Why does Round[2.75,0.1] return 2.800000000003? - wolfram-mathematica

Question

Mathematica 8.0.1
Any one could explain what would be the logic behind this result
In[24]:= Round[10.75, .1]
Out[24]= 10.8
In[29]:= Round[2.75, .1]
Out[29]= 2.8000000000000003
I have expected the second result above to be 2.8?
EDIT 1:
I was trying to do the above for formatting purposes only to make the number fit in the space. I ended up doing the following to get the result I want:
In[41]:= NumberForm[2.75,2]
Out[41] 2.8
I wish Mathematica has printf() like formatting function. I find formatting numbers in Mathematica for exact field width and form a little awkward compared to using printf() formatting rules.
EDIT 2:
I tried $MaxExtraPrecision=1000 on some number I was trying for format/round, but it did not work, that is why I posted this question. Here it is
In[42]:= $MaxExtraPrecision=1000;
Round[2035.7520395261859,.1]
Out[43]= 2035.8000000000002
In[46]:= $MaxExtraPrecision=50;
Round[2.75,.1]
Out[47]= 2.8000000000000003
EDIT 3:
I found this way, to format a number to one decimal point only. Use Numberform, but first need to find what n-digit precision to use by counting the number of digits to the left of the decimal point, then adding 1.
In[56]:= x=2035.7520395261859;
NumberForm[x,IntegerLength[Round#x]+1]
Out[57]//NumberForm= 2035.8
EDIT 4:
The above (Edit 3) did not work for numbers such as
a=2.67301785 10^7
After some trials, I found Accounting Form to do what I want. AccountingForm gets rid of the 10^n form which NumberForm did not:
In[76]:= x=2035.7520395261859;
AccountingForm[x,IntegerLength[Round#x]+1]
Out[77]//AccountingForm= 2035.8
In[78]:= x=2.67301785 10^7;
AccountingForm[x,IntegerLength[Round#x]+1]
Out[79]//AccountingForm= 26730178.5
For formatting numerical values, the best language I found was Fortran, followed COBOL and also by those languages that use or support printf() standard formatting. With Mathematica, one can do such formatting I am sure, but it sure seems too complicated to me. I never understood why Mathematics does not have Printf[].

Answer 1

Not all decimal (base 10) numbers with a finite number of digits are representable in binary (base 2) with a finite number of digits. E.g. 0.1 is not representable in binary, just like 1/3 ~= 0.33333... is not representable in decimal. Mathematica (and other software) will only use a limited number of decimal digits when showing the number to hide this effect. However, occasionally it might happen that enough decimal digits are shown that the mismatch becomes visible.
http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion_and_rounding
EDIT
This command will show you what happens when you find the closes binary representation of 0.1 using 20 binary digits, then convert it back to decimal:
RealDigits[FromDigits[RealDigits[1/10, 2, 20], 2], 10]

Answer 2

The number is stored in base 2, rather than base 10 (decimal). It's impossible to represent 2.8 in base 2, so it uses the closest value: 2.8000000000000003

Answer 3

Number/AccountingForm can take a list in the second argument, the second item of which is how many digits after the decimal place to show:
In[61]:= x=2035.7520395261859;
In[62]:= AccountingForm[x,{Infinity,3}]
Out[62]//AccountingForm= 2035.752
Perhaps this is useful.