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                                        <img border="0" src="../../../../../CHAPID=knet-v214aCH47501/RLOID=knet-v214aRLO47520/RIOID=knet-v214aRIO125207/knet/v214aclixsi144/line450.gif" width="450" height="1"></p>
                                    <p class="smtext">
                                        <img border="0" src="../../../../../CHAPID=knet-v214aCH47501/RLOID=knet-v214aRLO47520/RIOID=knet-v214aRIO125207/knet/v214aclixsi144/line450.gif" width="450" height="1"></p>
                                </blockquote>
<p align="left">
                                    <span class="resourceSectionTitle"><span class="resourceRed">Step 1 - Decimal Numbers.</span></span>
                                </p>
                                <blockquote>
                                    <p class="smtext">
                                        <b>Explanation:</b> We are most familiar with "decimal" numbers (Base 10). The decimal numbering system is based on the powers of 10. This exercise will help develop an understanding of the exponentiation or "powers" of numbers using the Base 10 number system. The Base 10 system is what our arithmetic and money system is based on. With Base 10, the right-most position has a value of 1 (same as Base 2). Each position moving to the left is worth 10 times more. 10 to the zero power (10^0) is one, 10 to the first power (10^1 or 10 x 1) is 10, 10 to the second power (10^2 or 10 x 10) is 100 and 10 to the third power (10^3 or 10 x 10 x 10) is 1,000 and so on. Just multiply the number in each position times the value of each position (for example, 400 = 4 x 10^2 or 4 x 100). Remember any number to the zero power is 1.</p>
                                    <p class="smtext">
                                        <b>Decimal Number Conversion Example.</b>
                                    </p>
                                    <p class="smtext">The following chart shows how the decimal number system represents the number 352,481. This will help in understanding the binary numbering system.</p>
                                    <table border="1" cellpadding="2" cellspacing="0" width="81%" bordercolor="#000000">
<tr>
<td class="smtext" colspan="1" rowspan="1"><b>Exponent</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>6</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>5</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>4</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>3</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>2</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>1</sup></td>
<td class="smtext" align="right" colspan="1" rowspan="1">10<sup>0</sup></td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1"><b>Position</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1">7</td>
<td class="smtext" align="right" colspan="1" rowspan="1">6</td>
<td class="smtext" align="right" colspan="1" rowspan="1">5</td>
<td class="smtext" align="right" colspan="1" rowspan="1">4</td>
<td class="smtext" align="right" colspan="1" rowspan="1">3</td>
<td class="smtext" align="right" colspan="1" rowspan="1">2</td>
<td class="smtext" align="right" colspan="1" rowspan="1">1</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1"><b>Value</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1">1000000</td>
<td class="smtext" align="right" colspan="1" rowspan="1">100000</td>
<td class="smtext" align="right" colspan="1" rowspan="1">10000</td>
<td class="smtext" align="right" colspan="1" rowspan="1">1000</td>
<td class="smtext" align="right" colspan="1" rowspan="1">100</td>
<td class="smtext" align="right" colspan="1" rowspan="1">10</td>
<td class="smtext" align="right" colspan="1" rowspan="1">1</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1"><b>Number</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1">0</td>
<td class="smtext" align="right" colspan="1" rowspan="1">3</td>
<td class="smtext" align="right" colspan="1" rowspan="1">5</td>
<td class="smtext" align="right" colspan="1" rowspan="1">2</td>
<td class="smtext" align="right" colspan="1" rowspan="1">4</td>
<td class="smtext" align="right" colspan="1" rowspan="1">8</td>
<td class="smtext" align="right" colspan="1" rowspan="1">1</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1">聽</td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>0 x 1,000,000</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>3 x 100,000</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>5 x 10,000</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>2 x 1,000</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>聽4 x 100</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>8 x 10</b></td>
<td class="smtext" align="right" colspan="1" rowspan="1"><b>1 x 1</b></td>
</tr>
</table>
                                    <p class="smtext">
                                        <b>The number 352,481 if read from right to left would be (1 x 1) + (8 x 10) + (4 x 100) + (2 x 1,000) + (5 x 10,000) + (3 x 100,000) for a total of 352,481 (a six-digit number).</b>
                                    </p>
                                    <p class="smtext">
                                        <b>Here is another way to look at it that makes it easier to add up the decimal number values:</b>
                                    </p>
                                    <table border="1" cellpadding="2" cellspacing="0" width="81%" bordercolor="#000000">
<tr>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="22%"><b>Position of digit (from right)</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="26%"><b>Value of bit position (10^X or ten to the power of)</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="17%"><b>Number value from 0 to 9</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="18%"><b>Calculation</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="18%"><b>Decimal Value</b></td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">1st Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%">10^ 0 or 1</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">1</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">1 x 1</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%">1</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">2nd Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%">10^ 1 or 10</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">8</td>
<td class="smtext" colspan="1" rowspan="1" width="18%"> 8 x 10</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%">80</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">3rd Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%"> 10^ 2 or 100</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">4</td>
<td class="smtext" colspan="1" rowspan="1" width="18%"> 4 x 100</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%"> 400</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">4th Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%">10^ 3 or 1000</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">2</td>
<td class="smtext" colspan="1" rowspan="1" width="18%"> 2 x 1,000</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%">聽2,000</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">5th Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%">10^ 4 or 10000</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">5</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">5 x 10,000</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%">52,000</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%">6th Decimal Digit</td>
<td class="smtext" colspan="1" rowspan="1" width="26%">10^ 5 or 100000</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">3</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">3 x 100,000</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%">300,000</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>Decimal Value (Total of 6 digits)</b> </td>
<td class="smtext" colspan="1" rowspan="1" width="26%">聽</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">聽</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">聽</td>
<td class="smtext" align="right" colspan="1" rowspan="1" width="18%"><b>352,481</b> </td>
</tr>
</table>
                                </blockquote>
<p align="left">
                                    <span class="resourceSectionTitle"><span class="resourceRed">Step 2 - Binary Numbers</span></span>
                                </p>
                                <blockquote>
                                    <p class="smtext">
                                        <b>Explanation:</b> Binary means "two" and each digit in a binary number can only have two values, 0 (zero) or 1. It is also called a Base 2 numbering system. Binary numbers are the key to understanding how routers work and how packets get from one workstation (host) to another server (host) on a TCP/IP network. Internet addresses are made up of 32 bits or four groups of eight bits known as "OCTETS". Each bit of each octet has a value based on its position. Of the eight bits in an octet, the left-most bit is worth 128 (2^7) and the right most bit is worth 1 (2^0). The value of each bit is based on the powers of 2.</p>
                                    <p class="smtext">The binary numbering system is based on the powers of 2. This exercise will help develop an understanding of exponentiation or "powers" of numbers using the Base 2 number system, which is what all computers and data communications use. With Base 2, the right-most position has a value of 1 as with Base 10. Each position moving to the left is worth 2 times more. 2 to the zero power (2^0) is one, 2 to the first power (2^1 or 2 x 1) is 2. 2 to the second power (2^2 or 2 x 2) is 4 and 2 to the third power (2^3 or 2 x 2 x 2) is 8, and so on. Just multiply the number in each position (either a 0 [zero] or a 1) times the value of each position (for example, 8 = 1 x 2^3 or 1 x 8) and add up the total. Remember any number to the zero power is 1. Convert the following binary numbers to decimal numbers. In the first exercise you will convert a binary number to a decimal number. Starting from the right, the first binary digit is a zero which is calculated as zero times 2^0 (2 to the zero power or 0 x 1). Anything to the zero power is 1. The second position from the left is also a zero so this is zero times 2^1 (or 0 x 2). The third binary number from the right is a 1. This is 1 times 2^2 (2 to the 2nd power, or 4).</p>
                                    <p class="smtext">
                                        <b>Binary Number Conversion Example.</b>
                                    </p>
                                    <p class="smtext">The following table shows the detail calculations (starting from the right side) to convert the binary number 10011100 into a decimal number.</p>
                                    <table border="1" cellpadding="2" cellspacing="0" width="81%" bordercolor="#000000">
<tr>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="22%"><b>Position of digit (from right)</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="26%"><b>Value of bit position (two to the power of)</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="17%"><b>Is bit a One (on) or a Zero (Off)</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="18%"><b>Calculation</b></td>
<td class="smtext" align="left" colspan="1" rowspan="1" valign="top" width="18%"><b>Decimal Value</b></td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>1st Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 0 or 1</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">0</td>
<td class="smtext" colspan="1" rowspan="1" width="18%"> 0 x 1</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">0</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>2nd Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 1 or 2</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">0</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">0 x 2</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">0</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>3rd Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 2 or 4</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">1</td>
<td class="smtext" colspan="1" rowspan="1" width="18%"> 1 x 4</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">4</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>4th Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 3 or 8</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">1</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">1 x 8</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">8</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>5th Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 4 or 16</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">1</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">1 x 16</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">16</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>6th Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 5 or 32</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">0</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">0 x 32</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">0</td>
</tr>
<tr>
<td class="smtext" colspan="1" rowspan="1" width="22%"><b>7th Binary Digit</b></td>
<td class="smtext" colspan="1" rowspan="1" width="26%">2^ 6 or 64</td>
<td class="smtext" colspan="1" rowspan="1" width="17%">0</td>
<td class="smtext" colspan="1" rowspan="1" width="18%">0 x 64</td>
<td class="smtext" align="left" colspan="1" rowspan="1" width="18%">0</td>