Decimal parts of multiples

For a real number $x$ , let $f(x)$ be a function such that $f(x) = x - \lfloor x \rfloor$ ; and let $A(x)$ be a set such that $A(x) = \{f(xn) : n \in \mathbb{N} \}$ .

Prove the following:

(1) $x$ is rational if and only if $A(x)$ is finite;

(2) $x$ is irrational if and only if $A(x)$ is dense in the interval $[0,1)$ .

Have fun :)

#Algebra #Sets #IrrationalNumbers #Decimals #FloorFunction

Easy Math Editor

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Comments

Suppose $A(x)$ is finite. Then by the Pigeonhole Principle, there are two distinct integers $m$ and $n$ such that $f(xm) = f(xn)$ . So $xm-xn = k$ is an integer. Then $x = \frac{k}{m-n}$ is rational.

Suppose $A(x)$ is dense in $[0,1)$ . Then it is certainly infinite. So $x$ can't be rational, by (1).

What remains is to show that if $x$ is irrational, then $A(x)$ is dense in $[0,1)$ .

Patrick Corn - 6 years ago

You can do this by the Pigeonhole Principle as well:

http://math.stackexchange.com/questions/272545/multiples-of-an-irrational-number-forming-a-dense-subset

Patrick Corn - 6 years ago

We see that $f(x) = \{x\}$ , i.e. it is the fractional part of $x$ .

(1) $\longrightarrow$ In this direction we assume that $x$ is rational and try to prove $A(x)$ is finite. Since $x$ is rational, we can write it as $x = \frac{p}{q}$ where $p$ and $q$ are coprime integers and $q$ is not zero.

Claim: $A(x)$ will have exactly $q$ elements.

The elements in $A(x)$ will be $\big \{\frac{p}{q} \big \}, \big \{\frac{2p}{q} \big \}, \big \{\frac{3p}{q} \big \}, \big \{\frac{4p}{q} \big \}, \ldots$ .

Since $p$ and $q$ are integers, these terms can be written as $\frac{p ~\bmod~ q}{q}, \frac{2p ~\bmod ~ q}{q}, \frac{3p ~\bmod ~ q}{q}, \frac{4p ~\bmod~ q}{q}, \ldots$ . There can be only $q$ such terms, viz $\frac{0}{q}, \frac{1}{q}, \frac{2}{q}, \ldots \frac{q-1}{q}$ . Since $q$ is a finite number, $A(x)$ also has a finite number of elements.

Pranshu Gaba - 6 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$