (subtracted with 1, if (D \< 0))
87+
88+## Conversion routines
89+
90+- function FloatToFraction(Value, Precision: Double): TFraction;
91+ - returns an approximation of Value in the form of a fraction, e.g. FloatTofraction(0.5,0.01) returns 1/2 (one half)
92+ - Precision defines the acceptable value for which (Abs(Result) - Abs(Value) \<= Precision.
93+ - Precision is bound to Value by a magnitude of 15 at most
94+ - FloatToFraction in fact is a function varable which is initialized to the MF_FloatTofraction() function. You can assign your own function to it if wanted.
95+- function TryFloatToFraction(Value, Precision: Double; out F: TFraction; AcceptPrecisionError: Boolean): Boolean;
96+ - return value depends on AcceptPrecisionError value:
97+ - if (AcceptPrecisionError = True) then the function retruns True if an approxiamtion can be found
98+ - if (AcceptPrecisionError = False) the functions returns True if an approximation can be found AND Abs(Abs(Result)-Abs(Value)) \<= Precision
99+ - the value of F only makes sense if the function returns True
100+- function FloatToFractionDef(Value, Precision: Double; Def: TFraction; AcceptPrecisionError: Boolean): TFraction;
101+ - returns the found approximation (F) if TryFloatToFraction(Value, Precision, F, AcceptPrecisionError: Boolean) succeeds, otherwise returns Def
102+- function StrToFraction(const S: String): TFraction;
103+ - returns fraction that is represented in S, raises an EConvertError upon failure
104+- function TryStrToFraction(const S: String; out F: TFraction): Boolean;
105+ - retruns True if S can be converted to a fraction, otherwise returns False
106+ - the value of F only makes sense f the function returns True
107+- function StrToFractionDef(const S: String; Def: TFraction): TFraction;
108+ - returns the fraction represented in S if conversion succeeds, otherwise returns Def
109+
110+## Example program
111+
112+``` pascal
113+program example;
114+
115+{$mode objfpc}{$H+}
116+{$apptype console}
117+
118+uses
119+ Classes, sysutils, fractions, math;
120+
121+var
122+ F1, F2: TFraction;
123+ D, Prec: Double;
124+ i: Integer;
125+begin
126+ F1 := Fraction(1,1,3); // 1 1/3
127+ F2 := Fraction(4,3); // 4/3
128+ writeln('F1.ToString = ',F1.ToString); // '4/3'
129+ writeln('F1.Resolve = ',F1.Resolve); // '1 1/3'
130+ writeln('F1.ToFloat = ',F1.ToFloat:16:16); // 1.3333333333333333
131+ writeln('F2.ToString = ',F2.ToString); // '4/3'
132+ writeln('(F1 = F2) = ',F1=F2); //True
133+
134+ F1 := Fraction(1,2);
135+ F2 := Fraction(1,3);
136+ writeln(F1.ToString,' * ',F2.ToString,' = ',(F1*F2).Resolve); // '1/6'
137+ writeln(F1.ToString,' / ',F2.ToString,' = ',(F1/F2).Resolve); // '1 1/2'
138+ writeln(F1.ToString,' + ',F2.ToString,' = ',(F1+F2).Resolve); // '5/6'
139+ writeln(F1.ToString,' - ',F2.ToString,' = ',(F1-F2).Resolve); // '1/6'
140+ writeln(F1.ToString,' ** 2 = ',(F1**2).Resolve); // '1/6'
141+
142+ D := 0.25;
143+ F1 := FloatToFraction(D, 0.000001);
144+ writeln('FloatTofraction(0.25) -> ',F1.ToString); // '1/4'
145+ writeln;
146+ writeln('Approximations of Pi:');
147+ writeln(' [Pi = ',Pi:16:16,']');
148+ Prec := 1.0;
149+ for i := 1 to 10 do
150+ begin
151+ Prec := Prec / 10;
152+ F2 := FloatTofraction(Pi, Prec);
153+ writeln('FloatTofraction(Pi,',Prec:10:10,') = ',Format('%-13s',[F2.Resolve]),' [',F2.ToFloat:16:16,']');
154+ end;
155+end.
156+```
157+
158+This outputs:
159+
160+ F1.ToString = 4/3
161+ F1.Resolve = 1 1/3
162+ F1.ToFloat = 1.3333333333333333
163+ F2.ToString = 4/3
164+ (F1 = F2) = TRUE
165+ 1/2 * 1/3 = 1/6
166+ 1/2 / 1/3 = 1 1/2
167+ 1/2 + 1/3 = 5/6
168+ 1/2 - 1/3 = 1/6
169+ 1/2 ** 2 = 1/4
170+ FloatTofraction(0.25) -> 1/4
171+
172+ Approximations of Pi:
173+ [Pi = 3.1415926535897932]
174+ FloatTofraction(Pi,0.1000000000) = 3 1/7 [3.1428571428571428]
175+ FloatTofraction(Pi,0.0100000000) = 3 1/7 [3.1428571428571428]
176+ FloatTofraction(Pi,0.0010000000) = 3 16/113 [3.1415929203539825]
177+ FloatTofraction(Pi,0.0001000000) = 3 16/113 [3.1415929203539825]
178+ FloatTofraction(Pi,0.0000100000) = 3 16/113 [3.1415929203539825]
179+ FloatTofraction(Pi,0.0000010000) = 3 16/113 [3.1415929203539825]
180+ FloatTofraction(Pi,0.0000001000) = 3 4703/33215 [3.1415926539214212]
181+ FloatTofraction(Pi,0.0000000100) = 3 4703/33215 [3.1415926539214212]
182+ FloatTofraction(Pi,0.0000000010) = 3 4703/33215 [3.1415926539214212]
183+ FloatTofraction(Pi,0.0000000001) = 3 14093/99532 [3.1415926536189365]
184+
185+---
186+*From the Free Pascal wiki, page "Fractions", as it stood on 2022-08-06 after a change by Kai Burghardt, out of the wiki's dump of 2024-05-26. Copyright stays with the author who wrote it, according to the old wiki's [[Licensing]] page. If you have concerns about this material, [message the admin](/wiki-contact).*

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