Equivalence between binary and decimal precisions

We have binary number with fixed precision(e.g. after comma 7 binary digits). How to find the minimum number of decimal significant digits after period sufficient for obtaining equivalent precision? e.g. 11.1001001; equivalent decimal precision ? (3.???)

• If your precision is 7 digits, i.e $1111111_2$, what is this number in decimal? This number in decimal is of course $2^7 -1 = 127$ and this has three digits, so maybe this could be a clue ... – Matti P. Mar 21 '18 at 10:45
• I want to carry out Khachian's Ellipsoid Algorithm for Linear Strict Inequalities and Khachian proved suffices precision for binary representation, I want to generalize this result for decimal digits – Hovsep Sargis Papoyan Mar 21 '18 at 10:51
• Take the maximum number that is smaller than $1$(because after that number it is irrelevant) :$$0.1111111_2\\\underbrace{0.1111111_2}_{0,-1,-2,-3,-4,-5,-6,-7}\\0.1111111_2=0\cdot 2^0+1\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}+1\cdot 2^{-4}+1\cdot 2^{-5}+1\cdot 2^{-6}+1\cdot 2^{-7}=0.9921875$$ Hence to be sure you have $100\%$ accuracy you need at least $7$ digits. – Holo Mar 21 '18 at 10:59
• What do you mean with equivalent? If you want to represent your binary numbers exactly in base 10, you need need 7 decimal digits to represent $2^{-7} = 0.0078125.$ – gammatester Mar 21 '18 at 10:59
• I want to implement the Khachian's Ellipsoid method and the problem came from that side. Are you aware about this method and what you can kindly suggest ? – Hovsep Sargis Papoyan Mar 21 '18 at 11:22