Changing number bases for expansion of fractions

Question

Okay, so my question is, say if I have $141_{10}$ and I convert it into hexadecimal, I get $8D_{16}$ but if it I have $0.141_{10}$ and I convert it to hexadecimal, I get $0.24189374BC6A7EF9E$, why is there this discrepancy?

Answer 1

Suppose that the hexadecimal fraction is $f=.a_1a_2a_3..._{16}$ Then $$16f = a_1.a_2a_3.._{16}.$$ So, if we multiply the fraction by $16,$ the integer part of the result is the first hexadecimal digit. $$16\cdot.141=2.256,$$ so the first hex digit after the "hexadecimal point" is $2.$ To get the second digit, ignore the integer part of the last result, and multiply by $16$ again.$$16\cdot.256=4.096,$$ so the second digit is $4$. Then $$16\cdot.096=1.536$$ and so on. $$\boxed{.141_{10}=.241..._{16}}$$