Task: Imaginary base numbers
C#: using System;
using System.Linq;
using System.Text;

namespace ImaginaryBaseNumbers {
    class Complex {
        private double real, imag;

        public Complex(int r, int i) {
            real = r;
            imag = i;
        }

        public Complex(double r, double i) {
            real = r;
            imag = i;
        }

        public static Complex operator -(Complex self) =>
            new Complex(-self.real, -self.imag);

        public static Complex operator +(Complex rhs, Complex lhs) =>
            new Complex(rhs.real + lhs.real, rhs.imag + lhs.imag);

        public static Complex operator -(Complex rhs, Complex lhs) =>
            new Complex(rhs.real - lhs.real, rhs.imag - lhs.imag);

        public static Complex operator *(Complex rhs, Complex lhs) =>
            new Complex(
                rhs.real * lhs.real - rhs.imag * lhs.imag,
                rhs.real * lhs.imag + rhs.imag * lhs.real
                );

        public static Complex operator *(Complex rhs, double lhs) =>
             new Complex(rhs.real * lhs, rhs.imag * lhs);

        public static Complex operator /(Complex rhs, Complex lhs) =>
            rhs * lhs.Inv();

        public Complex Inv() {
            double denom = real * real + imag * imag;
            return new Complex(real / denom, -imag / denom);
        }

        public QuaterImaginary ToQuaterImaginary() {
            if (real == 0.0 && imag == 0.0) return new QuaterImaginary("0");
            int re = (int)real;
            int im = (int)imag;
            int fi = -1;
            StringBuilder sb = new StringBuilder();
            while (re != 0) {
                int rem = re % -4;
                re /= -4;
                if (rem < 0) {
                    rem = 4 + rem;
                    re++;
                }
                sb.Append(rem);
                sb.Append(0);
            }
            if (im != 0) {
                double f = (new Complex(0.0, imag) / new Complex(0.0, 2.0)).real;
                im = (int)Math.Ceiling(f);
                f = -4.0 * (f - im);
                int index = 1;
                while (im != 0) {
                    int rem = im % -4;
                    im /= -4;
                    if (rem < 0) {
                        rem = 4 + rem;
                        im++;
                    }
                    if (index < sb.Length) {
                        sb[index] = (char)(rem + 48);
                    } else {
                        sb.Append(0);
                        sb.Append(rem);
                    }
                    index += 2;
                }
                fi = (int)f;
            }
            string reverse = new string(sb.ToString().Reverse().ToArray());
            sb.Length = 0;
            sb.Append(reverse);
            if (fi != -1) sb.AppendFormat(".{0}", fi);
            string s = sb.ToString().TrimStart('0');
            if (s[0] == '.') s = "0" + s;
            return new QuaterImaginary(s);
        }

        public override string ToString() {
            double real2 = (real == -0.0) ? 0.0 : real;  
            double imag2 = (imag == -0.0) ? 0.0 : imag;  
            if (imag2 == 0.0) {
                return string.Format("{0}", real2);
            }
            if (real2 == 0.0) {
                return string.Format("{0}i", imag2);
            }
            if (imag2 > 0.0) {
                return string.Format("{0} + {1}i", real2, imag2);
            }
            return string.Format("{0} - {1}i", real2, -imag2);
        }
    }

    class QuaterImaginary {
        internal static Complex twoI = new Complex(0.0, 2.0);
        internal static Complex invTwoI = twoI.Inv();

        private string b2i;

        public QuaterImaginary(string b2i) {
            if (b2i == "" || !b2i.All(c => "0123.".IndexOf(c) > -1) || b2i.Count(c => c == '.') > 1) {
                throw new Exception("Invalid Base 2i number");
            }
            this.b2i = b2i;
        }

        public Complex ToComplex() {
            int pointPos = b2i.IndexOf(".");
            int posLen = (pointPos != -1) ? pointPos : b2i.Length;
            Complex sum = new Complex(0.0, 0.0);
            Complex prod = new Complex(1.0, 0.0);
            for (int j = 0; j < posLen; j++) {
                double k = (b2i[posLen - 1 - j] - '0');
                if (k > 0.0) {
                    sum += prod * k;
                }
                prod *= twoI;
            }
            if (pointPos != -1) {
                prod = invTwoI;
                for (int j = posLen + 1; j < b2i.Length; j++) {
                    double k = (b2i[j] - '0');
                    if (k > 0.0) {
                        sum += prod * k;
                    }
                    prod *= invTwoI;
                }
            }

            return sum;
        }

        public override string ToString() {
            return b2i;
        }
    }

    class Program {
        static void Main(string[] args) {
            for (int i = 1; i <= 16; i++) {
                Complex c1 = new Complex(i, 0);
                QuaterImaginary qi = c1.ToQuaterImaginary();
                Complex c2 = qi.ToComplex();
                Console.Write("{0,4} -> {1,8} -> {2,4}     ", c1, qi, c2);
                c1 = -c1;
                qi = c1.ToQuaterImaginary();
                c2 = qi.ToComplex();
                Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2);
            }
            Console.WriteLine();
            for (int i = 1; i <= 16; i++) {
                Complex c1 = new Complex(0, i);
                QuaterImaginary qi = c1.ToQuaterImaginary();
                Complex c2 = qi.ToComplex();
                Console.Write("{0,4} -> {1,8} -> {2,4}     ", c1, qi, c2);
                c1 = -c1;
                qi = c1.ToQuaterImaginary();
                c2 = qi.ToComplex();
                Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2);
            }
        }
    }
}

VB: Imports System.Text

Module Module1

    Class Complex : Implements IFormattable
        Private ReadOnly real As Double
        Private ReadOnly imag As Double

        Public Sub New(r As Double, i As Double)
            real = r
            imag = i
        End Sub

        Public Sub New(r As Integer, i As Integer)
            real = r
            imag = i
        End Sub

        Public Function Inv() As Complex
            Dim denom = real * real + imag * imag
            Return New Complex(real / denom, -imag / denom)
        End Function

        Public Shared Operator -(self As Complex) As Complex
            Return New Complex(-self.real, -self.imag)
        End Operator

        Public Shared Operator +(lhs As Complex, rhs As Complex) As Complex
            Return New Complex(lhs.real + rhs.real, lhs.imag + rhs.imag)
        End Operator

        Public Shared Operator -(lhs As Complex, rhs As Complex) As Complex
            Return New Complex(lhs.real - rhs.real, lhs.imag - rhs.imag)
        End Operator

        Public Shared Operator *(lhs As Complex, rhs As Complex) As Complex
            Return New Complex(lhs.real * rhs.real - lhs.imag * rhs.imag, lhs.real * rhs.imag + lhs.imag * rhs.real)
        End Operator

        Public Shared Operator /(lhs As Complex, rhs As Complex) As Complex
            Return lhs * rhs.Inv
        End Operator

        Public Shared Operator *(lhs As Complex, rhs As Double) As Complex
            Return New Complex(lhs.real * rhs, lhs.imag * rhs)
        End Operator

        Public Function ToQuaterImaginary() As QuaterImaginary
            If real = 0.0 AndAlso imag = 0.0 Then
                Return New QuaterImaginary("0")
            End If
            Dim re = CType(real, Integer)
            Dim im = CType(imag, Integer)
            Dim fi = -1
            Dim sb As New StringBuilder
            While re <> 0
                Dim rm = re Mod -4
                re \= -4
                If rm < 0 Then
                    rm += 4
                    re += 1
                End If
                sb.Append(rm)
                sb.Append(0)
            End While
            If im <> 0 Then
                Dim f = (New Complex(0.0, imag) / New Complex(0.0, 2.0)).real
                im = Math.Ceiling(f)
                f = -4.0 * (f - im)
                Dim index = 1
                While im <> 0
                    Dim rm = im Mod -4
                    im \= -4
                    If rm < 0 Then
                        rm += 4
                        im += 1
                    End If
                    If index < sb.Length Then
                        sb(index) = Chr(rm + 48)
                    Else
                        sb.Append(0)
                        sb.Append(rm)
                    End If
                    index += 2
                End While
                fi = f
            End If
            Dim reverse As New String(sb.ToString().Reverse().ToArray())
            sb.Length = 0
            sb.Append(reverse)
            If fi <> -1 Then
                sb.AppendFormat(".{0}", fi)
            End If
            Dim s = sb.ToString().TrimStart("0")
            If s(0) = "." Then
                s = "0" + s
            End If
            Return New QuaterImaginary(s)
        End Function

        Public Overloads Function ToString() As String
            Dim r2 = If(real = -0.0, 0.0, real) 
            Dim i2 = If(imag = -0.0, 0.0, imag) 
            If i2 = 0.0 Then
                Return String.Format("{0}", r2)
            End If
            If r2 = 0.0 Then
                Return String.Format("{0}i", i2)
            End If
            If i2 > 0.0 Then
                Return String.Format("{0} + {1}i", r2, i2)
            End If
            Return String.Format("{0} - {1}i", r2, -i2)
        End Function

        Public Overloads Function ToString(format As String, formatProvider As IFormatProvider) As String Implements IFormattable.ToString
            Return ToString()
        End Function
    End Class

    Class QuaterImaginary
        Private Shared ReadOnly twoI = New Complex(0.0, 2.0)
        Private Shared ReadOnly invTwoI = twoI.Inv()

        Private ReadOnly b2i As String

        Public Sub New(b2i As String)
            If b2i = "" OrElse Not b2i.All(Function(c) "0123.".IndexOf(c) > -1) OrElse b2i.Count(Function(c) c = ".") > 1 Then
                Throw New Exception("Invalid Base 2i number")
            End If
            Me.b2i = b2i
        End Sub

        Public Function ToComplex() As Complex
            Dim pointPos = b2i.IndexOf(".")
            Dim posLen = If(pointPos <> -1, pointPos, b2i.Length)
            Dim sum = New Complex(0.0, 0.0)
            Dim prod = New Complex(1.0, 0.0)
            For j = 0 To posLen - 1
                Dim k = Asc(b2i(posLen - 1 - j)) - Asc("0")
                If k > 0.0 Then
                    sum += prod * k
                End If
                prod *= twoI
            Next
            If pointPos <> -1 Then
                prod = invTwoI
                For j = posLen + 1 To b2i.Length - 1
                    Dim k = Asc(b2i(j)) - Asc("0")
                    If k > 0.0 Then
                        sum += prod * k
                    End If
                    prod *= invTwoI
                Next
            End If
            Return sum
        End Function

        Public Overrides Function ToString() As String
            Return b2i
        End Function
    End Class

    Sub Main()
        For i = 1 To 16
            Dim c1 As New Complex(i, 0)
            Dim qi = c1.ToQuaterImaginary()
            Dim c2 = qi.ToComplex()
            Console.Write("{0,4} -> {1,8} -> {2,4}     ", c1, qi, c2)
            c1 = -c1
            qi = c1.ToQuaterImaginary()
            c2 = qi.ToComplex()
            Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2)
        Next
        Console.WriteLine()
        For i = 1 To 16
            Dim c1 As New Complex(0, i)
            Dim qi = c1.ToQuaterImaginary()
            Dim c2 = qi.ToComplex()
            Console.Write("{0,4} -> {1,8} -> {2,4}     ", c1, qi, c2)
            c1 = -c1
            qi = c1.ToQuaterImaginary()
            c2 = qi.ToComplex()
            Console.WriteLine("{0,4} -> {1,8} -> {2,4}", c1, qi, c2)
        Next
    End Sub

End Module