Q:

Mathematica numerical "error" for simple multiplication

In[71]:= 0.6*0.8048780487804877`

Out[71]= 0.482927

In[72]:= 0.3*0.8414634146341463`

Out[72]= 0.252439

In[74]:= (0.6*0.8048780487804877`) + 0.3*0.8414634146341463`

Out[74]= 0.735366

In[75]:= 0.6*0.8048780487804877`+0.3*0.8414634146341463`

Out[75]= 0.406365

Why the brackets In[74] and In[75] have effect? I think it should have no difference.
I use mathematica 12.1.
Thank you for your help.

A:

0.8048780487804877`+0.3 is an arbitrary-precision number with precision 0.3.
With the parentheses, the 0.3 does not specify the precision, but stands as a number.  The second line is equivalent to
0.6 * (0.8048780487804877`+0.3) * 0.8414634146341463`

A:

Because the In[75] is completely different than you think.
0.6*0.8048780487804877`+0.3*0.8414634146341463`
(* 0.406365 *)

0.6*0.8048780487804877` + 0.3*0.8414634146341463`
(* 0.735366 *)

Notice the space in the second example: in your case +0.3 is a specification for precision for tick `
Easier example:
(* in this case it's specification of precision *)
1.2`+30
(* 1.20000000000000000000000000000 *)

(* in this case it's 1.2 with $MachinePrecision, plus 30 *)
1.2` + 30
(* 31.2 *)

A:

It's not hte parentheses; it's the missing whitespace!
There is a tiny difference between the meanings of
 0.8048780487804877`+0.3

0.*10^-1

and
 0.8048780487804877` +0.3

1.10488

We have
 0.8048780487804877`+0.3 == 0.8048780487804877`0.3

and the number behind the backtick denotes the number of significant digits. In particular, no addition is performed during evaluation of in case of 0.8048780487804877`+0.3

