<p>Did you know that the number system you use now is not the only one? Did you know that
computers don't use the numbers we use? Do you know why an hour has 60 minutes? All of questions
and many more find their answers in number bases. This unit will introduce you to numbers bases. </p>
</p>
</p>
<h2>Understanding Exponents</h2>
<p>Before you can understand number bases you need to a basic understanding of an exponent. An
exponent is a way of expressing a number multiplied by itself. There are two parts to a number written
in exponential form, the base and the exponent (base^exponent). This is best explained
using an example:</p>
<p>5³=5x5x5=125</p>
<p>In this example, the base (5) is multiplied by itself the number of times (3) indicated in the exponent
resulting in 125.</p>
<p>Evaluate the following (use your mouse to highlight the area to the right of the equals sign to see the
solution):</p>
<p>2²=<span style="color: #ffffff;">4</span></p>
<p>10³=<span style="color: #ffffff;">1000</span></p>
<p><span style="color: #000000;">52¹=<span style="color: #ffffff;">52</span></span></p>
<p><span style="color: #000000;">2⁵=<span style="color: #ffffff;">32</span></span></p>
<p><span style="color: #000000;">The one exponent that is special is zero. Any number raised to the
zero exponent is equal to 1. This is explained in the following video:</span></p>
<p><object height="315" width="420"><param name="movie"
value="//www.youtube.com/v/b9q24AS2mR0?version=3&amp;hl=en_US" /><param
name="allowFullScreen" value="true" /><param name="allowscriptaccess" value="always" /><embed
src="https://www.youtube.com/v/b9q24AS2mR0?version=3&amp;hl=en_US" type="application/xshockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="315" width="420"
/></object></p>
<p><span style="color: #000000;">Additional information on exponents can be found on the <a
href="http://www.mathsisfun.com/exponent.html">Math is Fun</a> website or you can watch
following video:</span></p>
<p><iframe src="https://www.youtube.com/embed/spa_dfxxlp4" allowfullscreen="" frameborder="0"
height="315" width="560"></iframe></p>
<p></p>
<h2>Place Value</h2>
<p>A second key idea for understanding number bases is place value. In the number system we
normally use the "2" in the following numbers are not of equal value. </p>
<table border="0">
<tbody>
<tr>
<td>Number</td>
<td>Value of the "2"</td>
</tr>
<tr>
<td>2</td>
<td>Two (2)</td>
</tr>
<tr>
<td>29</td>
<td>Twenty (20)</td>
</tr>
<tr>
<td>247</td>
<td>Two Hundred (200)</td>
</tr>
<tr>
<td>124 358</td>
<td>Twenty Thousand (20000)</td>
</tr>
</tbody>
</table>
<p></p>
<p>We could therefore look at the "2" in any position as equal to two times the value of place it
occupies. Rewriting our table it now looks like:</p>
<table border="0">
<tbody>
<tr>
<td>Number</td>
<td>Value of the "2"</td>
</tr>
<tr>
<td>2</td>
<td>2X1 </td>
</tr>
<tr>
<td>29</td>
<td>2x10</td>
</tr>
<tr>
<td>247</td>
<td>2x100</td>
</tr>
<tr>
<td>124 358</td>
<td>2x10000<br /><br /></td>
</tr>
</tbody>
</table>
<p></p>
<p>It is also important to note that a digit must be placed in each position even if that position has no
values. This is best explained by an example. Let's look at the number 203. The number in this case is
two hundred three. However, if the zero is omitted the number becomes 23 which is twenty three. A
totally different number.</p>
<p>You may ask why the number 203 is not written as 0203 to show there is no value in the thousands
place. By convention it is assumed the numbers to the left of any digit greater than one are zero unless
stated otherwise. If this convention were not followed an infinite number of zeros would need to placed
to the left of 203. This is the case for all number bases.</p>
<p>Now let's tie the ideas of exponents and place value together by looking at scientific notation. </p>
<h2>Scientific Notation</h2>
<p>Scientific notation is a way of writing numbers that are very large or very small. Numbers expressed
using scientific notation can be expressed as <em>a x 10^b. </em></p>
<p>The following video explains scientific notation and why it is used</p>
<p></p>
<p><object height="315" width="560"><param name="movie"
value="//www.youtube.com/v/WwmJ5nMmigQ?hl=en_US&amp;version=3" /><param
name="allowFullScreen" value="true" /><param name="allowscriptaccess" value="always" /><embed
src="https://www.youtube.com/v/WwmJ5nMmigQ?hl=en_US&amp;version=3" type="application/xshockwave-flash" allowscriptaccess="always" allowfullscreen="true" height="315" width="560"
/></object></p>
<table border="0">
<tbody>
<tr>
<td>Number</td>
<td>Value of the "2"</td>
<td>Scientific Notation</td>
</tr>
<tr>
<td>2</td>
<td>2X1 </td>
<td>2x10⁰</td>
</tr>
<tr>
<td>29</td>
<td>2x10</td>
<td>2.9x10¹^{<br />}</td>
</tr>
<tr>
<td>247</td>
<td>2x100</td>
<td>2.47x10²</td>
</tr>
<tr>
<td>124 358</td>
<td>2x10000<br /><br /></td>
<td>1.24358x10^{<span>7</span>}</td>
</tr>
</tbody>
</table>
<p>Using what we know about place value we know that a number like 526 is actually 500+20+6 which
can be written as 5x10² + 2x10¹+6x10⁰. It is also important to
note that each position MUST have a numeral. This means that with a number like 203 the "0" actually
means "no tens" or can be written as 2x10² + 0x10¹+3x10⁰. If
you understand these ideas you are ready to express numbers using different bases.</p>
<h2><strong>Base 2 (Binary Numbers)<br /></strong></h2>
<p>Lets have a look at base 10 and base 2. One of the key things to keep in mind is that you can not
have more digits than the number of digits in the base. This is true for any base. Therefore for base 10
there are ten possible digits, for base 5 there are five possible digits and for base 2 there are two
possible digits. Read on to find out more.</p>
<p>Using base 10 there are 10 possible digits we can use: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 for each of the place
values. Therefore the largest single digit we can have 9. The largest possible 2 digit number is 99, the
largest possible 3 digit number is 999 and so on.</p>
<p>How does this work for other bases?</p>
<p>Using base 2 there are two possible digits: 0, 1. Therefore the largest possible singe digit number is
"1". The largest possible 2 digit number is "11" which is bigger than the other two alternatives of "10"
and "01". But what does the "11" mean in this case? To explain this we need to use what we know about
powers. Using the table below we can see that each position has a value.</p>
<p><strong>Table 1</strong>: Base 10 to Base 2 Conversion Table</p>
<table border="3">
<tbody>
<tr>
<td>
<p>Base 10</p>
<p>Exponential Form</p>
</td>
<td>10⁸</td>
<td>10⁷</td>
<td>10⁶</td>
<td>10⁵</td>
<td>10⁴</td>
<td>10³</td>
<td>10²</td>
<td>10¹</td>
<td>10⁰</td>
</tr>
<tr>
<td>Value</td>
<td>100 000 000</td>
<td>10 000 000</td>
<td>1 000 000</td>
<td>100 000</td>
<td>10 000</td>
<td>1000</td>
<td>100</td>
<td>10</td>
<td>0</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>
<p>Base 2</p>
<p>Exponential Form</p>
</td>
<td>2⁸</td>
<td>2⁷</td>
<td>2⁶</td>
<td>2⁵</td>
<td>2⁴</td>
<td>2³</td>
<td>2²</td>
<td>2¹</td>
<td>2⁰</td>
</tr>
<tr>
<td>Value</td>
<td>256</td>
<td>128</td>
<td>64</td>
<td>32</td>
<td>16</td>
<td>8</td>
<td>4</td>
<td>2</td>
<td>1</td>
</tr>
</tbody>
</table>
<p></p>
<p>In base 10 we are used to the second position from the right being worth the number times ten. In
base 2 it is the value of the number times 2. So a number like "11" is worth a different amount
depending on the base. For example:</p>
<p>"11" in base 10 is (1x10¹)+(1x10⁰)=10+1=11</p>
<p>"11" in base 2 = (1x2¹)+(1x2⁰) = 2+1=3 in base 10</p>
<p>What would the following numbers be equal to in base 2 (use your mouse to highlight the area to
the right of the equals sign to see the solution):</p>
<p>111111111 = <span style="color: #ffffff;">(1x2</span><span style="color:
#ffffff;">⁸</span><span style="color: #ffffff;">)+</span><span style="color:
#ffffff;">(1x2⁷)+(1x2⁶)+(1x2⁵)+(1x2⁴)+(1x2<sup>3</
sup>)+(1x2²)+(1x2¹)+(1x2⁰) =
256+128+64+32+16+8+4+2+1=511</span></p>
<p>111011110 = <span style="color:
#ffffff;">(1x2⁸)+(1x2⁷)+(0x2⁶)+(1x2⁵)+(1x2<sup>4</
sup>)+(1x2³)+(1x2²)+(1x2¹)+(0x2⁰) =
256+128+64+32+16+8+4+2+1=446</span></p>
<p>110110101 = <span style="color:
#ffffff;">(1x2⁸)+(0x2⁷)+(1x2⁶)+(1x2⁵)+(0x2<sup>4</
sup>)+(1x2³)+(1x2²)+(0x2¹)+(1x2⁰) = the
256+128+64+32+16+8+4+2+1= 365</span></p>
<h2>Computers and Binary Numbers</h2>
<p>Curious how computers use binary numbers? Do you know your bits from your bytes? Check out the
video below find out how computers use binary to store information.</p>
<p><a href="http://www.youtube.com/watch?v=p7bwd8cQgOo">Explaining Binary Numbers and
Computing</a></p>
<p>Bit is a shortened version of the word BInary digiT. For a binary number there are only 2 possible
values 0 or 1 and therefore there are only two types of bits, 1 and 0.</p>
<p>Bytes are composed of bits usually in groups of eight. In the case of a 64 bit computer there would b
64 bits or 8 bytes.</p>
<h2>Check Your Understanding</h2>
<p>Test your understanding by answering the following questions:</p>
<p>1) What is the largest base 10 number than can be expressed using 8 digits in base 2?</p>
<p>2) In base 10, one million can be expressed using 7 digits. How many digits are needed to express a
million in base 2?</p>
<p>3) Mr. Williams said he could look at any number expressed in base 2 and determine instantly if it
was an even or odd number. Is this possible? If so how does he do it? If not how do you know?</p>
<p>4) Many computers express "characters" using a version of the binary (base 2) code called ASCII. This
is an example of 8 bit binary code which is explained on the <a
href="http://computer.howstuffworks.com/bytes1.htm">How Stuff Works</a> website. In this
computer code the first 32 numbers are reserved for commands the computer can understand. As a
result the number 0 in base 10 is not expressed as "00000000" as we would expect in binary. Examine
the table below. Can you tell what the binary number "00000000" would represent? How would you
represent 0? How would you spell your first name using ASCII code?</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/8/85/ASCII_Code_ChartQuick_ref_card.jpg" height="419" width="579" /></p>
<p>5) ?</p>
<p><strong>Other Bases</strong></p>
<p>Numbers can be written using other bases. A couple points to keep in mind:</p>
<p>1) Each base only has as many numbers as the base. So base 10 has 10 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9),
base 2 has 2 (0,1), base 3 has 3 (0,1,2), etc.</p>
<p>2) Each position (place value) must have a numeral in it.</p>
<p>3) A table like the one above for converting from base 10 to base 2 can help you convert numbers
from one base to another.</p>
<p></p>
<p>Using what you have learned try to write the following numbers from base 10 to base 3 and base
5:</p>
<table border="3">
<tbody>
<tr>
<td>Base 10</td>
<td>Base 3</td>
<td>Base 5</td>
</tr>
<tr>
<td>123</td>
<td></td>
<td></td>
</tr>
<tr>
<td>683</td>
<td></td>
<td></td>
</tr>
<tr>
<td>104</td>
<td></td>
<td></td>
</tr>
<tr>
<td>999</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p></p>
<p><strong>Extension Questions:</strong></p>
<p>1) Cultures through the centuries have used other number bases. Conduct some research to find
some examples?</p>
<p>2) It has been said that the way we tell time is based on number bases. Conduct some research to
explain why this statement is true or false.</p>
<p>3) Computers do all calculations using base 2. Use what you know about addition using base 2 to
explain addition in base 2.</p>
<p></p>
<p></p>
<p></p>