How to best deal with pesky$\pmod 1$ calculations? I have just began studying Dynamical Systems, where the unit circle is introduced as  $[0, 1]/\sim$, and addition is done $\pmod 1$. As a result, I have often confronted with proving identities such as
$$(t+x) \pmod 1 \cdot k \pmod 1 = (tx+tk) \pmod 1, \hspace 1cm k \in \mathbb{Z}$$
I don't have the intuition to say whether such expressions are true or not, and going into cases is painful.
Rather than go into cases, I think it would be a good idea to prove a few simple propositions that I can memorize, and would allow me to manipulate practically any expression involving $\pmod 1$. Two candidates for this are
$$[x \pmod1 + y \pmod 1] \pmod 1 = [x+y] \pmod 1$$
$$[x \pmod 1 \cdot k] \pmod 1 = kx \pmod 1 \hspace 1cm k \in \mathbb{Z}$$
(The second rule is inspired by the above example.) My question can be broken down into several pieces:

*

*Are there any other proposition that are very useful in $\pmod 1$ arithmetic?


*Are my own propositions too complex (can they be derived from simpler ones?)


*Is there a better way to deal with these pesky calculations that I haven't thought of?
 A: TL;DR: if the identity only involves addition, subtraction, and zero, it's probably valid. Since multiplication by an integer is defined in terms of addition, subtraction, and zero, it is also fine. However, identities involving multiplication by an arbitrary real number are definitely not fine.
Your book introduces the circle as $[0, 1] / \sim$. Let's go a bit further and define the circle slightly differently. If you're familiar with the "quotient group" construction, this should be very straightforward.
Define a new relation $\simeq$ on $\mathbb{R}^2$ as follows:
$x \simeq y$ iff $x - y$ is an integer
Clearly, this relation is an equivalence relation. Now, define $S = \mathbb{R} / \mathbb{Z}$ to be the quotient $\mathbb{R} / \simeq$, with projection $\pi : \mathbb{R} \to \mathbb{R} / \mathbb{Z}$.
We can define an operation $+$ on $S$ by $\pi(x) + \pi(y) = \pi(x + y)$, where the $+$ on the left is addition in $S$ and the $+$ on the right is addition in $\mathbb{R}$. This operation respects the equivalence relation $\simeq$ and is therefore well-defined.
Similarly, we can define an operation $-$ on $S$ by $-\pi(x) = \pi(-x)$. Once again, this operation is well-defined.
Finally, we have $0 \in S$, where $0 = \pi(0)$.
$S$ is nothing more than the circle.
Because of the way that $+$, $-$, and $0$ are defined on $S$, they inherit all their properties from $\mathbb{R}$. For example, we have for any $a, b, c \in S$,
$a + 0 = 0 + a = a$
$a + -a = -a + a = 0$
$a + (b + c) = (a + b) + c$
$a + b = b + a$
Because we can write $a, b, c$ as $\pi(x)$, $\pi(y)$, and $\pi(z)$ and directly verify each identity.
However, $S$ does not have a general notion of the multiplication of 2 elements nor a notion of scalar multiplication by a real number. It can't inherit these notions from $\mathbb{R}$ because these notions don't respect the equivalence $\simeq$.
$S$ does, however, have a notion of multiplication by $\mathbb{Z}$. Multiplication by an integer is just repeated addition (or subtraction). Another way to see it is by defining $k \pi(x) = \pi(kx)$ and noting that this respects the equivalence relation $\simeq$ whenever $k$ is an integer.
