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I was going over a paper regarding linear algebra and its relation to chaos. One particular section focuses on Arnold's cat map. It briefly mention modular arithmetic.

In this case, it dealt with the generic form of $x \hspace{1mm} mod \hspace{1mm} 1$.

More specifically, $x \hspace{1mm} mod \hspace{1mm} 1$ denotes the unique number in the interval $[0,1)$ that differes from $x$ by an integer.

That said, the paper gave a few examples:

$2.3 \hspace{1mm} mod \hspace{1mm} 1 = 0.3$
$0.9\hspace{1mm}mod \hspace{1mm}1 = 0.9$
$-3.7\hspace{1mm}mod\hspace{1mm}1 = 0.3$
$2.0\hspace{1mm}mod\hspace{1mm}1 = 0$
$(2.3, -7.9)\hspace{1mm}mod\hspace{1mm}1 = (0.3, 0.1)$

Based on those examples, what is the fundamental reason behind $0.9\hspace{1mm}mod \hspace{1mm}1 \hspace{1mm} equaling \hspace{1mm} 0.9$ and $0.3\hspace{1mm}mod \hspace{1mm}1 \hspace{1mm} equaling \hspace{1mm} 0.3$? In more detail, why couldn't the latter statement be equal to $0.7$?

Obviously, that isn't the answer, but is it a concept similar to why we round 7.5 up to 8 (i.e. the 0.5)?

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I think what you call $x$ mod $1$ is also called the fractional part of $x$, denoted $\{x\}$. Not sure if the latter is the usual terminology though. It is basically the difference of $x$ and the greatest integer function evaluated at $x$. –  Rankeya Nov 24 '12 at 8:30
    
Also, I find the terminology modular arithmetic slightly strange. –  Rankeya Nov 24 '12 at 8:37

2 Answers 2

up vote 1 down vote accepted

$$x \equiv a \pmod{1}$$ is sometimes used a short hand notation to denote the fractional part of $x$, where $a$ denotes the fractional part of $x$ i.e. $$x = \lfloor x \rfloor + \{x\}$$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ and $\{x\} \in [0,1)$ is the fractional part of $x$.

Another interpretation of $x \pmod{1}$ is the distance of $x$ from the largest integer no larger than $x$ i.e. the largest integer that falls to the left of $x$ or $x$ itself on the real number line. I have tried to illustrate this with couple of diagrams below.

The first picture indicates $x\pmod{1}$ when $x > 0$. enter image description here

The second picture indicates $x\pmod{1}$ when $x <0$. enter image description here

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0.3 mod 1 can't be 0.7 because 0.3 and 0.7 don't differ by an integer. They differ by 0.4.

-0.3 mod 1 would be 0.7.

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Makes sense. Thanks! –  Mlagma Nov 24 '12 at 8:33

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