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NUMBER SYSTEMS!
<html>
For anyone who was confused by the lesson on “Number Systems”, here are some extra examples to help you all to understand the seemingly complex, yet very basic, logic behind it: <font face=“pa”> <table border=“1” cellspacing=“0” bordercolor=“#000000” cellpadding=“3” width=“400” align=“center”>
<caption>Base Correlation Chart</caption>
<tr>
<td align="middle" bgcolor="#94dafc"><em>Binary</em> Base <strong>2</strong> </td>
<td align="middle" bgcolor="#d7e6e4"><em>Octal</em> Base <strong>8</strong> </td>
<td align="middle" bgcolor="#94dafc"><em>Decimal</em> Base <strong>10</strong> </td>
<td align="middle" bgcolor="#d7e6e4"><em>Hexacedimal</em> Base <strong>16</strong> </td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">0</td>
<td align="right" bgcolor="#fefeda">0</td>
<td align="right" bgcolor="#e0f4fe">0</td>
<td align="right" bgcolor="#fefeda">0</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1</td>
<td align="right" bgcolor="#fefeda">1</td>
<td align="right" bgcolor="#e0f4fe">1</td>
<td align="right" bgcolor="#fefeda">1</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10</td>
<td align="right" bgcolor="#fefeda">2</td>
<td align="right" bgcolor="#e0f4fe">2</td>
<td align="right" bgcolor="#fefeda">2</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11</td>
<td align="right" bgcolor="#fefeda">3</td>
<td align="right" bgcolor="#e0f4fe">3</td>
<td align="right" bgcolor="#fefeda">3</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100</td>
<td align="right" bgcolor="#fefeda">4</td>
<td align="right" bgcolor="#e0f4fe">4</td>
<td align="right" bgcolor="#fefeda">4</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">101</td>
<td align="right" bgcolor="#fefeda">5</td>
<td align="right" bgcolor="#e0f4fe">5</td>
<td align="right" bgcolor="#fefeda">5</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">110</td>
<td align="right" bgcolor="#fefeda">6</td>
<td align="right" bgcolor="#e0f4fe">6</td>
<td align="right" bgcolor="#fefeda">6</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">111</td>
<td align="right" bgcolor="#fefeda">7</td>
<td align="right" bgcolor="#e0f4fe">7</td>
<td align="right" bgcolor="#fefeda">7</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1000</td>
<td align="right" bgcolor="#fefeda">10</td>
<td align="right" bgcolor="#e0f4fe">8</td>
<td align="right" bgcolor="#fefeda">8</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1001</td>
<td align="right" bgcolor="#fefeda">11</td>
<td align="right" bgcolor="#e0f4fe">9</td>
<td align="right" bgcolor="#fefeda">9</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1010</td>
<td align="right" bgcolor="#fefeda">12</td>
<td align="right" bgcolor="#e0f4fe">10</td>
<td align="right" bgcolor="#fefeda">A</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1011</td>
<td align="right" bgcolor="#fefeda">13</td>
<td align="right" bgcolor="#e0f4fe">11</td>
<td align="right" bgcolor="#fefeda">B</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1100</td>
<td align="right" bgcolor="#fefeda">14</td>
<td align="right" bgcolor="#e0f4fe">12</td>
<td align="right" bgcolor="#fefeda">C</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1101</td>
<td align="right" bgcolor="#fefeda">15</td>
<td align="right" bgcolor="#e0f4fe">13</td>
<td align="right" bgcolor="#fefeda">D</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1110</td>
<td align="right" bgcolor="#fefeda">16</td>
<td align="right" bgcolor="#e0f4fe">14</td>
<td align="right" bgcolor="#fefeda">E</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">1111</td>
<td align="right" bgcolor="#fefeda">17</td>
<td align="right" bgcolor="#e0f4fe">15</td>
<td align="right" bgcolor="#fefeda">F</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10000</td>
<td align="right" bgcolor="#fefeda">20</td>
<td align="right" bgcolor="#e0f4fe">16</td>
<td align="right" bgcolor="#fefeda">10</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10001</td>
<td align="right" bgcolor="#fefeda">21</td>
<td align="right" bgcolor="#e0f4fe">17</td>
<td align="right" bgcolor="#fefeda">11</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10010</td>
<td align="right" bgcolor="#fefeda">22</td>
<td align="right" bgcolor="#e0f4fe">18</td>
<td align="right" bgcolor="#fefeda">12</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10011</td>
<td align="right" bgcolor="#fefeda">23</td>
<td align="right" bgcolor="#e0f4fe">19</td>
<td align="right" bgcolor="#fefeda">13</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10100</td>
<td align="right" bgcolor="#fefeda">24</td>
<td align="right" bgcolor="#e0f4fe">20</td>
<td align="right" bgcolor="#fefeda">14</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10101</td>
<td align="right" bgcolor="#fefeda">25</td>
<td align="right" bgcolor="#e0f4fe">21</td>
<td align="right" bgcolor="#fefeda">15</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10110</td>
<td align="right" bgcolor="#fefeda">26</td>
<td align="right" bgcolor="#e0f4fe">22</td>
<td align="right" bgcolor="#fefeda">16</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">10111</td>
<td align="right" bgcolor="#fefeda">27</td>
<td align="right" bgcolor="#e0f4fe">23</td>
<td align="right" bgcolor="#fefeda">17</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11000</td>
<td align="right" bgcolor="#fefeda">30</td>
<td align="right" bgcolor="#e0f4fe">24</td>
<td align="right" bgcolor="#fefeda">18</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11001</td>
<td align="right" bgcolor="#fefeda">31</td>
<td align="right" bgcolor="#e0f4fe">25</td>
<td align="right" bgcolor="#fefeda">19</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11010</td>
<td align="right" bgcolor="#fefeda">32</td>
<td align="right" bgcolor="#e0f4fe">26</td>
<td align="right" bgcolor="#fefeda">1A</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11011</td>
<td align="right" bgcolor="#fefeda">33</td>
<td align="right" bgcolor="#e0f4fe">27</td>
<td align="right" bgcolor="#fefeda">1B</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11100</td>
<td align="right" bgcolor="#fefeda">34</td>
<td align="right" bgcolor="#e0f4fe">28</td>
<td align="right" bgcolor="#fefeda">1C</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11101</td>
<td align="right" bgcolor="#fefeda">35</td>
<td align="right" bgcolor="#e0f4fe">29</td>
<td align="right" bgcolor="#fefeda">1D</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11110</td>
<td align="right" bgcolor="#fefeda">36</td>
<td align="right" bgcolor="#e0f4fe">30</td>
<td align="right" bgcolor="#fefeda">1E</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">11111</td>
<td align="right" bgcolor="#fefeda">37</td>
<td align="right" bgcolor="#e0f4fe">31</td>
<td align="right" bgcolor="#fefeda">1F</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100000</td>
<td align="right" bgcolor="#fefeda">40</td>
<td align="right" bgcolor="#e0f4fe">32</td>
<td align="right" bgcolor="#fefeda">20</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100001</td>
<td align="right" bgcolor="#fefeda">41</td>
<td align="right" bgcolor="#e0f4fe">33</td>
<td align="right" bgcolor="#fefeda">21</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100010</td>
<td align="right" bgcolor="#fefeda">42</td>
<td align="right" bgcolor="#e0f4fe">34</td>
<td align="right" bgcolor="#fefeda">22</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100011</td>
<td align="right" bgcolor="#fefeda">43</td>
<td align="right" bgcolor="#e0f4fe">35</td>
<td align="right" bgcolor="#fefeda">23</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100100</td>
<td align="right" bgcolor="#fefeda">44</td>
<td align="right" bgcolor="#e0f4fe">36</td>
<td align="right" bgcolor="#fefeda">24</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100101</td>
<td align="right" bgcolor="#fefeda">45</td>
<td align="right" bgcolor="#e0f4fe">37</td>
<td align="right" bgcolor="#fefeda">25</td></tr>
<tr>
<td align="right" bgcolor="#e0f4fe">100110</td>
<td align="right" bgcolor="#fefeda">46</td>
<td align="right" bgcolor="#e0f4fe">38</td>
<td align="right" bgcolor="#fefeda">26</td></tr></table>
</font>
Let's use the example of the Octal value 24. Why is Octal 24 equal to the number 20 in our typical “base 10” lives? Because equivalent values existing in the <i>different</i> bases are… well… <i>different</i>. There is no existant “8”, or “9” digit in an Octal system, therefore we lose the ability to use those digits. “19” and “98”, for example, can not exist without having access to the digits 8 and 9. This means that we need to “carry” or extend our work to the next column over when we reach this imaginary “ceiling”, if you will, of the base. <br><br>
Now lets try switching bases, but preserving the <i>SAME</i> value (we'll work in base 10 since that is what most people are most familiar with). This can be done by recognizing that a “carry” must happen once we have reached our maximum digit in any given base but still need to continue counting. With Base 10 we do this after we reach the digit 9. We increase the value of the left-most value by one to continue counting upward.<br><br>
Here's an Example:<br><br>
When dealing with the digit 9410 (Base 10), we can represent any values <i>preceeding</i> the 9 as 0's. The number 000094 is the same as just plain 94. If placed in front of something, 0 has <i>NO</i> value.<br><br>
Understanding this, we should be able to recognize that once we reach 99, and need to continue, we will be able to roll one of the imaginary leading zeros up to our next digit and reset everything that follows.<br><br>
Counting upward in Base 10:<br><br> 000094<br> 000095<br> 000096<br> 000097<br> 000098<br> 000099<br> 000100 - We simply call this 100, and typically ignore the laws behind why it's a 1 followed by two 0's (The same<br> 000101 goes for 10, or 1,000, 10,000, and so on…)<br> 000102<br><br>
We're going to increase the value of our column to the left by 1 factor each time we reach our maximum in the column to it's right. If we apply this concept while counting in any base then we should be able to list the numbers of any number system consecutively. Becoming familiar is simply a matter of practice. </html>
From Tom: I also found a nice conversion chart at http://www.interfacebus.com/Glossary-of-Terms-gray-code-circuit.html.
For all the byte enthusiasts, here's a pretty cool explanation chart for byte sizes in binary and decimal: http://www.zdnet.com/blog/storage/snow-leopard-fixes-disk-capacity-bug/589.
And finally, a nice conversion chart from ASCII to binary (scroll down page a “bit” :) — http://www.cs.ndsu.nodak.edu/~adenton/ExpandingHorizons/EH2005/Expanding%20Horizons%2005.html.
