Do trailing zeroes after the radix point matter in binary?

Question

In decimal I can discard zeros after the radix point, e.g.: $$ 0.250_{10} = 0.25_{10} $$ It seems to me that I can do the same with binary: $$ 0.10_2 = 0.1_2 $$ Because

$$ 1\times\frac{1}{2}+0\times\frac{1}{4} = 1\times\frac{1}{2} $$ Am I right?

Answer 1

Yes it's exactly the same system. The only restriction on decimal relative to binary is no digits 2-9. As per your proof, removing a trailing zero is always equivalent to subtracting zero and therefore has no effect.

Answer 2

While Roberts Frost's answer is perfectly correct there is however one small catch!

When engaged in practical computations which seek to obtain an approximation $A$ to the solution $T$ of a complicated equation, then there is a profound difference between the statement $T \approx 1$ and the statement $T \approx 1.0$. In the first case, we implicitly state the error $E = T-A$ satisfies $|E| \leq 5\times10^{-1}$. In the second case, we implicitly make the stronger statement that $|E| \leq 5 \times 10^{-2}$.

By dropping the "extra" 0 we are selling ourselves short because we give the wrong impression of the quality of the approximation.