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Our professor has started a little scavenger hunt via mail and asked us to detect the concept of $\bmod$ in calculus so he can add some more definitions to his calculus lecture scriptum. Right off the bat, I could more or less only think of the periodicity in $\sin$ and $\cos$ that are always taken $\bmod 2\pi$ and $\bmod \pi$ respectively. Could you possibly think of any other concepts in calculus that make use of this modular arithmetic? We are talking very basic college calculus, so we're not looking at very advanced stuff.

John Hamford
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    Time itselft, calendars... Also used as a tool to compute certain operations. More info here: https://www.irishtimes.com/news/science/modular-arithmetic-you-may-not-know-it-but-you-use-it-every-day-1.3268649 :) – nachosemu Mar 12 '21 at 13:18
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    Technically your example of $\sin$ and $\cos$ has nothing to do with modular arithmetic. Indeed, $\sin(x)\in [-1,1]$ and thus looking at $\sin(x)$ up to multiples of $2\pi$ is meaningless. What is true is that $\sin(x+2n\pi)=\sin(x)$ for all $x$. The latter just means that $\sin(x)$ is a $2\pi$-periodic function, that's very different from modular arithmetic. – Mathematician 42 Mar 12 '21 at 13:19
  • hmm I see. Could you personally think of any other concepts in calculus then that make use of modular arithmetic? I mean, this is something an professor asked us. Or is it possibly a prank to see who wits that there is no such concept applicable in calculus (?) @nacho oh yeah that I am aware of, but the question was aimed at the concepts of calculus, not the real world. – John Hamford Mar 12 '21 at 13:21
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    If you had to calculate the $n$th derivative of the function $sin$, you could classify your answers mod $4$. – MasB Mar 12 '21 at 13:25
  • Calculus deals with real numbers (advanced versions also with complex numbers) , modulo-calculations with integers. Those are very different concepts. – Peter Mar 12 '21 at 13:25
  • I know, but I'm looking for examples such as the one mentioned by Bernard. @Bernard: what actually do you mean by that? Classifying the results or the functions themselves? – John Hamford Mar 12 '21 at 13:37
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    @Mathematician42 Not true - it has much to do with modular arithmetic - see the circle group. – Bill Dubuque Mar 12 '21 at 14:08
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    @Johh Hamford, The different derivatives will be $\cos,-\sin,-\cos,\sin,\cos,-\sin,-\cos...$ The $n$th derivative will depend on the value of $n \mod4$. – MasB Mar 12 '21 at 14:16
  • Oh thanks, I got it now! @Bill then, would you say it is justified I can suggest that concept of the circle group in the email? After all, the concept of a group is still an algebraic one. – John Hamford Mar 12 '21 at 14:19
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    I think perhaps the professor is encouraging free thinking, while many of the answers here are discouraging it. – johnnyb Mar 12 '21 at 15:01
  • In perhaps the opposite direction we can find basic calculus in modular arithmetic through Hensel's lemma. Solving $f(x)\equiv 0 \mod p$ and $f'(x) \not \equiv 0 \mod p$ allows us to use Newton's method $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ to converge to solutions to $f(x) \equiv 0 \mod p^k$. – Merosity Mar 13 '21 at 13:38

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