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.NET 1.1+

# Rational Number Arithmetic

Standard arithmetic uses integers or floating-point values for operands and results. Rational numbers, or fractions, are not amongst the standard data types. In this article, we will create a structure for fractions and add arithmetic functionality.

### Finding the Greatest Common Divisor

The GCD of two values can be determined using the Euclidean Algorithm. This algorithm follows an iterative process. The first value is divided by the second value and the remainder determined. If the remainder is zero, the GCD is the first value. If the remainder is non-zero, the process is repeated using the second term and the remainder from the calculation. The algorithm and its proof are explained in the above link.

To implement the algorithm, add the following GetGCD method. The method calls itself iteratively until the greatest common divisor is determined. As it uses to instance fields it can be declared as static:

private static int GetGCD(int term1, int term2)
{
if (term2 == 0)
return term1;
else
return GetGCD(term2, term1 % term2);
}

When creating a new rational number, the constructor reduces it to its lowest terms using the two added methods. This can now be tested using the following code in the Main method of the Program class:

RationalNumber rn = new RationalNumber(5, 10);
Console.WriteLine("{0}/{1}", rn.Numerator, rn.Denominator);     // Outputs "1/2"

Now that the rational number structure is able to hold a fraction in its lowest terms, we can add the arithmetic functions. To achieve this, we will overload the operators for +, -, * and / with new versions that supply the correct functionality for addition, subtraction, multiplication and division of fractions. The first to be created is the addition operator. To add two rational numbers together, the following formula is used:

This formula can be added to the structure by adding the following overloaded operator, which simply creates a new rational number using the equation shown above. The act of creating the new fraction will also reduce it to its lowest terms automatically.

public static RationalNumber operator +(RationalNumber r1, RationalNumber r2)
{
return new RationalNumber((r1.Numerator * r2.Denominator)
+ (r2.Numerator * r1.Denominator), r1.Denominator * r2.Denominator);
}

Subtraction of rational numbers uses a similar formula to addition. It is:

To create the overloaded subtraction operator, add the following code to the structure.

public static RationalNumber operator -(RationalNumber r1, RationalNumber r2)
{
return new RationalNumber((r1.Numerator * r2.Denominator)
- (r2.Numerator * r1.Denominator), r1.Denominator * r2.Denominator);
}

### Adding the Multiplication and Division Operators

Finally we can add the operators for multiplication and division. To multiply, the numerators of the fraction are multiplied together, as are the denominators. For division, the fraction being divided by is inverted before performing a multiplication. The formulae for these operations are as follows:

To create the two operators, add the following code to the structure:

public static RationalNumber operator *(RationalNumber r1, RationalNumber r2)
{
return new RationalNumber(r1.Numerator * r2.Numerator, r1.Denominator * r2.Denominator);
}

public static RationalNumber operator /(RationalNumber r1, RationalNumber r2)
{
return new RationalNumber(r1.Numerator * r2.Denominator, r1.Denominator * r2.Numerator);
}

### Testing the Rational Number Arithmetic

The completed structure can now be tested using the Main method of the program class. Try the following calculations:

RationalNumber r1 = new RationalNumber(3, 4);
RationalNumber r2 = new RationalNumber(2, 3);