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Created July 25, 2013 15:35
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1reales
<p>%md</p>
<h2 id="set-11">Set 1.1</h2>
<p>En los siguientes ejercicios determine si el número es natural, entero, racional o irracional.</p>
<ol>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-1-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1" style="width: 10.955em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.11em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-2"><span class="mo" id="MathJax-Span-3" style="font-family: STIXGeneral-Regular;">−</span><span class="mn" id="MathJax-Span-4" style="font-family: STIXGeneral-Regular;">203</span><span class="mspace" id="MathJax-Span-5" style="height: 0.003em; vertical-align: 0.003em; width: 6.729em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.004em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-1">-203\hspace{3cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-2-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-6" style="width: 2.443em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.027em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.658em 1000.003em 3.098em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-7"><span class="mstyle" id="MathJax-Span-8"><span class="mrow" id="MathJax-Span-9"><span class="mfrac" id="MathJax-Span-10" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 1.729em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -3.211em; left: 50%; margin-left: -0.771em;"><span class="mn" id="MathJax-Span-11" style="font-family: STIXGeneral-Regular;">203</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mn" id="MathJax-Span-12" style="font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 1.729em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 2.575em; vertical-align: -0.925em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-2">\displaystyle\frac{203}{2}</script></li>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-3-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-13" style="width: 1.789em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.658em 1000.003em 3.098em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-14"><span class="mstyle" id="MathJax-Span-15"><span class="mrow" id="MathJax-Span-16"><span class="mfrac" id="MathJax-Span-17" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 1.193em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -3.211em; left: 50%; margin-left: -0.533em;"><span class="mn" id="MathJax-Span-18" style="font-family: STIXGeneral-Regular;">27</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mn" id="MathJax-Span-19" style="font-family: STIXGeneral-Regular;">4</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 1.193em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 2.575em; vertical-align: -0.925em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-3">\displaystyle\frac{27}{4}</script><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-20" style="width: 9.527em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.92em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.027em 1000.003em 2.384em -0.533em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-21"><span class="mspace" id="MathJax-Span-22" style="height: 0.003em; vertical-align: 0.003em; width: 7.86em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.146em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-4">\hspace{3.5cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-5-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-23" style="width: 3.217em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.682em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.479em 1000.003em 3.277em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-24"><span class="mstyle" id="MathJax-Span-25"><span class="mrow" id="MathJax-Span-26"><span class="msqrt" id="MathJax-Span-27"><span style="display: inline-block; position: relative; width: 2.622em; height: 0px;"><span style="position: absolute; clip: rect(2.443em 1000.003em 4.884em -0.414em); top: -3.985em; left: 1.134em;"><span class="mrow" id="MathJax-Span-28"><span class="mfrac" id="MathJax-Span-29" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 1.193em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -3.211em; left: 50%; margin-left: -0.533em;"><span class="mn" id="MathJax-Span-30" style="font-family: STIXGeneral-Regular;">27</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mn" id="MathJax-Span-31" style="font-family: STIXGeneral-Regular;">4</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 1.193em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -4.699em; left: 1.134em;"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.955em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.301em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.658em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(1.729em 1000.003em 4.586em -0.414em); top: -3.449em; left: 0.003em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 3.146em; vertical-align: -1.139em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-5">\displaystyle\sqrt{\frac{27}{4}}</script></li>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-6-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-32" style="width: 11.313em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.408em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.134em 1000.003em 2.384em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-33"><span class="msubsup" id="MathJax-Span-34"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.414em); top: -2.557em; left: 0.003em;"><span class="mn" id="MathJax-Span-35" style="font-family: STIXGeneral-Regular;">10</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; top: -2.676em; left: 1.074em;"><span class="mn" id="MathJax-Span-36" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">6</span><span style="display: inline-block; width: 0px; height: 2.265em;"></span></span></span></span><span class="mspace" id="MathJax-Span-37" style="height: 0.003em; vertical-align: 0.003em; width: 7.86em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.218em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-6">10^6\hspace{3.5cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-7-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-38" style="width: 2.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.479em 1000.003em 3.277em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-39"><span class="mstyle" id="MathJax-Span-40"><span class="mrow" id="MathJax-Span-41"><span class="mfrac" id="MathJax-Span-42" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 1.61em; height: 0px;"><span style="position: absolute; clip: rect(2.92em 1000.003em 4.17em -0.414em); top: -4.64em; left: 50%; margin-left: -0.771em;"><span class="msubsup" id="MathJax-Span-43"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.414em); top: -2.557em; left: 0.003em;"><span class="mn" id="MathJax-Span-44" style="font-family: STIXGeneral-Regular;">10</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; top: -2.676em; left: 1.074em;"><span class="mn" id="MathJax-Span-45" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">6</span><span style="display: inline-block; width: 0px; height: 2.265em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.92em 1000.003em 4.17em -0.414em); top: -3.152em; left: 50%; margin-left: -0.771em;"><span class="msubsup" id="MathJax-Span-46"><span style="display: inline-block; position: relative; width: 1.491em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.414em); top: -2.557em; left: 0.003em;"><span class="mn" id="MathJax-Span-47" style="font-family: STIXGeneral-Regular;">10</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; top: -2.676em; left: 1.074em;"><span class="mn" id="MathJax-Span-48" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">7</span><span style="display: inline-block; width: 0px; height: 2.265em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 1.61em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 3.075em; vertical-align: -1.139em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-7">\displaystyle\frac{10^6}{10^7}</script></li>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-8-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-49" style="width: 11.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.229em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-50"><span class="mn" id="MathJax-Span-51" style="font-family: STIXGeneral-Regular;">8.7</span><span class="mspace" id="MathJax-Span-52" style="height: 0.003em; vertical-align: 0.003em; width: 7.86em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.004em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-8">8.7\hspace{3.5cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-9-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-53" style="width: 1.67em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.372em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.313em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-54"><span class="mn" id="MathJax-Span-55" style="font-family: STIXGeneral-Regular;">8.</span><span class="munderover" id="MathJax-Span-56"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -2.557em; left: 0.003em;"><span class="mn" id="MathJax-Span-57" style="font-family: STIXGeneral-Regular;">7</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.086em 1000.003em 2.503em -0.533em); top: -3.211em; left: 0.003em;"><span class="mo" id="MathJax-Span-58" style=""><span style="display: inline-block; position: relative; width: 0.539em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.241em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.432em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-9">8.\overline{7}</script></li>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-10-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-59" style="width: 11.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.348em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-60"><span class="mn" id="MathJax-Span-61" style="font-family: STIXGeneral-Regular;">8.74</span><span class="mspace" id="MathJax-Span-62" style="height: 0.003em; vertical-align: 0.003em; width: 7.443em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.004em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-10">8.74\hspace{3.3cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-11-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-63" style="width: 2.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.313em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-64"><span class="mn" id="MathJax-Span-65" style="font-family: STIXGeneral-Regular;">8.</span><span class="munderover" id="MathJax-Span-66"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -2.557em; left: 0.003em;"><span class="mn" id="MathJax-Span-67" style="font-family: STIXGeneral-Regular;">74</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.086em 1000.003em 2.503em -0.533em); top: -3.211em; left: 0.003em;"><span class="mo" id="MathJax-Span-68" style=""><span style="display: inline-block; position: relative; width: 1.074em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.777em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.241em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.539em;">⎯<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.432em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-11">8.\overline{74}</script></li>
<li>(a) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-12-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-69" style="width: 2.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.908em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.253em 1000.003em 2.562em -0.473em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-70"><span class="mstyle" id="MathJax-Span-71"><span class="mrow" id="MathJax-Span-72"><span class="msqrt" id="MathJax-Span-73"><span style="display: inline-block; position: relative; width: 1.848em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-74"><span class="mn" id="MathJax-Span-75" style="font-family: STIXGeneral-Regular;">99</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.926em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.598em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.301em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.807em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.289em; vertical-align: -0.282em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-12">\displaystyle\sqrt{99}</script><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-13-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-76" style="width: 9.051em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.503em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(2.027em 1000.003em 2.384em -0.533em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-77"><span class="mspace" id="MathJax-Span-78" style="height: 0.003em; vertical-align: 0.003em; width: 7.443em; display: inline-block; overflow: hidden;"></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 0.146em; vertical-align: -0.068em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-13">\hspace{3.3cm}</script> (b) <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-14-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-79" style="width: 4.408em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.634em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em 1000.003em 2.86em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-80"><span class="msqrt" id="MathJax-Span-81"><span style="display: inline-block; position: relative; width: 1.848em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-82"><span class="mn" id="MathJax-Span-83" style="font-family: STIXGeneral-Regular;">99</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.985em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 1.074em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.598em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.301em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.866em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-84" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">+</span><span class="mn" id="MathJax-Span-85" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.289em; vertical-align: -0.211em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-14">\sqrt{99}+1</script></li>
<li><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-15-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-86" style="width: 5.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.705em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em 1000.003em 2.86em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-87"><span class="mn" id="MathJax-Span-88" style="font-family: STIXGeneral-Regular;">3</span><span class="msqrt" id="MathJax-Span-89"><span style="display: inline-block; position: relative; width: 2.384em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.414em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-90"><span class="mn" id="MathJax-Span-91" style="font-family: STIXGeneral-Regular;">101</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.985em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 1.61em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 1.134em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.36em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="font-family: STIXGeneral-Regular; position: absolute; top: -3.985em; left: 0.777em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.866em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-92" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">+</span><span class="mn" id="MathJax-Span-93" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.289em; vertical-align: -0.211em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-15">3\sqrt{101}+1</script></li>
<li><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-16-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-94" style="width: 9.11em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.562em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(1.551em 1000.003em 2.92em -0.473em); top: -2.557em; left: 0.003em;"><span class="mrow" id="MathJax-Span-95"><span class="mo" id="MathJax-Span-96" style="font-family: STIXGeneral-Regular;">(</span><span class="mn" id="MathJax-Span-97" style="font-family: STIXGeneral-Regular;">3</span><span class="mo" id="MathJax-Span-98" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">−</span><span class="msqrt" id="MathJax-Span-99" style="padding-left: 0.241em;"><span style="display: inline-block; position: relative; width: 1.313em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-100"><span class="mn" id="MathJax-Span-101" style="font-family: STIXGeneral-Regular;">2</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.985em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.063em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.866em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-102" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-103" style="font-family: STIXGeneral-Regular;">(</span><span class="mn" id="MathJax-Span-104" style="font-family: STIXGeneral-Regular;">3</span><span class="mo" id="MathJax-Span-105" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">+</span><span class="msqrt" id="MathJax-Span-106" style="padding-left: 0.241em;"><span style="display: inline-block; position: relative; width: 1.313em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-107"><span class="mn" id="MathJax-Span-108" style="font-family: STIXGeneral-Regular;">2</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.985em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.063em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.866em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-109" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.361em; vertical-align: -0.282em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-16">(3-\sqrt{2})(3+\sqrt{2})</script></li>
<li><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-17-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-110" style="width: 5.122em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.229em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.539em 1000.003em 3.098em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-111"><span class="mstyle" id="MathJax-Span-112"><span class="mrow" id="MathJax-Span-113"><span class="mfrac" id="MathJax-Span-114" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 3.932em; height: 0px;"><span style="position: absolute; clip: rect(1.551em 1000.003em 2.92em -0.473em); top: -3.211em; left: 50%; margin-left: -1.902em;"><span class="mrow" id="MathJax-Span-115"><span class="mo" id="MathJax-Span-116" style="font-family: STIXGeneral-Regular;">(</span><span class="msqrt" id="MathJax-Span-117"><span style="display: inline-block; position: relative; width: 1.313em; height: 0px;"><span style="position: absolute; clip: rect(1.67em 1000.003em 2.741em -0.473em); top: -2.557em; left: 0.777em;"><span class="mrow" id="MathJax-Span-118"><span class="mn" id="MathJax-Span-119" style="font-family: STIXGeneral-Regular;">5</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(2.979em 1000.003em 3.396em -0.533em); top: -3.985em; left: 0.777em;"><span style="display: inline-block; position: relative; width: 0.539em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.003em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; font-family: STIXGeneral-Regular; top: -3.985em; left: 0.063em;">‾<span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; clip: rect(2.86em 1000.003em 4.17em -0.473em); top: -3.866em; left: 0.003em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" id="MathJax-Span-120" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">+</span><span class="mn" id="MathJax-Span-121" style="font-family: STIXGeneral-Regular; padding-left: 0.241em;">1</span><span class="mo" id="MathJax-Span-122" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -1.842em; left: 50%; margin-left: -0.235em;"><span class="mn" id="MathJax-Span-123" style="font-family: STIXGeneral-Regular;">4</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 3.932em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 2.789em; vertical-align: -0.925em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-17">\displaystyle\frac{(\sqrt{5}+1)}{4}</script></li>
<li><span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-18-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-124" style="width: 4.051em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.336em; height: 0px; font-size: 120%;"><span style="position: absolute; clip: rect(0.658em 1000.003em 3.098em -0.414em); top: -2.199em; left: 0.003em;"><span class="mrow" id="MathJax-Span-125"><span class="mstyle" id="MathJax-Span-126"><span class="mrow" id="MathJax-Span-127"><span class="mfrac" id="MathJax-Span-128" style="padding-left: 0.122em; padding-right: 0.122em;"><span style="display: inline-block; position: relative; width: 3.039em; height: 0px;"><span style="position: absolute; clip: rect(1.729em 1000.003em 2.741em -0.533em); top: -3.211em; left: 50%; margin-left: -1.426em;"><span class="mn" id="MathJax-Span-129" style="font-family: STIXGeneral-Regular;">0.1234</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(1.67em 1000.003em 2.741em -0.533em); top: -1.842em; left: 50%; margin-left: -1.426em;"><span class="mn" id="MathJax-Span-130" style="font-family: STIXGeneral-Regular;">0.5677</span><span style="display: inline-block; width: 0px; height: 2.562em;"></span></span><span style="position: absolute; clip: rect(0.836em 1000.003em 1.253em -0.533em); top: -1.307em; left: 0.003em;"><span style="border-left-width: 3.039em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 1.25px; vertical-align: 0.003em;"></span><span style="display: inline-block; width: 0px; height: 1.074em;"></span></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="border-left-width: 0.004em; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 2.575em; vertical-align: -0.925em;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-18">\displaystyle\frac{0.1234}{0.5677}</script></li>
</ol>
<blockquote>
<p>Written with <a href="http://benweet.github.io/stackedit/">StackEdit</a>.</p>
</blockquote>
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