Fractions

This post was published 10 years, 10 months ago. Due to the rapidly evolving world of technology, some concepts may no longer be applicable.

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.)

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