Strictly, it's not because it's floating point, it's because it's binary floating point.

When decimal 1.51 is converted to a binary fraction the result is:

  1.1000_0010_1000_1111_0101_1100_0010_1000_1111_0101_1100_0010_1000_1111_0101_1100_0010_1000_1111_0101...

As you say, most decimal fractions are like this... so before any errors can be introduced by rounding and what not, most decimal fractions have a built-in "representation" error.

Most of the time you won't see the "representation" error, because conversion back to decimal rounds it off. This can lead to bafflement when two values look the same when printed out, but fail $x == $y. Addition and subtraction are the more difficult floating point operations, so you're more likely to see the problems there. Consider:

  print 1.09 - 1, "\n" ;
  print "0.84 - 0.34 == ", 0.84 - 0.34, 
       ( 0.84 - 0.34 == 0.5 ? " ==" : " but !=" ), " 0.5\n" ;
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  0.0900000000000001
  0.84 - 0.34 == 0.5 but != 0.5

This issue is caused by binary floating point. But, it is an artifact of any floating point notation. For example,

1/3 + 1/3 + 1/3 != 1
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for any finite number of digits using decimal representations.

Given any base, there will be some fractions that cannot be properly represented and would therefore cause this problem. (At least until such time as we have infinite precision floating point representations.<grin/>)

G. Wade

This problem can be solved with Continued Fractions. Unfortunately nobody seems to use continued fraction libraries.

I know that MJD wrote one in Perl, but he didn't polish it up and release it on CPAN. I don't think anyone else has released one either.

At least until such time as we have infinite precision floating point representations.

I believe there have been efforts to represent p-adic numbers. perlnumber makes a brief mention of "exotic" number operations with p-adic arithmetic.