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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Proof by Contradiction</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"> <p><img alt="" src="../../files/algebra/proof-by-contradiction/main.png" style="float: left; width: 100px; height: 100px;">"...when you have eliminated the impossible, whatever remains, <em>however improbable</em>, must be the truth?" Sherlock Holmes in The Sign of the Four by Sir Arthur Conan Doyle. Holmes was a wonderful detective and he would also have been excellent at carrying out proofs by contradiction, as this is exactly the process we use. To complete a proof by contradiction, we start by assuming the opposite is true. If we show that the consequences of this are impossible, our original statement must be true. This is a challenging topic in the course. The page will prepare you well.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Carry out proof by contradiction to show irrationality of roots, like <span class="math-tex">\(\sqrt{3}\)</span></li> <li>Carry out proof by contradiction to show that there are an infinite number of primes</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/revision-notes_proof_by_contradiction.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/proof-by-contradiction/revision-notes_proof_by_contradiction.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz to prove that if the square of a positive integer is odd, the number is odd</p> <div class="tib-quiz" data-stats="11-454-1734"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Prove that if the square of a positive integer, n² is odd, then the number n is odd</p><p> <span class="q-text-draggable draggable" draggable="true">This is a contradiction</span> <span class="q-text-draggable draggable" draggable="true">n² = 2(2k²)</span> <span class="q-text-draggable draggable" draggable="true">n = 2k, where k is an integer</span> <span class="q-text-draggable draggable" draggable="true">Hence, if n² is odd, then the number n is odd</span> <span class="q-text-draggable draggable" draggable="true">n² is even</span> <span class="q-text-draggable draggable" draggable="true">n is even</span> <span class="q-text-draggable draggable" draggable="true">n² = 4k²</span> </p></div><div class="q-answer"><p>Let n be a positive integer, where n² is odd</p><p>Assume the opposite, that <input type="text" style="height: auto;" data-c="n is even"> <span class="review"></span> </p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="n = 2k, where k is an integer"> <span class="review"></span> </p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="n² = 4k²"> <span class="review"></span> </p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="n² = 2(2k²)"> <span class="review"></span> </p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="n² is even"> <span class="review"></span> </p><p> <input type="text" style="height: auto;" data-c="This is a contradiction"> <span class="review"></span> </p><p> <input type="text" style="height: auto;" data-c="Hence, if n² is odd, then the number n is odd"> <span class="review"></span> </p></div><div class="q-explanation"><p>You could carry out this proof using a deductive proof, but if the exam question asks you to do a proof by contradiction, then you must do a proof by contradiction!</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> <hr class="hidden-separator"> <p>Here is a quiz to prove that <span class="math-tex">\(\sqrt{5}\)</span> is irrational</p> <div class="tib-quiz" data-stats="11-453-1734"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to prove that <span class="math-tex">\(\sqrt {5}\)</span> is<strong> irrational</strong></p><p><span class="q-text-draggable draggable" draggable="true">a is divisible by 5</span> <span class="q-text-draggable draggable" draggable="true">a is divisible by 5</span> <span class="q-text-draggable draggable" draggable="true">This is a contradiction</span> <span class="q-text-draggable draggable" draggable="true">5b² = a²</span> <span class="q-text-draggable draggable" draggable="true">a² is divisible by 5</span> <span class="q-text-draggable draggable" draggable="true">5b² = (5n)²</span> <span class="q-text-draggable draggable" draggable="true">b² = 5n²</span> <span class="q-text-draggable draggable" draggable="true">b is divisible by 5</span> <span class="q-text-draggable draggable" draggable="true">a and b are divisible by 5</span></p></div><div class="q-answer"><p>Assume that <span class="math-tex">\(\sqrt {5}\)</span> is <strong><em>rational</em></strong></p><p><span class="math-tex">\(\sqrt {5}=\frac{a}{b}\quad a,b\in\mathbb{Z}, b\neq0\)</span></p><p><span class="math-tex">\(\implies\)</span> <span class="math-tex">\(5=\frac {a²}{b²}\)</span></p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="5b² = a²"> <span class="review"></span></p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="a² is divisible by 5"> <span class="review"></span></p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="a is divisible by 5"> <span class="review"></span></p><p><span class="math-tex">\(\implies\)</span> a = 5n</p><p>Substitute this in 5b² = a²</p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="5b² = (5n)²"> <span class="review"></span></p><p><span class="math-tex">\(\implies\)</span> 5b² = 25n²</p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="b² = 5n²"> <span class="review"></span></p><p><span class="math-tex">\(\implies\)</span> b² is divisible by 5</p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="b is divisible by 5"> <span class="review"></span></p><p>This means that both <input type="text" style="height: auto;" data-c="a and b are divisible by 5"> <span class="review"></span></p><p><span class="math-tex">\(\implies \frac{a}{b}\)</span> has a common factor of 5</p><p> <input type="text" style="height: auto;" data-c="This is a contradiction"> <span class="review"></span></p><p><span class="math-tex">\(\implies \sqrt {5}\)</span> is<strong> irrational</strong></p></div><div class="q-explanation"><p>There are other ways to prove that <span class="math-tex">\(\sqrt{5}\)</span> is irrational, buut this technique is one that you can use for other roots of prime numbers. This quiz is something you can practise several times!</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> <hr class="hidden-separator"> <p>Here is a quiz to prove that there are infinitely many primes</p> <div class="tib-quiz" data-stats="11-455-1734"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>Complete the following to prove that there are an infinite number of primes</p><p> <span class="q-text-draggable draggable" draggable="true">be divided by p2, or p3, ... or pn</span> </p><p> <span class="q-text-draggable draggable" draggable="true">there are an infinite number of primes</span> </p><p> <span class="q-text-draggable draggable" draggable="true">(p1 x p2 x p3 x ...x pn) + 1</span> </p><p> <span class="q-text-draggable draggable" draggable="true">This is a contradiction that p1 , p2 , p3, ..., pn contains all the primes</span> </p><p> <span class="q-text-draggable draggable" draggable="true">p1 , p2 , p3, ..., pn</span> </p><p> <span class="q-text-draggable draggable" draggable="true">another prime or divisible by another prime not in our list of primes</span> </p><p> <span class="q-text-draggable draggable" draggable="true">by p1 (it leaves a remainder of 1)</span> </p><p> <span class="q-text-draggable draggable" draggable="true">finite number of primes</span> </p></div><div class="q-answer"><p>Assume that there are a <input type="text" style="height: auto;" data-c="finite number of primes"> <span class="review"></span> </p><p>We can list all the primes: <input type="text" style="height: auto;" data-c="p1 , p2 , p3, ..., pn"> <span class="review"></span> </p><p>Let N = <input type="text" style="height: auto;" data-c="(p1 x p2 x p3 x ...x pn) + 1"> <span class="review"></span> </p><p>N cannot be divided <input type="text" style="height: auto;" data-c="by p1 (it leaves a remainder of 1)"> <span class="review"></span> </p><p>N cannot <input type="text" style="height: auto;" data-c="be divided by p2, or p3, ... or pn"> <span class="review"></span> </p><p>N must be <input type="text" style="height: auto;" data-c="another prime or divisible by another prime not in our list of primes"> <span class="review"></span> </p><p><span class="math-tex">\(\implies\)</span> <input type="text" style="height: auto;" data-c="This is a contradiction that p1 , p2 , p3, ..., pn contains all the primes"> <span class="review"></span> </p><p>Hence, <input type="text" style="height: auto;" data-c="there are an infinite number of primes"> <span class="review"></span> </p></div><div class="q-explanation"><p>This is Euclid's classic proof. You should learn how to do this by heart.</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> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="854"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove by contradiction that, if n² is even, then n is even</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">Assume that n is odd, i.e. n = 2k + 1<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="856"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>a) Use a deductive proof to prove that even x even = even</p> <p>b) Similarly prove that odd x odd = odd</p> </div> <p>c) Hence, use proof by contradiction to prove that <span class="math-tex">\(log_25\)</span> is irrational</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) Assume that <span class="math-tex">\(log_25=\frac{a}{b}\)</span><content></content> and convert this into an index equation.</p> <p>You might find your proofs from part a and b useful</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="855"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove by contradiction that a rational number + an irrational number = irrational number.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content> <p>Remember the definition of a rational and irrational numbers</p> <ul> <li>Rational numbers can be expressed as fractions.</li> <li>Irrational numbers cannot be expressed as fractions.</li> </ul> <p>Assume that one number is rational and the sum is irrational (opposite of proof)</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="857"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> Prove that <span class="math-tex">\(\sqrt{3}\)</span> is irrational</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>There are LOTS of ways of proving this result. I have included 2 below. I prefer the first one because this can easily be adapted to prove other roots of prime numbers.</p> <p><content>The key is to use the Fundamental Theorem of Algebra. Have a look at the video for the proof of irrationality of <span class="math-tex">\(\sqrt{2}\)</span></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction4a.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction4a.pdf" width="640"></iframe></p> </section> <h4> </h4> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Alternate Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction4b.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction4b.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="858"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that <span class="math-tex">\(\sqrt[3]{5}\)</span> is irrational</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>You should be able to use the same method as for the proof that <span class="math-tex">\(\sqrt{3} \)</span> is irrational.</p> <p>See previous exam-style question or the video for proof that <span class="math-tex">\(\sqrt{2}\)</span> is irrational</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction5.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="860"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove by contradiction that the length of the hypotenuse of a right-angled triangle is less than the sum of the other two sides.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Let c be the length of the hypotenuse and a and b be the lengths of the other two sides.</p> <p><content>Show that c ≥ a + b leads to a contradiction</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction7.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction7.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="859"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove that <span class="math-tex">\(\sqrt[n]{p}\)</span> is irrational, given that p is prime.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">This is just an extension of other proof of irrationality of other roots<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction6.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction6.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 8</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="862"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Prove by contradiction that there are no rational roots to the equation <span class="math-tex">\(x^3+x+1=0\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Assume the opopsite, that x = <span class="math-tex">\(\frac{a}{b}\)</span></p> <p><content>and consider all the cases for a and b being odd/even</content></p> <p>That is</p> <p>1) a and b are both even</p> <p>2) a and b are both odd</p> <p>...etc</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction8.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/proof-by-contradiction/esq_proof_by_contradiction8.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="1734"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label 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