You don’t see overloaded operators used that much in custom code. And, in few business applications have I seen a request for fractions, but there was such a request made of me recently. Following a search on the web. I discovered news groups and Q&A sites where some of you did request a Fraction’s class. So, I thought I’d share a solution.
In this article, you will learn how to implement a Fraction’s class—which is ideally suited for overloaded operators, how to overload operators in Visual Basic, and get a quick reminder from your college days on the Euclidean algorithm for calculating the greatest common divisor.
Turning a decimal number into a fraction encompasses a couple of specific steps. (There may be a faster, smaller way to convert decimals into fractions, but this solution works and uses some pretty neat elements of the .NET framework.) To convert a decimal number into a fraction, first note that .2 is equivalent to 2/10ths; thus, to solve the problem programmatically you need to:
- Save the sign
- Store the whole number
- Find the numerator
- Find the denominator
- Find the greatest common divisor and reduce the fraction
To complete a Fraction class and make it generally useful, you also want to support basic arithmetic operations on fractions and mixed arithmetic on fractions and decimal (or double) numbers. All of these elements are described in the remainder of this article with a complete code listing at the end of the article.
Determining the Sign with Regular Expressions
You can start anywhere, but logically people in the West are oriented to reading from left to right, so that’s where you will start. Assuming you have a decimal (double or single) number or a string representing the same, you can strip and store the sign of the number. The following fragment uses a Regular Expression to determine whether a string—you can easily convert numeric strings to and from a string or numeric representation—contains a negative sign and then you store the sign as an integer—1 or -1.
Private Sub SetSign(ByVal value As String) If (Regex.IsMatch(value, "^-")) Then Sign = -1 End If End Sub
The Regular Expressions “^-” simply checks for a bo?= symbol at the beginning of the input string.
Storing the Whole Number
The whole number is as easy to store as the minus sign. You could convert the string to an integer that would do the job for you. You could use Math.Abs to get the number as an absolute value (because you stored the sign), or you could extract the substring, from left to right, up to the index of the decimal point. The latter is the technique demonstrated in the SetWholePart method that will be added to the Fraction class.
Private Sub SetWholePart(ByVal value As String, _ ByVal index As Integer) Dim whole As String = value.Substring(0, index) If (whole.Length > 0) Then WholeNumber = Convert.ToInt32(whole) End If End Sub
The index of the decimal point is returned from the CheckNoDecimalPoint method, shown next.
Private Function CheckNoDecimalPoint(ByVal number As Double) _ As Integer Dim index As Integer = number.ToString().LastIndexOf(".") If (index = -1) Then WholeNumber = Convert.ToInt32(number) End If Return index End Function
Finding the Numerator
The easy work is done. Next, you need to find the numerator (and denominator) from the mantissa or the decimal (or fractional) part of the number. The answer is simple: The decimal number without the decimal point is the numerator. For example, 3.2 has a mantissa of .2. The mantissa .2 is 2/10ths, so clearly the mantissa reveals the numerator of the fraction.
The method SetFractionalPart orchestrates setting the numerator and denominator by stripping the mantissa from your input value.
Private Sub SetFractionalPart(ByVal value As String, _ ByVal index As Integer) Dim fraction As String = value.Remove(0, index + 1) If (fraction.Length > 0) Then SetFractionalPart(fraction) _numerator = Convert.ToInt32(fraction) SetDenominator(fraction) ReduceWithGcd() End If End Sub
SetFractionalPart clearly shows that you Remove all of the input value up to and including the decimal point and then convert the remaining digits to the numerator store as an integer.
Finding the Denominator
The denominator is also pretty straightforward. The denominator is always 10nth where n is the length of the numerator. For example, a numerator of 234 has a denominator of 1,000 or 103. You can use Math.Exp and the length of the numerator to calculate the denominator (as shown next).
Private Sub SetDenominator(ByVal fraction As String) Denominator = Math.Pow(10, fraction.Length) End Sub
The field fraction is set in an overloaded SetFractionalPart method that truncates the length of the decimal number to eight characters to ensure that it fits in an integer. (If you need a longer fraction, use a long data type.) Here is the overloaded SetFractionalPart method.
Private Sub SetFractionalPart(ByRef fraction As String) If (fraction.Length > 8) Then fraction = fraction.Substring(0, 8) End If _fractionalNumber = _ Math.Round(Convert.ToDouble("." + fraction), 8) End Sub
Factoring and Reducing with the Euclidean Algorithm
The final step is to find the greatest common divisor and reduce the numerator and denominator by this divisor. You can use the Euclidean algorithm—discovered by Euclid around 300 BC—that uses division, modular arithmetic, and remainders to quickly resolve the greatest common divisor. Here is a non-recursive Euclidean Gcd algorithm and a helper function that reduces the fraction.
Private Function Gcd(ByVal num As Integer, _ ByVal den As Integer) As Integer If (den Mod num = 1) Then Return 1 While (den Mod num <> 0) Dim temp As Integer = num num = den Mod num den = temp End While Return num End Function Private Sub ReduceWithGcd() Dim divisor As Integer = Gcd(_numerator, _denominator) _numerator = _numerator / divisor _denominator = _denominator / divisor End Sub
That’s it. You are finished. I will wrap up the discussion with some examples of overloaded operators that will permit arithmetic operations on Fraction objects.
Implementing Custom Operators
Overloaded operators are shared methods that accept the number and type of arguments based on the operator count. For example, – (subtraction) is a binary operator, so to support Fraction subtraction you need a shared method that takes two Fraction arguments. Operators also use the operator keyword. Here is an implementation of the subtraction operator for your Fraction class.
Public Shared Operator -(ByVal lhs As Fraction, _ ByVal rhs As Fraction) _ As Fraction Return New Fraction(rhs.Number - lhs.Number) End Operator
Operators are often perceived to be hard, but they are pretty intuitive. For the most part, the rule is that you need to implement symmetric operations. For example, if you implement subtraction, you should then implement addition. The other rule is don’t add side effects or change the understood behavior of an operator. For example, the addition operator should perform some kind of arithmetic operation.