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435 lines
20 KiB
HTML
435 lines
20 KiB
HTML
<!doctype html>
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<head>
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<meta charset="UTF-8">
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<title>What is Infinity?</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">What is Infinity?</h1>
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<div class="center"><img src="../sets/images/infinity.svg" alt="infinity" height="80" width="198" ></div>
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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>Infinity ...</td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td>... it's not big ...</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td><b>... it's not huge ...</b></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td class="large">... it's not tremendously large ...</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td class="larger">... it's not extremely humongously enormous ...</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td>... it's ...</td>
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</tr>
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</tbody></table><br>
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<div style="font-size: 40px; color: blue;" align="center">Endless!</div><br>
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<h3>Infinity has no end</h3>
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<p>Infinity is the idea of something that has no end.</p>
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<p>In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.</p>
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<p>So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".</p>
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<p>If there is no reason something should stop, then it is infinite.</p>
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<p> </p>
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<h3>Infinity does not grow</h3>
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<p>Infinity is not "getting larger", it is already fully formed.</p>
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<p>Sometimes people (including me) say it "goes on and on" which sounds like it is growing somehow. But infinity does not <b>do</b> anything, it just <b>is</b>.</p>
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<h3>Infinity is not a real number</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/galaxies.jpg" alt="galaxies" height="154" width="301" ></p>
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<p>Infinity is not a real number, it is an idea. An idea of something without an end.</p>
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<p>Infinity cannot be measured.</p>
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<p>Even these faraway galaxies can't compete with infinity.</p>
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<p> </p>
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<h3>Infinity is Simple</h3>
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<p>Yes! It is actually simpler than things which <i>do</i> have an end. Because when something has an end, we have to define where that end is.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/line-segment-ray.svg" alt="line, line segment and ray" height="143" width="197" ></p>
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<h3>Example: in Geometry a Line has infinite length.</h3>
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<p>A Line goes in both directions <b>without end</b>.</p>
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<p>When there is one end it is called a Ray, and when there are two ends it is called a Line Segment, but they need <b>extra information</b> to define where the ends are.</p>
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<p>So a Line is actually simpler then a Ray or Line Segment.</p>
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</div>
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<p> </p>
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td colspan="2">
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<h3>More Examples:</h3> </td>
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</tr>
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<tr>
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<td align="center" valign="top" nowrap="nowrap">
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<p>{1, 2, 3, ...}</p>
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</td>
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<td>
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<p>The sequence of <a href="../whole-numbers.html">natural numbers</a> never ends, and is infinite.</p>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td width="16%"><img src="images/one-third.svg" alt="one-third" height="112" width="73" > </td>
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<td width="84%">
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<p>OK, <sup>1</sup>/<sub>3</sub> is a <b>finite</b> number (it is not infinite).</p>
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<p>But written as a <b>decimal number</b> the digit <b>3</b> repeats forever (we say "0.3 repeating"):</p>
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<p class="center large">0.3333333... (etc)</p>
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<p>There's no reason why the <b>3</b>s should ever stop: they <b>repeat infinitely</b>.</p>
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</td>
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</tr>
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<tr>
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<td align="center" width="16%"> </td>
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<td width="84%"> </td>
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</tr>
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<tr>
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<td class="large" align="center">0.999...</td>
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<td>
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<p>So, when we see a number like "0.999..." (i.e. a decimal number with an infinite series of 9s), there is <b>no end</b> to the number of 9s.</p>
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<p>You cannot say "but what happens if it ends in an 8?", because it simply does not end. (This is why <a href="../9recurring.html">0.999... equals 1</a>).</p>
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</td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td> </td>
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</tr>
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<tr>
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<td class="large" align="center" width="16%">AAAA...</td>
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<td width="84%">
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<p>An infinite series of "A"s followed by a "B" will NEVER have a "B".</p>
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</td>
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</tr>
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<tr>
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<td align="center" valign="top"> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="center" valign="top"><img src="../geometry/images/line-sm.gif" alt="line" height="44" width="82" ></td>
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<td>There are <a href="../sets/set-of-points.html">infinite points in a line</a>. Even a short line segment has infinite points.</td>
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</tr>
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</tbody></table>
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<p> </p>
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<h2>Big Numbers</h2>
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<p>There are some really impressively big numbers.</p>
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<p>A <b>Googol</b> is 1 followed by one hundred zeros (10<sup>100</sup>) :</p>
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<p class="center"><i>10,000,000,000,000,000,000,000,000,000,000,000,<br>
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000,000,000,000,000,000,000,000,000,000,000,<br>
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000,000,000,000,000,000,000,000,000,000,000</i></p>
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<p>A Googol is already bigger than the number of elementary particles in the known Universe, but then there is the <b>Googolplex</b>. It is 1 followed by<i> Googol zeros</i>. I can't even write down the number, because there is not enough matter in the known universe to form all the zeros:</p>
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<p class="center">10,000,000,000,000,000,000,000,000,000,000,000,000, ... (Googol number of Zeros)</p>
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<p>And there are even larger numbers that need to use "Power Towers" to write them down.</p>
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<p>For example, a Googolplex can be written as this power tower: <img src="../images/power-tower-googolplex.png" alt="power tower googolplex" height="22" width="55" ><br>
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That is ten to the power of (10 to the power of 100),</p>
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<p>But imagine an even bigger number like <img src="../images/power-tower-10-googolplex.png" alt="power tower 10 googolplex" height="27" width="73" > (which is a <b>Googolplexian</b>).</p>
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<p>And we can easily create much larger numbers than those!</p>
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<div class="center80">
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<h3>Finite</h3>
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<p>All of these numbers are "finite", we could eventually "get there".</p>
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<p>But none of these numbers are even close to infinity. Because they are finite, and infinity is ... <b>not</b> <b>finite!</b></p>
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</div>
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<h2>Using Infinity</h2>
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<p>We can sometimes use infinity <i>like</i> it is a number, but infinity does not behave like a real number.</p>
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<p>To help you understand, think "endless" whenever you see the "<span style="font-size: 20px; font-family: serif;">∞</span>":</p>
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<div class="example">
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<h3>Example: <span style="font-size: 20px; font-family: serif;">∞</span> + 1 = <span style="font-size: 20px; font-family: serif;">∞</span></h3>
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<p>It says that when something is endless, we can add 1 and it is still endless.</p>
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<p class="center large">But be careful with <span class="times">∞</span> in equations!<span class="times"></span></p>
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<p>Let us try to subtract <span style="font-size: 20px; font-family: serif;">∞</span> from both sides:</p>
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<div class="so"><span class="times">∞</span> − <span class="times">∞ </span>+ 1 = <span class="times">∞</span> − <span class="times">∞ </span></div>
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<div class="so">1 = 0</div>
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<p><b>Oh no!</b> Something is wrong here.</p>
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<p>In fact <span class="times">∞</span> − <span class="times">∞</span> is <b>undefined</b>.</p>
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<p>To avoid such mistakes:</p>
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<p class="center large">Imagine every <span class="times">∞</span> has a different value</p>
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</div>
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<p>We don't know how big infinity is, so we can't say two infinities are the same:</p>
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<div class="example">
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<h3>Example: Even Numbers</h3>
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<p>The set of natural numbers {1, 2, 3, ...} can be matched one-to-one with the set of even numbers {2, 4, 6, ...} like this:</p>
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<p class="center"><img src="../sets/images/infinity-1-1-even.svg" alt="natural 1-1 with even" height="114" width="286" ></p>
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<p><b>Both sets are infinite</b> (endless), but one <b>seems</b> to be twice as big as the other!</p>
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</div>
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<h2>Properties</h2>
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<p>The most important thing about infinity is that:</p>
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<div class="def">
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<p class="center large"><b>-<span class="times">∞</span> < x < <span class="times">∞</span></b></p>
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<p class="center">Where <b>x</b> is a <a href="real-numbers.html">real number</a></p></div>
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<p class="center">Which is mathematical shorthand for<br>
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<i>"<b>negative infinity</b> is less than any real number,<br>
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and <b>infinity</b> is greater than any real number"</i></p>
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<p>Here are some more properties:</p>
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<div class="simple">
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<table align="center" width="50%" border="0">
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<tbody>
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<tr>
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<th align="center">Special Properties of Infinity</th>
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</tr>
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<tr>
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<td style="text-align:center;"> <span style="font-size: 20px; font-family: serif;">∞</span> + <span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;">-<span style="font-size: 20px; font-family: serif;">∞</span> + -<span style="font-size: 20px; font-family: serif;">∞</span> = -<span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;"><span style="font-size: 20px; font-family: serif;">∞</span> × <span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;">-<span style="font-size: 20px; font-family: serif;">∞</span> × -<span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;">-<span style="font-size: 20px; font-family: serif;">∞</span> × <span style="font-size: 20px; font-family: serif;">∞</span> = -<span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b> + <span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b> + (-<span style="font-size: 20px; font-family: serif;">∞</span>) = -<span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b> - <span style="font-size: 20px; font-family: serif;">∞</span> = -<span style="font-size: 20px; font-family: serif;">∞</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b> − (-<span style="font-size: 20px; font-family: serif;">∞</span>) = <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center" height="16"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td align="left" height="16">For <b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span>>0 :</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center" height="16"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span> × <span style="font-size: 20px; font-family: serif;"></span><span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span> × <span style="font-size: 20px; font-family: serif;"></span>(-<span style="font-size: 20px; font-family: serif;">∞</span>) = <span style="font-size: 20px; font-family: serif;">-∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="left" height="16">For <b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span><0 :</td>
|
||
</tr>
|
||
<tr>
|
||
<td align="center" height="16"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span> × <span style="font-size: 20px; font-family: serif;"></span><span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">-∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">x</span></b><span style="font-size: 20px; font-family: serif;"></span> × <span style="font-size: 20px; font-family: serif;"></span>(-<span style="font-size: 20px; font-family: serif;">∞</span>) = <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
|
||
<h2>Undefined Operations</h2>
|
||
|
||
<p>All of these are "undefined":</p>
|
||
</div>
|
||
<div class="beach">
|
||
|
||
<table align="center" width="50%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<th align="center">"Undefined" Operations</th>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"> 0 × <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">0 × -<span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><span style="font-size: 20px; font-family: serif;">∞</span> + -<span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><span style="font-size: 20px; font-family: serif;">∞</span> - <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><span style="font-size: 20px; font-family: serif;">∞</span> / <span style="font-size: 20px; font-family: serif;">∞</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><span style="font-size: 20px; font-family: serif;">∞</span><sup>0</sup></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;"><b><span style="font-size: 18px; font-family: serif; font-style: italic;">1</span></b><sup><span style="font-size: 20px; font-family: serif;">∞</span></sup></td>
|
||
</tr>
|
||
</tbody></table><br>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Is <span class="intbl"><em>∞</em><strong>∞</strong></span> equal to 1?</h3>
|
||
<p>No, because we can't say that two infinities are the same.</p>
|
||
<p>For example <span style="font-size: 20px; font-family: serif;">∞</span> + <span style="font-size: 20px; font-family: serif;">∞</span> = <span style="font-size: 20px; font-family: serif;">∞</span>, so</p>
|
||
|
||
<table align="center" cellspacing="5" cellpadding="0" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<td> </td>
|
||
<td><span class="intbl"><em>∞</em><strong>∞</strong></span> = <span class="intbl"><em>∞ + ∞</em><strong>∞
|
||
</strong></span></td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><i>which looks like:</i></td>
|
||
<td><span class="intbl"><em>1</em><strong>1</strong></span> = <span class="intbl"><em>2</em><strong>1</strong></span></td>
|
||
<td><img src="../images/style/no.svg" alt="not" height="30" width="30" ></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>And that doesn't make sense!</p>
|
||
<p>So we say that <span class="intbl"><em>∞</em><strong>∞</strong></span> is undefined.</p>
|
||
</div>
|
||
|
||
|
||
<h2>Infinite Sets</h2>
|
||
|
||
<p>If you continue to study this subject you will find discussions about infinite sets, and the idea of <b>different sizes</b> of infinity.</p>
|
||
<p>That subject has special names like Aleph-null (how many Natural Numbers), Aleph-one and so on, which are used to measure the sizes of <a href="../sets/sets-introduction.html">sets</a>.</p>
|
||
|
||
<div class="example">
|
||
<p>For example, there are infinitely many <a href="../whole-numbers.html">whole numbers</a> {0, 1, 2, 3, 4,...},</p>
|
||
<p style="float:right; margin: 10px;"><img src="../sets/images/infinity-whole-real.svg" alt="infinity whole real" height="186" width="234" ></p>
|
||
<p> </p>
|
||
<p>But there are <b>more</b> <a href="real-numbers.html">real numbers</a> (such as 12.308 or 1.1111115) because there are infinitely many possible variations <i>after</i> the decimal place as well.</p>
|
||
<div style="clear:both">
|
||
</div>
|
||
<p>Or think of it this way: unlike integers, we can always discover new real numbers <b>in-between</b> other real numbers, no matter how small the gap.</p>
|
||
</div>
|
||
<p>But that is an advanced topic, and goes beyond the simple concept of infinity we discuss here.</p>
|
||
|
||
|
||
<h2>Conclusion</h2>
|
||
|
||
<p>Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.</p>
|
||
<p> </p>
|
||
|
||
<div class="related">
|
||
<a href="../calculus/limits.html">Limits</a>
|
||
<a href="index.html">Numbers Index</a>
|
||
</div>
|
||
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