An Inconsistency in Numerical Approximation Consider the expression 
$$
10^5 - \frac{10^{10}}{1+10^5}.
$$
Using the elementary properties of fractions we can evaluate the expression as 
$$
10^5 - \frac{10^{10}}{1+10^5} = \frac{10^5 + 10^{10} - 10^{10}}{1+10^5} = \frac{10^5}{1+10^5}\approx 1.
$$
Note that the approximation $10^5+1 \approx 10^5$ is used in the last step. Now suppose we use the same approximation, but apply it before we perform the subtraction. We get
$$
10^5 - \frac{10^{10}}{1+10^5} \approx 10^5 - \frac{10^{10}}{10^5} = 0.
$$
The same logic works for 
$$
10^p - \frac{10^{2p}}{1+10^p}
$$
for arbitrary large $p$, so it cannot be simply an issue with the accuracy of the approximation. 
Is there an easy explanation of what's going on here?

There is no paradox.
When you approximate $$\frac{10^5}{1+10^5}=1-0.000099999000\cdots$$ with $1$, the error is on the order of $10^{-5}$.
But in the second case, the same error is multiplied by $10^5$, so that it is no more negligible.
