Computer Science 101

with:  Erik Oosterwal

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Decimal Fractions to Binary Fractions


A visitor to this site send me an email asking if it were possible to use the Numeric Base Conversion algorithm to convert fractions between bases.  I thought it would be pretty straight forward to implement the change to the general algorithm, but found it was easier just to make a single purpose algorithm to convert from decimal to binary, since that was specifically what he asked for.  As time allows, I'll update the algorithm to make it general purpose to convert fractions between any two bases.

This algorithm borrows from the general Numeric Base Conversion algorithm, but it has been simplified to only convert from base 10 to base 2, and an additional portion that converts the fractional part of the input number to a binary fraction has been added.

Binary fractions are a bit tricky to work with.  Some values convert easily, such as 0.5, 0.375, and 0.75, which are 0.1, 0.011, and 0.11, respectively, in binary form.  Other fractional values, which are easily displayed in base 10, become long strings of seemingly random bits.  For instance, 1/10 = 0.1 (base 10) = 0.000110011001100110... (base 2), and 1/3 = 0.33333... (base 10) = 0.010011001100110011... (base 2), and 1/100 = 0.01 (base 10) = 0.000000101000111101... (base 2).

If you are already familiar with binary notation, you recognize that each binary digit represents a power of 2.  Starting at the least significant bit and moving towards the most significant bit, the bits represent 1, 2, 4, 8, 16, 32, 64, 128, etc.  Just as decimal fraction digits represent the inverses of the powers of 10: 1/10, 1/100, 1/1000, etc, binary fractional bits represent the inverses of the powers of 2:  1/2, 1/4, 1/8, 1/16, etc., which are 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.0078125, etc.  As you can see, we have to take a binary fraction out to 7 places before we get less than 0.01 (base 10).

The Visual Basic code below will only output 20 bits, including the decimal point.  You can modify this yourself by changing the value assigned to intNumBits near the top of the listing.  The embedded comments in the code should be fairly explanatory, but if you have questions or comments, feel free to send me an email at the address listed on the Home Page of this site.



Private Sub Dec2Bin_Click()

'   Subroutine to convert floating point decimal values to floating point binary values.
'
'   Written by: Erik Oosterwal
'   Started on: October 26, 2005
'   Completed:  October 27, 2005

    Dim txtNumericBaseData, txtOutputValue As String
    Dim intX, intNumBits As Integer
    Dim dblDecimalValue, dblFractionValue, dblBinaryFraction As Double
        
        
    intNumBits = 20             ' The number of bits in the output (including the decimal point)
    txtNumericBaseData = "01"   ' Output digits can be either "0" or "1"
    dblBinaryFraction = 0.5     ' Binary fraction bits are powers of 0.5
    txtOutput.Text = ""         ' Clear the display
    txtOutputValue = ""         ' Clear the working output variable
    
    
    dblDecimalValue = Int(CDbl(txtInput.Text))                  ' Get the integer portion of the input
    dblFractionValue = CDbl(txtInput.Text) - dblDecimalValue    ' Get the fractional portion of the input
        
        
'   Figure out the integer portion of the input.
    While dblDecimalValue > 0
        intX = Int(((dblDecimalValue / 2) - Int(dblDecimalValue / 2)) * 2 + 1.5)
        txtOutputValue = Mid(txtNumericBaseData, intX, 1) + txtOutputValue
        dblDecimalValue = Int(dblDecimalValue / 2)
    Wend
    
    
    If txtOutputValue = "" Then txtOutputValue = "0"    ' If there was no whole number, set it to "0"
    txtOutputValue = txtOutputValue + "."               '   then add a decimal point
    
    
'   Figure out the fractional portion of the input.
    While Len(txtOutputValue) < intNumBits And dblFractionValue > 0     ' As long as we're not exceeding the
                                                                        ' allowed number of bits in the output
                                                                        ' and there's still part of the input
                                                                        ' value to be converted...
        If dblFractionValue >= dblBinaryFraction Then                   ' If the input number is larger than the fraction bit,
            txtOutputValue = txtOutputValue + "1"                       '   add a "1" to the output and
            dblFractionValue = dblFractionValue - dblBinaryFraction     '   reduce the input number by the fraction bit's value.
        Else                                                            ' Otherwise...
            txtOutputValue = txtOutputValue + "0"                       '   just tag on a "0" to the end of the output.
        End If
        dblBinaryFraction = dblBinaryFraction / 2#                      ' The fraction bit must be cut in half.
    Wend
    
    txtOutput.Text = txtOutputValue                     ' Send the output value to the display.

End Sub
Discuss this algorithm at the Computer Algorithms blog page.



For a more generalized routine that can convert integer values between any two numeric bases, see the Numeric Base Conversion article from the main Computer Science 101 page.

You can also download a zip file that contains the listing given above as well as an executable file that demonstrates this routine.




Discuss computer algorithms and other computer science topics at the Computer Algorithms blog page.

All code and original algorithms are © Erik Oosterwal - 1987-2008
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