<s>[INST] 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?
 [/INST] The first approximation is fine. The second on is not, because, $10^5$ and $\dfrac{10^{10}}{1+10^5}$ are large numbers with approximately the same size. You are saying that since $10\,001$ is close to $10\,000$, then $1$ is close to $0$.
</s>