Adding random numbers I haven't taken statistics or calculus, but I'm curious: if you take $100$ perfectly random numbers from $-1$ to $1$ and add them, how close would it be to $0$? How about adding $10000$ perfectly random numbers from $-1$ to $1$? How about  infinity ($\infty$)?
Perfectly random should be defined. 

To answer your question we can say that if you repeat selecting $100$ numbers randomly from $-1$ to $1$, many times, and record the sums, then these sums are approximately normally distributed with the mean of $0$
If by perfectly random you mean that each number is independent of the other number and have identical probabilities of being some value, there is a famous law called "The Law of Large Numbers" that states if you have infinite samples, the empirical average of all of those numbers will equal the true average.

To be more specific, Let $X_i$ be the $i^{th}$ sample that you pick between $[-1,1]$. The empirical average is $\frac{1}{n}\underset{i=1}{\overset{n}{\sum}} X_i$ which goes to $0$ exponentially as $n$ grows large. The exact rate is given by Hoeffding's Inequality, which when applied to this problem states that

$$|\underset{i=1}{\overset{n}{\sum}}X_i| \geq n \sqrt{2 \ln \frac{2}{\delta}}$$  with probability $1 - \delta$. So if you just plug in the amount of error you allow into $\delta$ you can get the minimal number of samples you need.