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Converting Floating Decimals to Other Bases

Date: 11/27/2003 at 09:46:06
From: Ferran
Subject: Floating Base Conversions

Hi, how can I convert between bases but preserve the decimal places?

For example:

123.5 (decimal) = 7B.8 (Hexadecimal)
123.5 (decimal) = 1111011.1 (Binary)

I got these results from a calculator.  How are the operations done?

Date: 11/29/2003 at 22:09:06
From: Doctor Mike
Subject: Re: Floating Base Conversions

Hello Ferran,

Clearly from your example you already have some familiarity with
hexadecimal representation.  By your example it seems you may be 
interested in converting the fractional part of a number into hex.  To 
see how that is done, think about what the fractional part of a hex 
number means:  

The number 0.5C31 means 5/16 + 12/16^2 + 3/16^3 + 1/16^4.  

If you multiply that by 16, you get 5 + 12/16 + 3/16^2 + 1/16^3, which 
is the same as 5.C31 in hex.  Do you see what is happening? 
Remember that for decimals in base ten, multiplying by 10 moves the
"point" one place to the right.  In hexadecimal it is the same, in a
way.  If you multiply a hexadecimal representation number by "10 hex"
(which is decimal 16) you move the point one place to the right.  So,
.5C31 times "10 hex" becomes 5.C31.  The "5" which used to be the
first hex digit to the right of the point is now the whole number
part.  If you subtract off the 5 then the result is another fractional
part 0.C31, and you can do the same thing all over again.  To review,
the steps are:

    1. Multiply the base ten fractional part by 16.
    2. Use the whole number part of this as the next hex digit
    3. Subtract off this whole number part, leaving a new
        fractional part.
    4. If this fractional part is 0.0 you are done; if it's not, go
        back to step 1 and continue.

This repetitive process gives you the hex digits corresponding to the
base 10 digits in the original fractional decimal.  Here is an example: 

In base ten 1/9 = 0.1111111111111111111111...  Let's convert that to 
hexadecimal notation.  Here are the steps:

  * Multiply 0.11111111111111... by 16 to get 1.77777777777777...

  * The whole number part is "1" so that is the first hex digit.

  * Subtract 1 to get a new fractional part 0.77777777777777...

  * Multiply 0.77777777777777... by 16 to get 12.44444444444444...

  * The whole number part is "12" or "C" for the next hex digit.

  * Subtract 12 to get a new fractional part 0.44444444444444...

  * Multiply 0.44444444444444... by 16 to get 7.11111111111111...

  * The whole number part is "7" so that is the third hex digit.

  * Subtract 7 to get a new fractional part 0.11111111111111...

  * Multiply 0.11111111111111... by 16 to get 1.77777777777777...

and you see we are back where we started, so the hex decimal is going
to be an infinitely repeating one 0.1C71C71C7... just like the
original 0.111111111... was.  Neat, eh!

Notice that the whole numbers 1 and 9 are the same in hexadecimal as
they are in base ten, so we can write the following in hex: 
      1/9 = 0.1C71C71C7... 
Another interesting example is "pi minus 3", using the famous real
number pi.  Pi minus 3 is approximately 0.14159265358979324 and if you
follow this procedure 12 times, you get "pi - 3" in hexadecimal
accurate to 12 places, namely, 0.243F6A8885A3.  You will probably want
to use a calculator!!  Adding back in the 3, you get a really good
approximation for pi in hexadecimal:
      pi = 3.243F6A8885A3...
I enjoyed thinking about this.  I hope you did, too.  

- Doctor Mike, The Math Forum 

Date: 11/30/2003 at 04:22:11
From: Ferran
Subject: Thank you (Floating Base Conversions)

Thanks so much for the explanation.  Finally I can see the light!
Associated Topics:
High School Number Theory

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