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512 lines
16 KiB
HTML
512 lines
16 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- #BeginEditable "doctitle" -->
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<title>Binary Number System</title>
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</head>
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<body id="bodybg">
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<div class="bg">
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<div id="stt"></div>
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<header>
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<div id="hdr"></div>
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<div id="logo"><a href="index.html"><img src="images/style/logo.svg" alt="Math is Fun"></a></div>
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<script>document.write(getTrans());</script>
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</header>
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<nav>
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<script>document.write(getMenu(0));</script>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Binary Number System</h1>
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<p>A Binary Number is made up of only <b>0</b>s and <b>1</b>s.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr align="center">
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<td>
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<div class="huge">110100</div></td>
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</tr>
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<tr align="center">
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<td>Example of a Binary Number</td>
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</tr>
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</tbody></table>
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<p>There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!</p>
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<p>Binary numbers have many uses in mathematics and beyond.</p>
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<p class="center"><img src="numbers/images/digital-world.jpg" alt="digital world" height="144" width="360"><br>
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In fact the digital world uses <a href="binary-digits.html">binary digits</a>.</p>
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<h2>How do we Count using Binary?</h2>
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<p>It is just like counting in decimal except we reach 10 much sooner.</p>
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<div class="simple">
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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 class="large" align="center">Binary</td>
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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 class="large" align="center">0</td>
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<td> </td>
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<td>We start at 0 </td>
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</tr>
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<tr>
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<td class="large" align="center">1</td>
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<td> </td>
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<td>Then 1</td>
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</tr>
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<tr>
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<td class="large" align="center">???</td>
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<td> </td>
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<td>But then there is no symbol for 2 ... what do we do?</td>
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</tr>
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</tbody></table>
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</div>
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<p> </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 colspan="4"><span class="larger">Well how do we count
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in Decimal?</span></td>
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</tr>
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<tr>
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<td style="width:20px;"> </td>
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<td class="large" align="right">0</td>
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<td> </td>
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<td>Start at 0</td>
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</tr>
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<tr>
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<td> </td>
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<td class="large" align="right">...</td>
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<td> </td>
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<td>Count 1,2,3,4,5,6,7,8, and then...</td>
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</tr>
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<tr>
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<td> </td>
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<td class="large" align="right">9</td>
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<td> </td>
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<td>This is the <b>last digit</b> in Decimal</td>
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</tr>
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<tr>
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<td> </td>
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<td class="large" align="right"><span class="hilite">1</span>0</td>
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<td> </td>
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<td>So we start back at 0 again, but add <span class="hilite">1</span> on the left</td>
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</tr>
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</tbody></table>
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<p>The same thing is done in binary ...</p>
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<div class="simple">
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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> </td>
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<td class="large" align="center">Binary</td>
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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" align="right">0</td>
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<td> </td>
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<td>Start at 0</td>
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</tr>
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<tr>
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<td>•</td>
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<td class="large" align="right">1</td>
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<td> </td>
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<td>Then 1</td>
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</tr>
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<tr>
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<td>••</td>
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<td class="large" align="right"><span class="hilite">1</span>0</td>
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<td> </td>
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<td>Now start back at 0 again, but <b>add <span class="hilite">1</span> on the left</b></td>
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</tr>
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<tr>
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<td>•••</td>
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<td class="large" align="right">11</td>
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<td> </td>
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<td>1 more</td>
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</tr>
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<tr>
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<td>••••</td>
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<td class="large" align="right">???</td>
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<td> </td>
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<td>But NOW what ... ?</td>
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</tr>
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</tbody></table>
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</div>
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<p> </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 colspan="4" class="large"><span class="larger">What happens in Decimal?</span></td>
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</tr>
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<tr>
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<td class="large" align="center" width="20"> </td>
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<td class="large" align="right">99</td>
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<td> </td>
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<td>When we run out of digits, we ...</td>
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</tr>
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<tr>
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<td class="large" align="center"> </td>
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<td class="large" align="right">100</td>
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<td> </td>
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<td>... start back at 0 again, but add <span class="hilite">1</span> on the left<br>
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</td>
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</tr>
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</tbody></table>
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<p>And that is what we do in binary ...</p>
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<div class="simple">
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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> </td>
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<td class="large" align="center">Binary</td>
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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" align="right">0</td>
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<td> </td>
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<td>Start at 0 </td>
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</tr>
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<tr>
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<td>•</td>
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<td class="large" align="right">1</td>
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<td> </td>
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<td>Then 1</td>
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</tr>
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<tr>
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<td>••</td>
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<td class="large" align="right"><span class="hilite">1</span>0</td>
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<td> </td>
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<td>Start back at 0 again, but add 1 on the left</td>
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</tr>
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<tr>
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<td>•••</td>
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<td class="large" align="right">11</td>
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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" align="right"><span class="hilite">1</span>00</td>
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<td> </td>
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<td>start back at 0 again, and add one to the number on the left...<br>
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... but that number is already at 1 so it also goes back to 0 ...<br>
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... and 1 is added to the <i>next position</i> on the left</td>
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</tr>
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<tr>
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<td>•••••</td>
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<td class="large" align="right">101</td>
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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" align="right">110</td>
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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" align="right">111</td>
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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" align="right"><span class="hilite">1</span>000</td>
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<td> </td>
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<td>Start back at 0 again (for all 3 digits),<br>
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add 1 on the left</td>
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</tr>
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<tr>
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<td>•••••••••</td>
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<td class="large" align="right">1001</td>
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<td> </td>
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<td>And so on!</td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<p>See how it is done in this little demonstration (press play button):</p>
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<script>numberodometerMain(2);</script>
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<h2>Decimal vs Binary</h2>
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<p>Here are some equivalent values:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr align="center">
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<th>Decimal:</th>
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<td>0</td>
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<td>1</td>
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<td>2</td>
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<td>3</td>
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<td>4</td>
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<td>5</td>
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<td>6</td>
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<td>7</td>
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<td>8</td>
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<td>9</td>
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<td>10</td>
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<td>11</td>
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<td>12</td>
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<td>13</td>
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<td>14</td>
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<td>15</td>
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</tr>
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<tr align="center">
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<th>Binary:</th>
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<td>0</td>
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<td>1</td>
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<td>10</td>
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<td>11</td>
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<td>100</td>
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<td>101</td>
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<td>110</td>
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<td>111</td>
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<td>1000</td>
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<td>1001</td>
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<td>1010</td>
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<td>1011</td>
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<td>1100</td>
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<td>1101</td>
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<td>1110</td>
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<td>1111</td>
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</tr>
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</tbody></table>
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</div>
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<h3>Symmetry</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/style/yin-yang.gif" alt="yin yang" height="100" width="100"></p>
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<p>Binary numbers also have a beautiful and elegant pattern:</p>
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<p class="center"><img src="numbers/images/binary-symmetry.svg" alt="binary symmetry"></p>
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<div style="clear:both"></div><br>
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<p>Here are some larger values:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr align="center">
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<th>Decimal:</th>
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<td>20</td>
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<td>25</td>
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<td>30</td>
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<td>40</td>
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<td>50</td>
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<td>100</td>
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<td>200</td>
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<td>500</td>
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</tr>
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<tr align="center">
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<th>Binary:</th>
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<td>10100</td>
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<td>11001</td>
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<td>11110</td>
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<td>101000</td>
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<td>110010</td>
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<td>1100100</td>
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<td>11001000</td>
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<td>111110100</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center large">"Binary is as easy as 1, 10, 11."</p>
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<p>Now see how to use Binary to count past 1,000 on your fingers:</p>
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<div class="activity"><a href="numbers/binary-count-fingers.html">Activity: Binary Fingers</a></div>
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<p> </p>
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<h2>Position</h2>
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<p>In the <a href="decimals.html">Decimal System</a> there are Ones, Tens, Hundreds, etc</p>
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<p>In <b>Binary</b> there are Ones, Twos, Fours, etc, like this:</p>
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<p class="center"><img src="numbers/images/binary-number.svg" alt="binary number 1101.101"></p>
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<p class="center larger">This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8)<br>
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= <b>13.625 in Decimal</b></p>
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<p> </p>
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<p>Numbers can be placed to the left
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or right of the point, to show values greater than one
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and less than one.</p>
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<table align="center" width="485" border="0">
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<tbody>
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<tr>
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<td colspan="2" class="huge">10.1</td>
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</tr>
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<tr>
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<td class="larger" width="105"><img src="images/style/left-arrow.gif" alt="left arrow" height="46" width="46"><img src="images/style/dot-blue.gif" alt="dot blue" height="14" width="14"></td>
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<td class="larger">The number to the left of the point
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is a whole number (such as 10)</td>
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</tr>
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<tr>
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<td class="larger" width="105"> </td>
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<td class="larger"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td colspan="2" class="larger">As we move further left, every number place<br>
|
||
gets <b>2
|
||
times bigger</b>.</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" width="105"> </td>
|
||
<td class="larger"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" align="right" width="105"><img src="images/style/dot-blue.gif" alt="dot blue" height="14" width="14"><img src="images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></td>
|
||
<td class="larger">The first digit on the right means <b> halves </b>(1/2).</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" width="105"> </td>
|
||
<td class="larger"> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" width="105"> </td>
|
||
<td class="larger" align="right">As we move further right, every number place<br>
|
||
gets <b>2 times smaller</b> (half as big).</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="example">
|
||
<h3>Example: 10.1</h3>
|
||
<ul>
|
||
<li>The "10" means 2 in decimal,</li>
|
||
<li>The ".1" means half,</li>
|
||
<li>So "10.1" in binary is 2.5 in decimal</li>
|
||
</ul>
|
||
</div>
|
||
<p>You can do conversions at <a href="binary-decimal-hexadecimal-converter.html">Binary to Decimal to Hexadecimal Converter</a>.</p>
|
||
<h2>Words</h2>
|
||
<div class="words">
|
||
<p>The word <b>binary</b> comes from "Bi-" meaning two. We see "bi-" in words such as "bicycle" (two wheels) or "binocular" (two eyes).</p>
|
||
</div>
|
||
<table align="center" width="90%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/style/two-speech.gif" alt="speech" height="40" width="49"></td>
|
||
<td>When you <b>say</b> a binary number, pronounce each digit (example, the binary number "101" is spoken as <i>"one zero one"</i>, or sometimes <i>"one-oh-one"</i>). This way people don't get confused with the decimal number. </td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="words">
|
||
<p>A single binary digit (like "0" or "1") is called a "bit".</p>
|
||
<p>For example <b>11010</b> is five bits long.</p>
|
||
<p>The word<b> bit</b> is made up from the words "<b>b</b>inary dig<b>it</b>"</p>
|
||
</div>
|
||
<h2>How to Show that a Number is Binary</h2>
|
||
<p>To show that a number is a <i>binary</i> number, follow it with a little 2 like this: <b>101<sub>2</sub></b></p>
|
||
<p>This way people won't think it is the decimal number "101" (one hundred and one).</p>
|
||
<h2>Examples</h2>
|
||
<div class="example">
|
||
<h3>Example: What is 1111<b><sub>2</sub></b> in Decimal?</h3>
|
||
<ul>
|
||
<li>The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)</li>
|
||
<li>The next "1" is in the "2×2" position, so that means 1×2×2 (=4)</li>
|
||
<li>The next "1" is in the "2" position, so that means 1×2 (=2)</li>
|
||
<li>The last "1" is in the ones position, so that means 1</li>
|
||
<li>Answer: 1111 = 8+4+2+1 = 15 in Decimal</li>
|
||
</ul>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: What is 1001<b><sub>2</sub></b> in Decimal?</h3>
|
||
<ul>
|
||
<li>The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8)</li>
|
||
<li>The "0" is in the "2×2" position, so that means 0×2×2 (=0)</li>
|
||
<li>The next "0" is in the "2" position, so that means 0×2 (=0)</li>
|
||
<li>The last "1" is in the ones position, so that means 1</li>
|
||
<li>Answer: 1001 = 8+0+0+1 = 9 in Decimal</li>
|
||
</ul>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: What is 1.1<b><sub>2</sub></b> in Decimal?</h3>
|
||
<ul>
|
||
<li>The "1" on the left side is in the ones position, so that means 1.</li>
|
||
<li>The 1 on the right side is in the "halves" position, so that means 1×(1/2)</li>
|
||
<li>So, 1.1 is "1 and 1 half" = 1.5 in Decimal</li>
|
||
</ul>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: What is 10.11<b><sub>2</sub></b> in Decimal?</h3>
|
||
<ul>
|
||
<li>The "1" is in the "2" position, so that means 1×2 (=2)</li>
|
||
<li>The "0" is in the ones position, so that means 0</li>
|
||
<li>The "1" on the right of the point is in the "halves" position, so that means 1×(1/2)</li>
|
||
<li>The last "1" on the right side is in the "quarters" position, so that means 1×(1/4)</li>
|
||
<li>So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal</li>
|
||
</ul>
|
||
</div>
|
||
<p class="center large">"There are 10 kinds of people in the world,<br>
|
||
those who understand binary numbers, and those who don't."</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>getQ(1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000);</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="binary-digits.html">Binary Digits</a>
|
||
<a href="decimals.html">Decimal Numbers</a>
|
||
<a href="hexadecimals.html">Hexadecimal Number System</a>
|
||
<a href="binary-decimal-hexadecimal-converter.html">Binary to Decimal to Hexadecimal Converter</a>
|
||
<a href="binary-decimal-hexadecimal.html">Binary, Decimal and Hexadecimal Numbers</a>
|
||
</div>
|
||
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