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412 lines
16 KiB
HTML
412 lines
16 KiB
HTML
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<head>
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<title>Binary, Decimal and Hexadecimal Numbers</title>
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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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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Binary, Decimal and Hexadecimal Numbers</h1>
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<h2>Decimals</h2>
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<p>How <i>do</i> <a href="decimals.html">Decimal Numbers</a> work?</p>
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<p>Every digit in a decimal number has a "position", and the <b>decimal point</b> helps us to know which position is which: </p>
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<p align="center"><img src="numbers/images/decimal.svg" alt="decimal 17.591" /></p>
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<p>The position <i>just to the left</i> of the point is the "Ones" position. If we see a "7" there we know it means 7 ones.</p>
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<p>Every position further to the left is 10 times bigger, and every position further to the right is 10 times smaller</p>
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<p>This is <b> just a way of writing down a value</b>. Other ways include <a href="roman-numerals.html">Roman Numerals</a>, <a href="binary-number-system.html">Binary</a>, <a href="hexadecimals.html">Hexadecimal</a>, and more. You could even just draw dots on a sheet of paper!</p>
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<h2>Bases</h2>
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<p>The Decimal Number System is also called "Base 10", because it is based on the number 10, with these 10 symbols:</p>
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<p class="center large">0, 1, 2, 3, 4, 5, 6, 7, 8 and 9</p>
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<p>But notice something interesting: <b>there is no symbol for "ten"</b>. "10" is actually two symbols put together, a "1" and a "0":</p>
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<div class="center80">
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<p>In decimal you count "0,1,2,3,4,5,6,7,8,9,..." but then you run out of symbols!</p>
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<p>So you add <b>1 on the left</b> and then <b>start again at 0</b>: 10,11,12, ...</p>
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</div>
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Like this:
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td> </td>
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<td align="right" class="large">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 align="right" class="large">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 align="right" class="large">2</td>
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<td> </td>
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<td>Then 2</td>
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</tr>
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<tr>
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<td> </td>
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<td align="right" class="large">⋮</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 align="right" class="large">9</td>
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<td> </td>
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<td>Up to 9</td>
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</tr>
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<tr>
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<td>••••••••••</td>
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<td align="right" class="large"><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>••••••••••<br>
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•</td>
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<td align="right" class="large">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>••••••••••<br>
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••</td>
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<td align="right" class="large">12</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 align="right" class="large">⋮</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>••••••••••<br>
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•••••••••</td>
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<td align="right" class="large">19</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>••••••••••<br>
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••••••••••</td>
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<td align="right" class="large"><span class="hilite">2</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>••••••••••<br>
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••••••••••<br>
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•</td>
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<td align="right" class="large">21</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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</table>
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</div>
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<h2>Counting with Different Number Systems</h2>
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<p>But you don't <b>have to</b> use 10 as a "Base". You could use 2 ("Binary"), 16 ("Hexadecimal"), or any number you want to!</p>
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<div class="center80">
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<p>Example: In binary you count "0,1,..." but then you run out of symbols!</p>
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<p>So you add <b>1 on the left</b> and then <b>start again at 0</b>: 10,11 ...</p>
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</div>
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<p> </p>
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<p>See how to count dots using Bases from 2 to 16 (press Play button):</p>
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<script type="text/javascript">numberodometerMain(2);</script>
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<p> </p>
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<p style="float:right; margin: 0 0 9px 20px;"><img src="numbers/images/number-odometer1.gif" width="402" height="114" alt="number odometer" /><br>
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<b>Example: 1×16 + 1×8 + 1×1 = 16+8+1 = 25</b></p>
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<p> </p>
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<p><i>Try this:</i> select a Base, watch it count for a while, then press "||" (Pause). Now see if it has tallied the right number of dots, as in this example using base 2.</p>
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<p> </p>
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<p>So the general rule is:</p>
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<p class="center80"> Count up until just before the "Base Number", then start at 0 again, but first you add 1 to the number on your left.</p>
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<h2>Binary Numbers</h2>
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<p><a href="binary-number-system.html">Binary Numbers</a> are just "Base 2" instead of "Base 10". So you start counting at 0, then 1, then you run out of digits ... so you start back at 0 again, but increase the number on the left by 1.</p>
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<p>Like this:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td> </td>
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<td align="right" class="large">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 align="right" class="large">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 align="right" class="large"><span class="hilite">1</span>0</td>
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<td> </td>
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<td>there is no "2" in binary, so start back at 0 ...<br />
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... and add one to the number 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 align="right" class="large">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 align="right" class="large"><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 align="right" class="large">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 align="right" class="large">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 align="right" class="large">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 align="right" class="large"><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 align="right" class="large">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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</table>
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</div>
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<h2>Hexadecimal Numbers</h2>
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<p><a href="hexadecimals.html">Hexadecimal numbers</a> are interesting. There are 16 of them! </p>
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<p>They look the same as the decimal numbers up to 9, but then there are the letters ("A',"B","C","D","E","F") in place of the decimal numbers 10 to 15. </p>
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<p>So a single Hexadecimal digit can show 16 different values instead of the normal 10 like this:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<th width="150">Decimal:</th>
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<td width="21">0</td>
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<td width="21">1</td>
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<td width="21">2</td>
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<td width="21">3</td>
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<td width="21">4</td>
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<td width="21">5</td>
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<td width="21">6</td>
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<td width="21">7</td>
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<td width="21">8</td>
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<td width="21">9</td>
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<td width="21">10</td>
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<td width="21">11</td>
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<td width="21">12</td>
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<td width="21">13</td>
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<td width="21">14</td>
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<td width="21">15</td>
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</tr>
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<tr align="center">
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<th width="150">Hexadecimal:</th>
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<td width="21">0</td>
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<td width="21">1</td>
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<td width="21">2</td>
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<td width="21">3</td>
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<td width="21">4</td>
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<td width="21">5</td>
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<td width="21">6</td>
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<td width="21">7</td>
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<td width="21">8</td>
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<td width="21">9</td>
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<td width="21">A</td>
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<td width="21">B</td>
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<td width="21">C</td>
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<td width="21">D</td>
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<td width="21">E</td>
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<td width="21">F</td>
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</tr>
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</table>
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</div>
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<p>And we count in Hexadecimal like this:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td> </td>
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<td align="right" class="large">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 align="right" class="large">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 align="right" class="large">2</td>
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<td> </td>
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<td>Then 2</td>
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</tr>
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<tr>
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<td> </td>
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<td align="right" class="large">⋮</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>••••••••••<br>
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•••••</td>
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<td align="right" class="large">F</td>
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<td> </td>
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<td>Up to F</td>
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</tr>
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<tr>
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<td>••••••••••<br>
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••••••</td>
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<td align="right" class="large"><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>••••••••••<br>
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•••••••</td>
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<td align="right" class="large">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>••••••••••<br>
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••••••••</td>
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<td align="right" class="large">12</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 align="right" class="large">⋮</td>
|
|
<td> </td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr>
|
|
<td>••••••••••<br>
|
|
••••••••••<br>••••••••••<br>
|
|
•</td>
|
|
<td align="right" class="large">1F</td>
|
|
<td> </td>
|
|
<td> </td>
|
|
</tr>
|
|
<tr>
|
|
<td>••••••••••<br>
|
|
••••••••••<br>
|
|
••••••••••<br>
|
|
••</td>
|
|
<td align="right" class="large"><span class="hilite">2</span>0</td>
|
|
<td> </td>
|
|
<td>Start back at 0 again, but add 1 on the left</td>
|
|
</tr>
|
|
<tr>
|
|
<td>••••••••••<br>
|
|
••••••••••<br>
|
|
••••••••••<br>
|
|
•••</td>
|
|
<td align="right" class="large">21</td>
|
|
<td> </td>
|
|
<td>And so on!</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<p> </p>
|
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<p> </p>
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<script type="text/javascript">getQ(2001, 2002, 2003, 2004, 2005, 2006);</script>
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<a href="games/hex-drums.html">Hexadecimal Drum Machine</a>
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