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<!DOCTYPE html>
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<title>Rational Numbers</title>
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<h1 class="center">Rational Numbers</h1>
<p class="center">A <b>Rational Number</b> can be made by dividing an integer by an integer.<br>
<i>(An <a href="whole-numbers.html">integer</a> itself has no fractional part.)</i></p>
<div class="example">
<h3>Example:</h3>
<p><b>1.5 is a rational number</b> because 1.5 = 3/2 &nbsp; (3 and 2 are both integers)</p>
<p class="center"><img src="numbers/images/rational.svg" alt="Rational Number" height="82" width="192"></p>
</div> <p class="center larger">Most numbers we use in everyday life are Rational Numbers.</p>
<p>You can make a few rational numbers yourself using the sliders below:</p>
<div class="script" style="height: 130px;">
numbers/images/rational.js
</div>
<p>Here are some more examples:</p>
<div class="simple">
<table align="center" width="400" border="0">
<tbody>
<tr style="text-align:center;">
<th>Number</th>
<th>As a Fraction</th>
<th>Rational?</th>
</tr>
<tr style="text-align:center;">
<td height="9">5</td>
<td height="9">5/1</td>
<td height="9">Yes</td>
</tr>
<tr style="text-align:center;">
<td height="9">1.75</td>
<td height="9">7/4</td>
<td height="9">Yes</td>
</tr>
<tr style="text-align:center;">
<td>1000</td>
<td>1000/1</td>
<td>Yes</td></tr>
<tr style="text-align:center;">
<td>.001</td>
<td>1/1000</td>
<td>Yes</td>
</tr>
<tr style="text-align:center;">
<td>−0.1</td>
<td>−1/10</td>
<td>Yes</td>
</tr>
<tr style="text-align:center;">
<td>0.111...</td>
<td>1/9</td>
<td>Yes</td>
</tr>
<tr style="text-align:center;">
<td>√2<br>
(square root of 2)</td>
<td class="large">?</td>
<td><b class="large">NO !</b></td>
</tr>
</tbody></table> </div>
<p>Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are <b>not rational</b> they are called <a href="irrational-numbers.html">Irrational</a>.</p>
<p>Another famous <b>irrational</b> number is <a href="numbers/pi.html">Pi (<span class="times">π</span>)</a>:</p>
<p class="center"><img src="numbers/images/pi-irrational.svg" alt="Rational Number" height="79" width="322"></p>
<h2>Formal Definition of Rational Number</h2>
<p>More formally we say:</p>
<div class="def">
<p class="center"><i>A rational number is a number that can be in the form<b> p/q</b><br>
where <b>p</b> and <b>q</b> are <a href="whole-numbers.html">integers</a> and <b>q</b> is not equal to zero.</i></p>
</div>
<p>So, a rational number can be:</p>
<div class="center larger"><span class="intbl"><em>p</em><strong>q</strong></span></div>
<p>where q is not zero.</p>
<h3>Examples:</h3>
<div class="simple">
<table style="border: 0; margin:auto;">
<tbody>
<tr style="text-align:center;">
<th width="35">p</th>
<th width="35">q</th>
<th width="80">p / q</th>
<th>=</th>
</tr>
<tr style="text-align:center;">
<td height="9" width="35">1</td>
<td height="9" width="35">1</td>
<td style="width:80px;">1/1</td>
<td height="9">1</td>
</tr>
<tr style="text-align:center;">
<td height="9" width="35">1</td>
<td height="9" width="35">2</td>
<td style="width:80px;">1/2</td>
<td height="9">0.5</td>
</tr>
<tr style="text-align:center;">
<td height="9" width="35">55</td>
<td height="9" width="35">100</td>
<td style="width:80px;">55/100</td>
<td height="9"> 0.55</td>
</tr>
<tr style="text-align:center;">
<td style="width:35px;">1</td>
<td style="width:35px;">1000</td>
<td style="width:80px;">1/1000</td>
<td>0.001</td>
</tr>
<tr style="text-align:center;">
<td style="width:35px;">253</td>
<td style="width:35px;">10</td>
<td style="width:80px;">253/10</td>
<td>25.3</td>
</tr>
<tr style="text-align:center;">
<td style="width:35px;">7</td>
<td style="width:35px;">0</td>
<td style="width:80px;">7/0</td>
<td><b>No! "q" can't be zero!</b></td>
</tr>
</tbody></table>
</div>
<p>Just remember: q can't be zero.</p>
<h2>Using Rational Numbers</h2>
<table style="border: 0;">
<tbody>
<tr>
<td><img src="images/arithmetic.jpg" alt="add, subtract, multiply and divide" height="74" width="90"></td>
<td>
<p>If a rational number is still in the form "p/q" it can be a little difficult to use, so I have a special page on how to:</p>
<p class="center"><a href="algebra/rational-numbers-operations.html">Add, Subtract, Multiply and Divide Rational Numbers</a></p></td>
</tr>
</tbody></table>
<p>&nbsp;</p>
<div class="fun">
<h3>Fun Facts ....</h3>
<p>The ancient greek mathematician <i><b>Pythagoras</b></i> believed that all numbers were rational, but one of his students <i><b>Hippasus</b></i> proved (using geometry, it is thought) that you could <b>not</b> write the square root of 2 as a fraction, and so it was <i>irrational</i>.</p>
<p>But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!</p>
</div>
<p>&nbsp;</p>
<div class="questions">1667, 1668, 3984, 3983, 5347, 9002, 9072, 9000, 9001, 9071</div>
<div class="related">
<a href="irrational-numbers.html">Irrational Numbers</a>
<a href="scientific-calculator.html">Scientific Calculator</a>
</div>
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