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class="page_title"> Prior Knowledge - Sets of Numbers <a href="#" class="mark-page-favorite pull-right" data-pid="1058" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1057/number-and-algebra.html">Number and Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Prior Knowledge - Sets of Numbers</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <h2><img alt="" src="../../files/NA/Setsofnumbers/SN-1.jpg" style="width: 150px; height: 157px; float: left; border-width: 0px; border-style: solid;">The four subsets</h2> <p>This sub topic is about the 4 key sets of numbers that define our number system. Natural numbers, Integers, Rational numbers and Real numbers. This section starts by taking you through the concept of number sets, starting with sets that we have been using for years, like even numbers. Then there is a little about how the sets have evolved followed by some key definitions of the four sets and example of how they fit together. Then, as usual, the section finishes with some revision exercises asnd examples of how this concept is tested.</p> <hr> <h2>What are sets of numbers?</h2> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/135549984"></iframe></div> <div id="carousel-5" class="dynamic-gallery carousel slide" data-id="5"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="media/std-galleries/3-5/1.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="media/std-galleries/3-5/1.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/1b.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/1b.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/4.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/4.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/5.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/5.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/6.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/6.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/7.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/7.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/8.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/8.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-5/9.jpg" data-fancybox="gallery-5" title="" data-caption=""><img alt="" src="../../../std-galleries/3-5/9.jpg"></a></div></div><a class="left carousel-control" href="#carousel-5" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-5" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="media/std-galleries/3-5/1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/3-5/1b-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/3-5/4-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/3-5/5-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/3-5/6-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/3-5/7-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/3-5/8-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/3-5/9-thumb128.jpg"><li></li></ol> <hr> <h2>Defining the sets and subsets!</h2> <p>In the video below, you will get some clear definitions of the 4 subsets and how they fit together this should help you to be clear about these key properties of numbers and how to classify them.</p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/135549985"></iframe></div> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div id="carousel-6" class="dynamic-gallery carousel slide" data-id="6"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/3-6/4.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/4.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-6/5.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/5.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-6/6.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/6.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-6/7.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/7.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-6/8.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/8.jpg"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/3-6/9.jpg" data-fancybox="gallery-6" title="" data-caption=""><img alt="" src="../../../std-galleries/3-6/9.jpg"></a></div></div><a class="left carousel-control" href="#carousel-6" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-6" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/3-6/4-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/3-6/5-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/3-6/6-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/3-6/7-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/3-6/8-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/3-6/9-thumb128.jpg"><li></li></ol> </section> <h2><img alt="" src="../../files/NA/Setsofnumbers/SN-1.jpg" style="width: 80px; height: 84px; float: left;">Revise</h2> <p>This section offers a brief summary of the key points and then a quiz to help you check that you know what you need to know. Click the 'hidden box symbol' to see this section</p> <hr class="hidden-separator"> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Read through and check the following flashcards. You can come back to these as often as you like.</p> <p>Gallery</p> <p>Complete all of the questions in the following quiz before checking your answers. Check how you did and then read the help offered for any questions you didn't get. You can redo the quiz at anytime to check yourself. <em>(The order of the answers is reshuffled when the page is reloaded)</em></p> <div class="tib-quiz" data-stats="15-311-1058"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following best describes the set of natural numbers?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> All positive whole numbers</label></p><p><label class="checkbox"><input type="checkbox"> All positive numbers</label></p><p><label class="checkbox"><input type="checkbox"> All whole numbers</label></p></div><div class="q-explanation"><p>The set of Natural numbers is all <em><strong>positive, whole</strong></em> numbers. This does not include any negative numbers or fractions.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following best describes the set of rational numbers?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> All fractions and decimals</label></p><p><label class="checkbox"><input type="checkbox"> All numbers that aren't whole</label></p><p><label class="checkbox"><input class="c" type="checkbox"> All numbers that can be expressed as fractions</label></p></div><div class="q-explanation"><p>Rational numbers are all the numbers that be expressed as whole. This includes all of the integers because they can all be expressed as fractions too. eg <img align="middle" alt="4 equals 12 over 3" class="Wirisformula" data-mathml="«math style=¨font-family:Arial¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mo»=«/mo»«mfrac»«mn»12«/mn»«mn»3«/mn»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-18.php?formula=7f82a4f1f7a88d7bda700813c0a661ab.png"> . Not all decimals can be expressed as fractions. Irrational numbers are decimals whose places go on infinitely without any discernable pattern, so that cannot be expressed as fractions.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following statements are true about the set of irrational numbers ?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Recurring decimals are rational</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Some decimals are rational</label></p><p><label class="checkbox"><input class="c" type="checkbox"> All integers are rational</label></p><p><label class="checkbox"><input type="checkbox"> All real numbers are rational</label></p><p><label class="checkbox"><input type="checkbox"> All decimals are irrational</label></p><p><label class="checkbox"><input class="c" type="checkbox"> All fractions are rational</label></p><p><label class="checkbox"><input type="checkbox"> All decimals are rational</label></p></div><div class="q-explanation"><p>Not all decimals can be expressed as fractions. Irrational numbers are decimals whose places go on infinitely without any discernable pattern, so that cannot be expressed as fractions.</p><p>All rational numbers are real, but not the other way around.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following statements are true?</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Natural numbers are both integers and rational</label></p><p><label class="checkbox"><input type="checkbox"> All rational numbers are natural</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Integers are both rational and real</label></p><p><label class="checkbox"><input type="checkbox"> All integers are natural numbers</label></p><p><label class="checkbox"><input class="c" type="checkbox"> All natural numbers are integers</label></p><p><label class="checkbox"><input class="c" type="checkbox"> All integers are real numbers</label></p></div><div class="q-explanation"><p>Check the subsets digram to review these statements</p><p style="text-align: center;"><img alt="" src="../../files/NA/Setsofnumbers/5.jpg" style="width: 500px; height: 370px;"></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sets does 45.643 belong in?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Natural numbers</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Rational Numbers</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Real numbers</label></p><p><label class="checkbox"><input type="checkbox"> Integers</label></p></div><div class="q-explanation"><p>45.643 is neither natural or an integer because it is not a whole number</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following sets does <img align="middle" alt="straight pi" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-22.php?formula=6dae009a0e3a5eec27fc1b747bab3923.png"> belong in?</p></div><div class="q-answer"><p> <label class="checkbox"><input type="checkbox"> Rational Numbers</label></p><p> <label class="checkbox"><input type="checkbox"> Natural numbers</label></p><p> <label class="checkbox"><input type="checkbox"> Integers</label>Integers</p><p> <label class="checkbox"><input class="c" type="checkbox"> Real numbers</label></p></div><div class="q-explanation"><p><img align="middle" alt="straight pi" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»§#960;«/mi»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-22.php?formula=6dae009a0e3a5eec27fc1b747bab3923.png"> is neither natural or an integer because it is not whole. It is also not rational beacuse it cannot be expressed as a fraction</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following numbers are rational but not integers </p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> -0.967563</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <img align="middle" alt="2 over 3" class="Wirisformula" data-mathml="«math style=¨font-family:Arial¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-23.php?formula=d0bb11015f474d0ed495414700ecf5d1.png" style="width: 22px; height: 46px;"></label></p><p><label class="checkbox"><input type="checkbox"> 23.5 x 10<sup>4</sup></label></p><p><label class="checkbox"><input type="checkbox"> -6</label></p><p><label class="checkbox"><input class="c" type="checkbox"> 0.75</label></p><p><label class="checkbox"><input type="checkbox"> 54</label></p></div><div class="q-explanation"><p>All of the 'whole' numbers are excluded and 23.5 x 10<sup>4</sup> = 235000 which is also whole.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following diagrams shows the correct arrangement of the four main subsets?</p><p style="text-align: center;"><img alt="" src="../../files/NA/Setsofnumbers/Q1.jpg" style="width: 600px; height: 374px;"></p></div><div class="q-answer"><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input type="checkbox"> A</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input type="checkbox"> B</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input class="c" type="checkbox"> C</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input type="checkbox"> D</label></p></div><div class="q-explanation"><p>Diagram C is the correct diagram because it shows, correctly, that all natural numbers are integers, all integers are rational and that all rational numbers are real.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Is the following table correctly filled in?</p><p style="text-align: center;"><img alt="" src="../../files/NA/Setsofnumbers/Q2.jpg" style="width: 500px; height: 170px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> Yes</label></p><p><label class="checkbox"><input type="checkbox"> No</label></p></div><div class="q-explanation"><p><img align="middle" alt="square root of 3" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msqrt»«mn»3«/mn»«/msqrt»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-24.php?formula=18d0e52c351a213b87661c3117a37b6e.png"> is irrational and therefore only real. 7 is natural and therefore everything else. -6.1 is part of a number and negative so not an integer and not natural, but is a terminating decimal so it can be expressed as a fraction, so it is rational. Likewise <img align="middle" alt="7 over 9" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»7«/mn»«mn»9«/mn»«/mfrac»«/math»" src="../../../ckeditor/plugins/wiris/integration/showimage-25.php?formula=0f6c234d8a9b6229d193d03c1f0ffc83.png">is rational and therefore real.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which of the following is the most comprehensive description of the set that 3.456 x 10<sup>3</sup> is in?</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> Integer</label></p><p><label class="checkbox"><input type="checkbox"> real</label></p><p><label class="checkbox"><input class="c" type="checkbox"> Natural</label></p><p><label class="checkbox"><input type="checkbox"> rational</label></p></div><div class="q-explanation"><p>The answer is 'Natural' because if a number is natural then it is also a real, rational integer!</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> <hr> <h2><img alt="" src="../../files/NA/Setsofnumbers/SN-1.jpg" style="width: 80px; height: 84px; float: left;">Practice questions</h2> <p>This section will go through a couple of exam style questions on this topic. Click the 'hidden box symbol' to see this section</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><strong>Question 1</strong></p> <p style="text-align: center;"><br> <img alt="" src="../../files/NA/Setsofnumbers/PPQ1.jpg" style="width: 500px; height: 563px;"></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/135549986"></iframe></div> <p><strong>Question 2</strong></p> <p style="text-align: center;"><strong><img alt="" src="../../files/NA/Setsofnumbers/PPQ2.jpg" style="width: 500px; height: 382px;"></strong></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" 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