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Entering a value such as 27.8675309 into the "Decimal representation" field of the IEEE 754 Converter changes the value I entered to 27.86753 . Likewise, Java drops the last two digits when a parse a string with the same value.

Float.parseFloat("27.8675309") // Results in a float value of 27.86753
I am not sure what the "Decimal representation" of the IEEE converter actually is (is it a float?) but I would expect it to give me the biggest number possible that:
Is a float value
Does not exceed the original value I entered
I would expect Java to behave in a similar fashion, that is, I would expect the line of code above to return a float value equal to 27.8675308 or an even larger float value that is closer to my original input instead of just dropping decimal places.  What am I missing here?
This is as expected.
If you look at the binary representation of 27.8675309 (27.867530822753906 as a double):
01000001110111101111000010110100
the next highest value is:
01000001110111101111000010110101
which yields 27.867533 (27.86753273010254 as a double), and the next lowest value is:
01000001110111101111000010110011
which yields 27.867529 (27.867528915405273 as a double)
There simply aren't enough bits in the mantissa of a Float to represent any value in between, so your value is rounded down in decimal to 27.867530
                Also, just so I am clear, this isn't a matter of running out of decimal places, it is a matter of no more bits for the exponent and/or mantissa of the float, correct?  Some other float value could have more than 5 decimal places right?
– ubiquibacon
                Jan 31, 2012 at 17:33
                @typoknig it's not more bits for the mantissa - this number is small enough that the exponent doesn't matter.  And yes, smaller numbers could have more than five decimal places, but no more than 7 or 8 significant digits.
– Alnitak
                Jan 31, 2012 at 17:36
                @Alnitak 27.8675309 rounded to 24 bits = 11011.11011110000101101 = 27.86753082275390625. Rounded to 8 decimal digits, that's 27.867531, not 27.867530. To 7 digits, it's 27.86753, but then so are both the prev float (27.8675289154052734375) and the next float (27.8675327301025390625). Why is h-schmidt.net/FloatApplet/IEEE754.html rounding 27.8675309 to 7 digits, but prev and next to 8?
– Rick Regan
                Feb 21, 2012 at 15:17
                @RickRegan in IEE754 the 27 won't be part of the mantissa - you'll actually have 2^4 * 1.741720...
– Alnitak
                Feb 21, 2012 at 15:53
        
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