# Negative accuracy numerics ( 0-128 notation )

What does it mean to have a negative accuracy number?

I understand 0128 to mean "the number zero to 128 decimal points". Mathematica corroborates this:

In[1]:= 0128 < 1
(* Out[1]= True *)
In[2]:= 0128 < 0
(* Out[2]= False *)
In[3]:= Accuracy[0128]
(* Out[3]= 128. *)


But then I have no idea what "negative accuracy" is:

In[1]:= 0-128 < 1
(* Out[1]= False *)
In[2]:= 0-128 > 1
(* Out[2]= False *)
In[3]:= 0-128 == 1
(* Out[3]= True *)     (* Why????? *)
In[4]:= Accuracy[0-128]
(* Out[4]= -128. *)


Context: these values are showing up in a system of ~10000 equations, as the result of computing 100s of expressions of the form

$$\sum_i \frac{\textrm{numer}}{\textrm{denom}} \textrm{basis}_i,$$

where both numer and denom are randomly generated complex numbers, with 200 digit precision. I then collect coefficients of the (symbolic) $\textrm{basis}_i$'s to form algebraic equations.

The 0128 objects appear in only a handful of the total number of equations, so I don't yet know what causes the 0-128's to appear.

• From the docs: "...The accuracy of a number can be negative, indicating that all the correctly known digits are to the left of the decimal point..." – ciao May 6 '15 at 0:29
• Ah, thanks @ciao. I "ctrl+f" ' d through every page (except that one apparently). It's here, for posterity. – jjstankowicz May 6 '15 at 0:36

For the sake of an answer:

From the docs: "...The accuracy of a number can be negative, indicating that all the correctly known digits are to the left of the decimal point..." – ciao May 6 '15 at 0:29

In other words, the error in 0a is at most 10^-a. So 0-128 represents a numeric quantity at most 10^128 in magnitude.

Example:

N[10^148 (1 + 9*^-21), 20] - 10^148 // InputForm
(*  0-128.  *)


And the error is less than 10^128:

10^148 * 9 * 10^-21 // N
(*  9.*10^127  *)