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262 lines
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HTML
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<title>Irrational Numbers</title>
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<article id="content" role="main">
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<h1 class="center">Irrational Numbers</h1>
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<p class="center">An <b>Irrational Number</b> is a real number that <b>cannot</b> be written as a simple fraction:</p>
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<p class="center"><img src="numbers/images/rational-vs-irrational.svg" alt="rational vs irrational" style="max-width:100%" height="79" width="542"></p>
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<p> <b>1.5</b> is rational, but <span class="times"><b>π</b></span> is irrational</p>
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<p class="center large">Irrational means <b>not Rational</b> (no ratio)</p>
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<p>Let's look at what makes a number rational or irrational ...</p>
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<h3>Rational Numbers</h3>
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<p>A <b><a href="rational-numbers.html">Rational</a></b> Number <b>can</b> be written as a <b>Ratio</b> of two integers (ie a simple fraction).</p>
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<div class="example">
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<p>Example: <b>1.5</b> is rational, because it can be written as the ratio <b>3/2</b></p>
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</div>
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<div class="example">
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<p>Example: <b>7</b> is rational, because it can be written as the ratio <b>7/1</b></p>
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</div>
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<div class="example">
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<p>Example <b>0.333...</b> (3 repeating) is also rational, because it can be written as the ratio <b>1/3</b></p>
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</div>
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<p> </p>
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<h3>Irrational Numbers</h3>
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<p>But some numbers <b>cannot</b> be written as a ratio of two integers ...</p>
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<p class="center larger">...they are called <b>Irrational Numbers</b>.</p>
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<div class="example">
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<h3>Example: <span class="times"><b>π</b> </span><b><a href="numbers/pi.html">(Pi)</a></b> is a famous irrational number.</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="numbers/images/pi1.svg" alt="Pi" height="100" width="100"></p>
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<p class="center"><b><span class="times"><b>π</b></span> = 3.1415926535897932384626433832795... (and more)</b></p>
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<p>We <b>cannot</b> write down a simple fraction that equals Pi.</p>
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<p>The popular approximation of <span class="frac"><sup>22</sup>/<sub>7</sub></span> = 3.1428571428571... is close but <b>not accurate</b>.</p>
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</div>
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<p>Another clue is that the decimal goes on forever without repeating.</p>
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<h2>Cannot Be Written as a Fraction</h2>
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<p class="center fun"><i>It is <b>irrational</b> because it cannot be written as a <b>ratio</b> (or fraction),<br>
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not because it is crazy!</i></p>
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<p>So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.</p>
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<div class="example">
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<h3>Example: <b>9.5</b> can be written as a simple fraction like this:</h3>
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<p class="center larger">9.5 = <span class="intbl"><em>19</em><strong>2</strong></span></p>
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<p class="center large">So it is a <b>rational number</b> (and so is <b>not irrational</b>)</p>
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</div>
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<p>Here are some more examples:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th>Number</th>
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<th width="10"> </th>
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<th>As a Fraction</th>
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<th width="30"> </th>
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<th>Rational or<br>
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Irrational?</th>
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</tr>
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<tr style="text-align:center;">
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<td height="9">1.75</td>
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<td> </td>
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<td height="9"><span class="intbl"><em>7</em><strong>4</strong></span></td>
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<td> </td>
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<td height="9">Rational</td>
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</tr>
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<tr style="text-align:center;">
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<td>.001</td>
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<td> </td>
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<td><span class="intbl"><em>1</em><strong>1000</strong></span></td>
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<td> </td>
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<td>Rational</td>
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</tr>
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<tr style="text-align:center;">
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<td>√2<br>
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(square root of 2)</td>
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<td class="large"> </td>
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<td class="large">?</td>
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<td> </td>
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<td><b>Irrational !</b></td>
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</tr>
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</tbody></table>
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<div class="simple">
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<h2>Square Root of 2</h2>
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<p>Let's look at the square root of 2 more closely.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="numbers/images/square-root-2.svg" alt="square root 2" height="171" width="158"></td>
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<td>When we draw a square of size "1",<br>
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what is the distance across the diagonal?</td>
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</tr>
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</tbody></table>
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<p>The answer is the <b><a href="square-root.html">square root</a> of 2</b>, which is<b> 1.4142135623730950...(etc)</b></p>
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<p>But it is not a number like 3, or five-thirds, or anything like that ...</p>
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<div class="indent50px">
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<p class="center">... in fact we <b>cannot</b> write the square root of 2 using a ratio of two numbers ...</p>
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<p class="center">... (you can learn <b>why</b> on the <a href="numbers/irrational-finding.html">Is It Irrational?</a> page) ...</p>
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<p class="center">... and so we know it is <b>an irrational number</b>.</p>
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</div>
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</div>
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<h2>Famous Irrational Numbers</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="numbers/images/pi1.svg" alt="Pi" height="100" width="100"></td>
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<td> </td>
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<td>
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<p><b><a href="numbers/pi.html">Pi</a></b> is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this:</p>
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<p>3.1415926535897932384626433832795 (and more ...)</p>
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</td>
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</tr>
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<tr>
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<td><img src="numbers/images/e1.svg" alt="e (eulers number)" height="100" width="100"></td>
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<td> </td>
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<td>
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<p>The number <i><b>e</b></i> (<a href="numbers/e-eulers-number.html">Euler's Number</a>) is another famous irrational number. People have also calculated <i><b>e</b></i> to lots of decimal places without any pattern showing. The first few digits look like this:</p>
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<p>2.7182818284590452353602874713527 (and more ...)</p>
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</td>
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</tr>
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<tr>
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<td><img src="numbers/images/phi.svg" alt="phi" height="95" width="84"></td>
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<td> </td>
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<td>
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<p>The <a href="numbers/golden-ratio.html">Golden Ratio</a> is an irrational number. The first few digits look like this:</p>
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<p>1.61803398874989484820... (and more ...)</p>
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</td>
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</tr>
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<tr>
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<td><img src="numbers/images/radical.svg" alt="radical symbol" height="95" width="72"></td>
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<td> </td>
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<td>
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<p>Many square roots, cube roots, etc are also irrational numbers. Examples:</p>
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<div class="simple">
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>√3</td>
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<td>1.7320508075688772935274463415059 (etc)</td>
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</tr>
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<tr>
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<td>√99</td>
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<td>9.9498743710661995473447982100121 (etc)</td>
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</tr>
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</tbody></table>
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</div>
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</td>
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</tr>
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</tbody></table>
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<p>But √4 = 2 is rational, and √9 = 3 is rational ...</p>
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<p class="center">... so <b>not all</b> roots are irrational.</p>
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<h3> </h3>
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<h3>Note on Multiplying Irrational Numbers</h3>
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<p>Have a look at this:</p>
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<ul>
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<li><span class="times">π</span> × <span class="times">π</span> = <span class="times">π</span><sup>2</sup> is known to be <b>irrational</b></li>
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<li>But √2 × √2 = <b>2</b> is <b>rational</b></li>
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</ul>
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<p>So be careful ... multiplying irrational numbers <b>might</b> result in a rational number!</p>
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<p> </p>
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<div class="fun">
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<h3>Fun Facts ....</h3>
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<p>Apparently <b><i>Hippasus</i></b> (one of <i>Pythagoras'</i> students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead he proved the square root of 2 <i>could not</i> be written as a fraction, so it is <i>irrational</i>.</p>
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<p>But followers of <i><b>Pythagoras</b></i> 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>
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</div>
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<p> </p>
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<div class="questions">434,435,1064,2022,3987,1065,3988,2023,2990,2991</div>
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<div class="related">
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<a href="surds.html">Surds</a>
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<a href="square-root.html">Square Roots</a>
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<a href="scientific-calculator.html">Scientific Calculator</a>
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<a href="numbers/irrational-finding.html">Is It Irrational?</a>
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<a href="numbers/index.html">Numbers Index</a>
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