# Is 3.567 rational?

Apr 12, 2018

Yes. Irrational values, like $\pi$ or $\sqrt{2}$, have a never ending decimal value that does not repeat. But $3.567$ does have an end, and we can also write it as a fraction and have it remain accurate: $\frac{3567}{1000}$

There is no decimal or fractional value of $\pi$ that is accurate (not an approximation but a representation of the exact value of $\pi$), so $\pi$ is irrational

Apr 12, 2018

Yes! The decimal equivalent of a $\underline{\text{rational number}}$ is either

Terminating decimal $\to \text{example } 0.125$
Has a repeating cycle of values for ever$\to \text{ example } 0.33333 \ldots .$

Another example: $\text{ } 0.125125125125 \ldots$

#### Explanation:

$\textcolor{red}{\text{Just for the hell of it}}$ lets determine the fractional equivalent of
0.125125125....

Set $x = 0.125125125 \ldots . \text{ } E q u a t i o n \left(1\right)$

We need to 'get rid' of the decimal part so to do this

Multiply both sides by 1000 giving

$1000 x = 125.125125125 \ldots . \text{ } E q u a t i o n \left(2\right)$

Subtract Eqn(1) from Eqn(2) to 'get rid' if the decimal

$1000 x = 125.125125125 \ldots . .$
$\underline{\textcolor{w h i t e}{1000} x = \textcolor{w h i t e}{12} 0.125125125 \ldots \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{1} 999 x = 125$

Divide both sides by 999

$x = \frac{125}{999} \textcolor{w h i t e}{\text{d") =color(white)("d}} 0.125125125 \ldots .$