<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-rational-versus-irrational-ii">Rational versus Irrational II Exercise</a></p><p>Now that we've discussed non-terminating, non-repeating decimals, we can introduce the <strong><span style="color:#27ae60;">Second Definition of Irrational Numbers</span></strong>:</p>
<ul>
<li><span style="color:#e74c3c;">First Definition</span>: Any number that cannot be represented as a fraction.
<ul>
<li>Recall that $\frac{\pi}{2}$ is <strong><u>not</u></strong> a fraction.</li>
</ul>
</li>
<li><span style="color:#27ae60;">Second Definition</span>: Any number that, when written as a decimal, is non-terminating and non-repeating.
<ul>
<li>For example, $\sqrt{6} = 2.449489742...$</li>
<li>Notice how there is no pattern there.</li>
</ul>
</li>
</ul>
<p><span style="font-size:20px;"><span style="color:#8e44ad;">Wanna Hear Some Crazy Shit?</span></span></p>
<p>When irrational numbers are written in decimal form, the digits after the decimal point go on forever, infinitely, in no pattern at all. This means that, if you look long enough, you can find <u><strong>any</strong></u> sequence of numbers you wish. Looking for the number $9$,$876$,$543$,$210$? It's in there! Looking for every phone number that you've ever had written in back-to-back form? It's in there! Looking for every integer from $1$ to $10$ billion, in order? It's in there!</p>
<p>Frickin' mind boggling, I'm telling you.</p>