How do I subtract times? I have an embarrassingly basic modular arithmetic question. I understand that I can subtract, for example, 1 hour from 10 o'clock to get 9 o'clock, or even 2 hours from 1 o'clock to get 11 o'clock; but how do I subtract 2 o'clock from 11 o'clock to get 3 hours. 
That is, I want a function that takes two times on the face of a clock, and gives me the interval between them.

For example: I have two angular values measured in "hours". One value is the right ascension of the apparent sun, $\alpha(t)$, while the other is the right ascension of the "mean sun", $\langle\alpha\rangle(t)$; both are mod 24. I'm interested in the angular difference between these two values 
$$E(t) = \langle\alpha\rangle(t) - \alpha(t)$$
Is this just inherently ambiguous, so that I need to impose some additional constraint based on information about the system (e.g., here that the values can be positive or negative, and are always small, so that, say $\pm 22$ should be interpreted as $\mp 2$), or is there some simple systematic way to ensure that I get the correct values?
 A: Just subtract. Your answer will only be well-defined mod 12 (because maybe 15 hours passed between 11 and 2 o clock). If you want a "canonical" answer, you can always take the least residue mod 12. In this case, we get $(2 - 11) = -9 = 3.$
In fancy language, the times on a clock are a torsor for $\frac{\mathbb{Z}}{12}$. John Baez has a great blog entry on this.
A: 2 o'clock - 11 o'clock = -9 = 3 mod 12 ?
A: $$
2-11=-9\equiv 3 \mod 12
$$
you can take the absolute value of that, or work with the $24$ hours format and use the fact that $2$ is also $14$ and then have
$$
14-11=3
$$
A: If you just have times with no dates attached, you can't tell +2 from -22.  You can choose to have the result be in the range $[-12,+12)$ or any other range of length $24$ like $(-5,19]$ that you like.  Based on what you day, I would opt for one centered at zero.
A: What you're doing is also a problem in directional statistics with measuring angles.
If the times are $t_1$ and $t_2$, then I think you want 
\begin{cases}
    |t_1 - t_2| & \text{If}\; |t_1 - t_2| \le 12\\
    24 - |t_1 - t_2| & \text{If}\; |t_1 - t_2|>12 \\
\end{cases}
