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<title>Decimals</title>
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<h1 class="center">Decimals</h1>
<div class="center"><span class="larger">A Decimal Number (<i>based on the number <b>10</b></i>) contains a <b>Decimal Point.</b></span></div>
<p>&nbsp;</p>
<div class="example">
<h3>First, let's have an example:</h3>
<p style="float:right; margin: 5px;"><img src="numbers/images/decimal-number.svg" alt="decimal number 45.6" /></p>
<p>Here&nbsp;is&nbsp;the number <span class="large"><i>&quot;forty-five and six-tenths&quot;</i></span> written as a decimal number:</p>
<p>&nbsp;</p>
<p>The decimal point goes between Ones and Tenths.</p>
<p>&nbsp;</p>
<p><b>45.6</b> has 4 Tens, 5 Ones and 6 Tenths, like this:</p>
<p class="center"><img src="numbers/images/decimal-number-circles.svg" alt="decimal number 45.6 = 40 + 5 + 6/10" /></p>
</div>
<p>Now, let's discover how it all works ...</p>
<h2>Place Value</h2>
<p>It is all about <a href="place-value.html">Place Value</a> !</p>
<div class="center80">
<p class="center larger">When we write numbers, the <b>position</b> (or &quot;<b>place</b>&quot;) of each digit<br>
is important.</p>
</div>
<p>In the number <span class="large">327</span>:</p>
<ul>
<li>the &quot;7&quot; is in the <b>Ones</b> position, meaning 7 ones (which is 7),</li>
<li>the &quot;2&quot; is in the <b>Tens</b> position meaning 2 tens (which is twenty),</li>
<li>and the &quot;3&quot; is in the <b>Hundreds</b> position, meaning 3 hundreds.</li>
</ul>
<table style="border: 0; margin:auto;">
<tr>
<td style="text-align:right;"><img src="numbers/images/place-value.svg" alt="Place Value" /></td>
</tr>
<tr>
<td style="text-align:right;">&quot;Three Hundred Twenty Seven&quot;</td>
</tr>
</table>
<table width="100%" border="0">
<tr>
<td width="8%"><span class="larger"><img src="images/style/left-arrow.gif" alt="keft" width="46" height="46" /></span></td>
<td width="92%">As we move left, each position is <span class="large">10 times bigger!</span></td>
</tr>
<tr>
<td>&nbsp;</td>
<td><b>Tens</b> are 10 times bigger than <b>Ones<br>
Hundreds</b> are 10 times bigger than <b>Tens</b></td>
</tr>
</table>
<p class="center">... and ...</p>
<table width="100%" border="0">
<tr>
<td width="92%" align="right">As we move right, each position is <span class="large">10 times <b>smaller</b></span>.</td>
<td width="8%"><span class="larger"><img src="images/style/right-arrow.gif" alt="right" width="46" height="46" /></span></td>
</tr>
<tr>
<td style="text-align:right;">From <b>Hundreds</b>, to <b>Tens</b>, to <b>Ones</b></td>
<td>&nbsp;</td>
</tr>
</table>
<p align="right"></p>
<table style="border: 0; margin:auto;">
<tr>
<td><img src="numbers/images/place-value-smaller.svg" alt="decimals-tenths" /></td>
<td style="width:20px;">&nbsp;</td>
<td><p>But what if we continue past Ones?</p>
<p>What is <b>10 times smaller</b> than Ones?</p>
<p><span class="large"><span class="intbl"><em>1</em><strong>10</strong></span>ths (Tenths) are!</span></p></td>
</tr>
</table>
<p>&nbsp;</p>
<table style="border: 0;">
<tr>
<td>But we must first put a <b>decimal point</b>,<br>
so we know exactly where the Ones position is:</td>
<td>&nbsp;</td>
<td><img src="numbers/images/place-value-tenths.svg" alt="tenths" /></td>
</tr>
<tr>
<td colspan="3" align="right"><p>&quot;three hundred twenty seven <b>and four tenths</b>&quot;</p>
<p>but we usually just say &quot;three hundred twenty seven <b>point four</b>&quot;</p></td>
</tr>
</table>
<p align="center" class="largest">And <b>that</b> is a Decimal Number!</p>
<p>&nbsp;</p>
<p>We can continue with smaller and smaller values, from <b>tenths</b>, to <b>hundredths</b>, and so on, like in this example:</p>
<p class="center"><img src="numbers/images/decimal.svg" alt="decimal 17.591" /><br></p>
<p>Have a play with decimal numbers yourself:</p>
<script>decimalchgMain();</script>
<h2>Large and Small</h2>
<p>So, our Decimal System lets us write numbers as large or as small as
we want, using the decimal point. Digits can be placed to the left
or right of a decimal point, to show values greater than one
or less than one.</p>
<p>The <b>decimal point</b> is the most important part of a Decimal Number. Without it we are lost, and don't know what each position means.</p>
<table style="border: 0; margin:auto;">
<tr>
<td colspan="2" align="right" class="larger"><span class="huge">17</span></td>
<td valign="bottom" class="larger"><img src="images/style/dot-blue.gif" width="14" height="14" alt="dot blue" /></td>
<td colspan="2" class="larger"><span class="huge">591</span></td>
</tr>
<tr>
<td align="right" class="larger">&nbsp;</td>
<td align="right" class="larger">&nbsp;</td>
<td valign="bottom" class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
</tr>
<tr>
<td align="right" class="larger">On the left of the decimal point
is a<br>
<a href="whole-numbers.html">whole number</a> (such as 17)</td>
<td align="right" class="larger"><img src="images/style/left-arrow.gif" width="46" height="46" alt="right arrow" /></td>
<td valign="bottom" class="larger"><img src="images/style/dot-blue.gif" width="14" height="14" alt="dot blue" /></td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
</tr>
<tr>
<td align="center" class="larger">As we move further left,<br>
every place gets <b>10
times bigger</b>.</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
</tr>
<tr>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
</tr>
<tr>
<td align="right" class="larger">&nbsp;</td>
<td align="right" class="larger">&nbsp;</td>
<td align="right" valign="bottom" class="larger"><img src="images/style/dot-blue.gif" width="14" height="14" alt="dot blue" /></td>
<td align="right" class="larger"><img src="images/style/right-arrow.gif" width="46" height="46" alt="right arrow" /></td>
<td class="larger">The first digit on the right means<br>
<b> tenths </b>(1/10).</td>
</tr>
<tr>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td class="larger">&nbsp;</td>
<td align="center" class="larger">As we move further right,<br>
every place
gets <b>10 times smaller</b><br>
(one tenth as big).</td>
</tr>
</table>
<h2>Zoom into decimals ...</h2>
<p class="center">See decimals on the <a href="numbers/number-line-zoom.html">Zoomable Number Line</a><br>
<a href="numbers/number-line-zoom.html"><img src="numbers/images/number-line-decimals.gif" width="400" height="37" border="0" alt="number line decimals" /></a></p>
<h2>Definition of Decimal</h2>
<div class="def">
<p><img src="images/style/blue-speech.gif" style="float:left; margin: 0 10px 5px 0;" width="48" height="41" alt="speech" />The word &quot;Decimal&quot; really means &quot;based on 10&quot; (From Latin <i>decima</i>: <i>a tenth part</i>).</p>
<p>We sometimes
say &quot;decimal&quot; when we mean anything to do with our numbering
system, but a &quot;Decimal Number&quot; usually means there is
a Decimal Point.</p>
</div>
<h2>Ways to think about Decimal Numbers ...</h2>
<h3>... as a Whole Number Plus Tenths, Hundredths, etc</h3>
<p>We can think of a decimal number as a whole number plus tenths, hundredths, etc:</p>
<div class="example">
<h3>Example 1: What is 2.3 ?</h3>
<ul>
<li>On the left side is &quot;2&quot;, that is the whole number
part.</li>
<li>The 3 is in the &quot;tenths&quot; position, meaning &quot;3
tenths&quot;, or 3/10</li>
<li>So, 2.3 is &quot;2 and 3 tenths&quot;</li>
</ul>
</div>
<div class="example">
<h3>Example 2: What is 13.76 ?</h3>
<ul>
<li>On the left side is &quot;13&quot;, that is the whole number
part.</li>
<li>There are two digits on the right side, the 7 is in the &quot;tenths&quot;
position, and the 6 is the &quot;hundredths&quot; position</li>
<li>So, 13.76 is &quot;13 and 7 tenths and 6 hundredths&quot;</li>
</ul>
<div class="simple"> </div>
</div>
<h3>... as a Decimal Fraction</h3>
<p>Or we can think of a decimal number as a Decimal Fraction.</p>
<div class="def">
<p>A Decimal Fraction is a fraction where the denominator (the bottom number) is a number such as 10, 100, 1000, etc (in other words a <a href="exponent.html">power of ten</a>)</p>
</div>
<table style="border: 0; margin:auto;">
<tr>
<td style="text-align:right;">So &quot;2.3&quot; looks like:</td>
<td width="120" align="center"><span class="intbl larger">
<em>23</em>
<strong>10</strong>
</span></td>
</tr>
<tr>
<td style="text-align:right;">&nbsp;</td>
<td width="120" align="center">&nbsp;</td>
</tr>
<tr>
<td style="text-align:right;">And &quot;13.76&quot; looks like:</td>
<td width="120" align="center"><span class="intbl larger">
<em>1376</em>
<strong>100</strong>
</span></td>
</tr>
</table>
<p>&nbsp;</p>
<h3>... as a Whole Number and Decimal Fraction</h3>
<p>Or we can think of a decimal number as a Whole Number plus a Decimal Fraction.</p>
<table style="border: 0; margin:auto;">
<tr>
<td style="text-align:right;">So &quot;2.3&quot; looks like:</td>
<td width="120" align="center"> <span class="larger">2 and</span> <span class="intbl larger">
<em>3</em>
<strong>10</strong>
</span></td>
</tr>
<tr>
<td style="text-align:right;">&nbsp;</td>
<td width="120" align="center">&nbsp;</td>
</tr>
<tr>
<td style="text-align:right;">And &quot;13.76&quot; looks like:</td>
<td width="120" align="center"> <span class="larger">13 and</span> <span class="intbl larger">
<em>76</em>
<strong>100</strong>
</span></td>
</tr>
</table>
<p>Those are all good ways to think of decimal numbers.</p>
<p>&nbsp;</p>
<div class="questions">
<script>getQ(950, 2529, 2530, 2531, 3449, 3450, 949, 1328, 3451, 3452);</script>&nbsp;
</div>
<div class="related">
<a href="decimals-menu.html">Decimals Index</a>
<a href="worksheets/decimals.html">Decimal Worksheets</a>
<a href="index-notation-powers.html">Powers of 10</a>
<a href="rounding-numbers.html">Rounding Numbers</a>
<a href="adding-decimals.html">Adding Decimals</a>
<a href="subtracting-decimals.html">Subtracting Decimals</a>
<a href="multiplying-decimals.html">Multiplying Decimals</a>
<a href="dividing-decimals.html">Dividing Decimals</a>
<a href="converting-decimals-fractions.html">Convert Decimals to Fractions</a>
<a href="converting-fractions-decimals.html">Converting Fractions to Decimals</a>
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