Fractions

For many people the mere mention of fractions elicits a wince, and while these lovely math constructs played a notable role in our early years of math, they are reasonably simple entities. While most humans revisit fractions over many years and often still fail to grasp the concept, for a computer, the elementary operations with fractions (addition, subtraction, multiplication, division, and the necessary greatest common factor (GCF) and lowest common multiple (LCM)) can be coded in just a few lines.

While by no means a perfect class (quite possibly a rather poorly coded class), the following provides some basic functions necessary for working with fractions.

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<?php
class frac{
public $d;
public $n;
 
	public function __construct($num, $den) {
		$this->n = $num;
		$this->d = $den;
	}
 
	public function gcf($n1, $n2){
		if ($n2>$n1){
			$tmp = $n1;
			$n1=$n2;
			$n2=$tmp;
		}
		do{
			$rem = $n1 % $n2;
			$n1 = $n2;
			$n2 = $rem;
		}while($rem!=0);
		return $n1;
	}
 
	public function lcm($n1, $n2){
		return $n1*($n2/frac::gcf($n1,$n2));
	}
 
	public function reduce (){
		$g = $this->gcf($this->n,$this->d);
		$this->n /= $g;
		$this->d /= $g;
	}
 
	public function multiply (frac $n1, frac $n2){
		$f = new frac($n1->n*$n2->n,$n1->d*$n2->d);
		$f->reduce();
		return $f;
	}
 
	public function divide (frac $n1, frac $n2){
		return frac::multiply($n1, new frac($n2->d,$n2->n));
	}
 
	public function add (frac $n1, frac $n2){
		$g = frac::lcm($n1->d,$n2->d);
		$f= new frac($n1->n*($g/$n1->d)+$n2->n*($g/$n2->d),$g);
		$f->reduce();
		return $f;
	}
 
	public function subtract (frac $n1, frac $n2){
		return frac::add($n1, new frac(-1*$n2->n,$n2->d));
	}
 
	public function display(){
		return $this->n . "/" . $this->d;
	}
}
?>

Examples of use:
1/3 + 1/2:

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<?php
$f1 = new frac(1,3);
$f2 = new frac(1,2); 
$ans = frac::add($f1,$f2); 
echo $ans->display();
?>

1/8 * 2/5

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<?php
$f1 = new frac(1,8);
$f2 = new frac(2,5);
$ans = frac::multiply($f1,$f2);
echo $ans->display();
?>

The gcf function uses Euclid’s algorithm, and the lcm function (used to find the common denominator) calls the gcf function.

Given the significant disparity between the ease with which a computer can ‘learn’ fractions, and the difficulty encountered by most students, perhaps it is time to consider teaching fractions as a series of concrete steps – an algorithm – instead of the current method. (Granted, most current methods do provide a method for arriving at an answer, but especially for the determination of the lowest common denominator (or reducing fractions), a procedural methodology (e.g. prime factoring, Euclid’s method, etc) is rarely given.)