## Saturday, September 10, 2011

### Handling Binary Fractions Conversions

Understanding integer binaries is simple. But how to handle the fractions and their binaries. Here is how it works -

Converting Decimal Fractions to Binary Fractions

There is a simple, step-by-step method for computing the binary expansion on the right-hand side of the point. We will illustrate the method by converting the decimal value .625 to a binary representation..

1. Step 1: Begin with the decimal fraction and multiply by 2. The whole number part of the result is the first binary digit to the right of the point.
1. Because .625 x 2 = 1.25, the first binary digit to the right of the point is a 1.
2. So far, we have .625 = .1??? . . . (base 2) .
2. Step 2: Next we disregard the whole number part of the previous result (the 1 in this case) and multiply by 2 once again. The whole number part of this new result is the second binary digit to the right of the point. We will continue this process until we get a zero as our decimal part or until we recognize an infinite repeating pattern.
1. Because .25 x 2 = 0.50, the second binary digit to the right of the point is a 0.
2. So far, we have .625 = .10?? . . . (base 2) .
3. Step 3: Disregarding the whole number part of the previous result (this result was .50 so there actually is no whole number part to disregard in this case), we multiply by 2 once again. The whole number part of the result is now the next binary digit to the right of the point.
1. Because .50 x 2 = 1.00, the third binary digit to the right of the point is a 1.
2. So now we have .625 = .101?? . . . (base 2) .
4. Step 4: In fact, we do not need a Step 4. We are finished in Step 3, because we had 0 as the fractional part of our result there.
1. Hence the representation of .625 = .101 (base 2) .

Converting Binary Fractions to Decimal Fractions

X-n means 1/Xn. So, 2-2 = 1/4 and 2-3 = 1/8.

To convert an expression in base two notation to base ten notation, just do the arithmetic. Here is 100.101 converted from binary representation to decimal representation:
You should double-check our result by expanding the binary representation.

 1 0 0 . 1 0 1 1×22 0×21 0×20 . 1×2-1 0×2-2 1×2-3 1×4 + 0×2 + 0×1+ . 1×0.5 + 0×0.25 + 1×0.125 4 + 0 + 0 + . 0.5 + 0 + 0.125 4 . 625

References

http://programmedlessons.org/AssemblyTutorial/Chapter-29/ass29_5.html
http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm