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233 lines
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HTML
<!DOCTYPE html>
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<!-- Mirrored from www.mathsisfun.com/numbers/algebraic-numbers.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:59:03 GMT -->
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<title>Algebraic Number</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Algebraic Number</h1>
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<p class="center">Most numbers we use every day are Algebraic Numbers<br>
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But some are <b>not</b>, such as <span class="times">π</span> and <i><b>e</b></i></p>
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<h2>Algebraic Number</h2>
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<div class="center80">
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<p><b> Put simply</b>, when we have a polynomial equation like (for example)</p>
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<h3 align="center">2x<sup>2</sup> + 4x − 7 = 0</h3>
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<p>whose coefficients (the numbers 2, 4 and −7) are <a href="../rational-numbers.html">rational numbers</a> (whole numbers or simple fractions) ...
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</p>
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<p class="center">... then <b>x</b> is <b>Algebraic</b>.</p>
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</div>
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<p>We can imagine all kinds of polynomials:</p>
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<ul>
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<li><b>x − 1 = 0</b> has x = <b>1</b>,</li>
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<li><b>x + 1 = 0</b> has x = <b>−</b><b>1</b>,</li>
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<li><b>2x − 1 = 0</b> has x = <b>½</b>,</li>
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<li><b>x<sup>2 </sup>− 2 = 0</b> has x = <b>√2</b>,</li>
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<li>and so on</li></ul>
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<p>In each case <b>x</b> is algebraic.</p>
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<p>In fact all <a href="../whole-numbers.html">integers</a>, all <a href="../rational-numbers.html">rational numbers</a>, some <a href="../irrational-numbers.html">irrational numbers</a> (such as √2) are Algebraic.</p>
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<div class="example">
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<h3>Testing Game</h3>
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<p>We can make a game of it!</p>
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<p>Start with our number, and we have to get it to <b>zero </b>using only:</p>
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<ul>
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<li>whole numbers</li>
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<li>add, subtract, multiply (but not divide)</li>
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<li>whole number exponents</li></ul>
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<p>Here are some examples:</p>
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<div class="tbl">
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<div class="row">
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<span class="lt">Number</span><span class="rt">Getting it to Zero</span>
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</div>
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<div class="row">
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<span class="lt">5</span><span class="rt">5 − 5 = 0</span>
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</div>
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<div class="row">
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<span class="lt">−7</span><span class="rt">−7 + 7 = 0</span>
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</div>
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<div class="row">
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<span class="lt">½</span><span class="rt">2(½) − 1 = 0</span>
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</div>
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<div class="row">
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<span class="lt">√2</span><span class="rt">(√2)<sup>2</sup> − 2 = 0</span>
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</div>
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<div class="row">
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<span class="lt">3√2</span><span class="rt">(3√2)<sup>2</sup> − 18 = 0 </span>
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</div>
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</div><p>All those numbers are algebraic!</p>
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<p>Try some yourself and see how you go.</p>
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<p>Now try <span class="times">π</span> (pi) and see if you have any success.</p>
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</div>
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<h2>More Formally</h2>
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<div class="words">
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<p>To be algebraic, a number must be a root of a non-zero polynomial equation with <a href="../rational-numbers.html">rational</a> <a href="../algebra/definitions.html">coefficients</a>.</p>
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</div>
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So <b>x</b> is algebraic in this example:
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<h3 align="center">2x<sup>3</sup> − 5x + 39 = 0</h3>
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<p>Because all conditions are met:</p>
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<ul>
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<li>2x<sup>3</sup> − 5x + 39 is a non-zero polynomial (a polynomial which is not just "0")</li>
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<li><b>x</b> is a root (i.e. <b>x</b> gives the result of <b>zero</b> for the function 2x<sup>3</sup> − 5x + 39)</li>
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<li>the <a href="../algebra/definitions.html">coefficients</a> (the numbers 2, −5 and 39) are <a href="../rational-numbers.html">rational numbers</a></li>
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</ul>
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<p>So we know <b>x</b> is algebraic</p>
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<p>But let's see its value anyway:</p>
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<div class="example">
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<h3>Example: 2x<sup>3</sup> − 5x + 39 = 0</h3>
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<p>We need to find the value of <b>x</b> where <b>2x<sup>3</sup> − 5x + 39 is equal to 0</b></p>
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<p>Well <b>x = −3</b> works, because 2(−3)<sup>3</sup> − 5(−3) + 39 = −54+15+39 = 0</p>
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</div>
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<p>Let's try another polynomial (remember: the coefficients must be rational).</p>
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<div class="example">
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<h3>Example: 2x<sup>3</sup> − ¼ = 0</h3>
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<p>The coefficients are 2 and −¼, both rational numbers. So <b>x is an Algebraic Number</b></p>
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<p>We can also discover that <b>x = 0.5</b>, because 2(0.5)<sup>3</sup> − ¼ = 0</p></div>
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<p>In fact:</p>
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<p class="center larger">All integers and rational numbers are algebraic,<br>
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but an <a href="../irrational-numbers.html">irrational</a> number <b>may or may not</b> be algebraic</p>
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<h2>Not Algebraic? Then Transcendental!</h2>
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<p>When a number is not algebraic, it is called <a href="transcendental-numbers.html">transcendental</a>.</p>
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<p>Back in 1844 Joseph Liouville created a number:</p>
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<p class="center larger">0.110001000000000000000001000000...</p>
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<p class="center">(it has a 1 in every <a href="factorial.html">factorial</a> numbered position such as 1, 2, 6, 24, etc)</p>
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<p>And he showed it is <b>not algebraic</b> ... he had broken mathematics!</p>
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<p>Just kidding. But it did <i>transcend</i> algebraic numbers and was</p>
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<p class="center larger">the first known transcendental number</p>
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<p>Then in 1873 Charles Hermite proved that <i><b>e</b></i> (<a href="e-eulers-number.html">Euler's number</a>) is transcendental, and in 1882 Ferdinand von Lindemann proved that <span class="times">π</span> (<a href="pi.html">pi</a>) is transcendental.</p>
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<p>It is actually hard to prove that a number is transcendental.</p>
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<h2>More</h2>
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<p>Let's investigate a few more numbers</p>
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<div class="example">
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<h3>Example: the unit <a href="imaginary-numbers.html">imaginary number</a> <i><b>i</b></i></h3>
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<p>Well, we know that <i><b>i</b></i><sup>2</sup> = −1, so <i><b>i</b></i> is the solution to:</p>
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<p class="center"><b>x<sup>2</sup> + 1 = 0</b></p>So the imaginary number <b><i>i</i> is an Algebraic Number</b>
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<p>Note: using the "testing game": <i><b>i</b></i><sup>2</sup> + 1 = 0</p>
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</div>
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<div class="example">
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<h3>Example: <span style="font-size:20px; ">φ</span> (the <a href="golden-ratio.html">Golden Ratio</a>)</h3>
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<p><span style="color:gold; font-size:20px; ">φ</span> is the solution to x<sup>2</sup> − x − 1 = 0</p>
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<p>So <span style="color:gold; font-size:20px; ">φ</span><b> is an Algebraic Number</b></p>
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</div>
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<div class="example">
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<h3>Example: x<sup>2</sup> + 2x + 10 = 0</h3>
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<p>The solutions to this <a href="../algebra/quadratic-equation.html">quadratic equation</a> are <a href="complex-numbers.html">complex numbers</a> :</p>
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<ul>
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<li>x = −1 + 3<i><b>i</b></i></li>
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<li>x = −1 − 3<i><b>i</b></i></li>
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</ul>
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<p>(Try putting them into the equation, and remember that <b><i>i</i></b><sup>2</sup> = −1)</p>
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<p>They are both <b>Algebraic Numbers</b></p>
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</div>
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<h2>Which is More Common?</h2>
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<p>It might seem that transcendental numbers are rare, but:</p>
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<p class="center large"><b>almost all</b> real and complex numbers are <b>transcendental.</b></p>
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<p>Why? Well, imagine some random real number where <b>each digit is randomly chosen</b>, and you get something like 7.17493614485672... (on for infinity). It is almost certain to be transcendental.</p>
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<p>But in every day life we use carefully chosen numbers like 6 or 3.5 or 0.001, so most numbers we deal with (except <span class="times">π</span> and <i><b>e</b></i>) are algebraic, but any truly randomly chosen real or complex number is almost certain to be transcendental.</p>
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<h2>Properties</h2>
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<p>All algebraic numbers are computable and so they are definable.</p>
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<p>The set of algebraic numbers is <b>countable</b>. Put simply, the list of <a href="../whole-numbers.html">whole numbers</a> is "countable", and you can arrange the algebraic numbers in a 1-to-1 manner with whole numbers, so they are also countable.</p>
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<p> </p>
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<div class="related">
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<a href="transcendental-numbers.html">Transcendental Numbers</a>
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<a href="../irrational-numbers.html">Irrational Numbers</a>
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<a href="../algebra/definitions.html">Basic Definitions in Algebra</a>
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</div>
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