Significant Digits

OneSeventh=1/7
OneSeventhChopped=Sum[2^(-3j),{j,1,18}]
OneSeventhChoppedMore=Sum[2^(-3j),{j,1,17}]

Abs[OneSeventhChopped-OneSeventhChoppedMore]
2^(-18*3)

-N[Log2[Abs[(OneSeventhChopped-OneSeventhChoppedMore)/OneSeventhChopped]]]

AE[pstar_,p_]:=Abs[pstar-p]
RE[pstar_,p_]:=AE[pstar,p]/Abs[p]

AE[OneSeventhChopped,OneSeventh]
re=RE[OneSeventhChopped,OneSeventh]
N[-Log10[re]]