I'm not sure what that has to do with it. Even if you are ok with the idea that integers and "decimal numbers" are different, it's still confusing that 0.1 + 0.2 != 0.3.

It's confusing because it is very difficult to look at a decimal number and know whether it can be represented exactly as base-2 floating point. It's especially confusing because you get no feedback about it! Here is a Ruby session:

    irb(main):001:0> 0.1
    => 0.1
    irb(main):002:0> 0.2
    => 0.2
    irb(main):003:0> 0.3
    => 0.3
    irb(main):004:0> 0.4
    => 0.4
    irb(main):005:0> 0.5
    => 0.5

They all look exact, right? Well actually only 0.5 is (IIRC?). All the others are actually approximations internally.

The problem here is that Ruby is lying to you – if you enter 0.3 as a literal, you will not get that value.

I would challenge you to name any software system that reliably shows you the precise value of a floating point number.

I couldn't find any (like not a single one), so I wrote a program myself to do it: http://blog.reverberate.org/2012/11/dumpfp-tool-to-inspect-f...

Python, JavaScript V8, Julia, Java.

The precise value of double(0.1) is 0.1000000000000000055511151231257827021181583404541015625. That is precise, not an approximation.

If you know of a program in any of these languages that will print this value for "0.1" using built-in functionality, please let me know because I would love to know about it.

Likewise the precise value of double(1e50) is 100000000000000007629769841091887003294964970946560. Anything else is an approximation of its true value.

In another message you said that what's really important is that the string representation uniquely identifies the precise value. While that will help you reconstruct the value later, it does not help you understand why 0.1 + 0.2 != 0.3.

Python does it:

  >>> from decimal import Decimal
  >>> Decimal(0.1)
  Decimal('0.1000000000000000055511151231257827021181583404541015625')
  >>> Decimal(1e50)
  Decimal('100000000000000007629769841091887003294964970946560')

Great to know, thanks!

Haskell will tell you that

    toRational 0.1 = 3602879701896397 % 36028797018963968
    toRational 1e50 = 100000000000000007629769841091887003294964970946560 % 1

Both of those values are precise, although I admit the first isn't really useful for human beings.

The easiest way to see this exact value in Julia is to convert the Float64 value 0.1 to BigFloat and then print that:

    julia> big(0.1)
    1.000000000000000055511151231257827021181583404541015625e-01

    julia> big(1e50)
    1.0000000000000000762976984109188700329496497094656e+50

Note that this is exactly why you don't normally want to construct BigFloats this way. Instead you want to do this:

    julia> BigFloat("0.1")
    1.000000000000000000000000000000000000000000000000000000000000000000000000000002e-01

    julia> BigFloat("1e50")
    1e+50

It helps because 0.1 + 0.2 produces 0.30000000000000004 for 64-bit floats – so at least you can see that this value isn't the same as 0.3. In Ruby you just get two values that print the same yet aren't equal, which is way more confusing. I agree that printing the minimal number of digits required for reconstruction does not help with explaining why 0.1, 0.2 and 0.3 in 64-bit floats aren't the real values 1/10, 2/10 and 3/10.

Python doesn't:

    rasky at monocle in ~
    ↪ python
    Python 2.7.5 (default, Sep  2 2013, 05:24:04)
    [GCC 4.2.1 Compatible Apple LLVM 5.0 (clang-500.0.68)] on darwin
    Type "help", "copyright", "credits" or "license" for more information.
    >>> 0.1
    0.1
    >>> ^D
    rasky at monocle in ~
    ↪ python3
    Python 3.3.3 (default, Dec 24 2013, 13:54:32)
    [GCC 4.2.1 Compatible Apple LLVM 5.0 (clang-500.2.79)] on darwin
    Type "help", "copyright", "credits" or "license" for more information.
    >>> 0.1
    0.1

It used to, but it was changed to reduce users' confusion.

We may be speaking across each other here. Ruby is lying in the sense that there are multiple distinct float values that it will print as 0.3 – in particular, confusion ensues when two values look the same but are unequal. These other languages print each distinct float value differently, using just enough decimal digits to reconstruct the exact binary value. Ruby doesn't give you enough digits to reconstruct the value you have. Nobody actually prints the full correct value because it's fifty digits long and is completely redundant given that you know you're dealing with a 64-bit float.

You're right.

    Python 3.3.5rc1 (default, Feb 23 2014, 17:44:29)  
    >>> 0.3 == 0.2 + 0.1
    False
    >>> 0.2 + 0.1
    0.30000000000000004
    >>> 0.3
    0.3

Yeah, that's my point. 0.1 and 0.2 do add exactly to 0.3, but in any finite representation of real numbers you'll get rounding errors on some equations like this. If you use an infinite representation then equality is no longer computable.