7379 /www/nrich/html/content/id/7379/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>In the following exhibits we give an advanced or alternative way of thinking about mathematics concepts which are likely to be known in a more familiar form.<br></br> <br></br> Explore these structures and experiment by substituting particular values such as $0, \pm 1$. Can you work out what they represent?<br></br> <br></br> <span style="font-weight: bold;">Exhibit A</span><br></br> All pairs of integers such that:<br></br> $$(a, b) + (c, d) = (ad+bc, bd)\quad\quad (Na, Nb) \equiv (a, b) \mbox{ for all } N\neq 0$$<br></br> Can you find two pairs which add up to give $(0, N)$ or $(0, M)$ for various values of $N$, $M$?<br></br>  <br></br> <br></br>  </p> <div>Next explore the properties of these structures:</div> <div> </div> <div><span style="font-weight: bold;">Exhibit B</span></div> <br></br> <p>A set of ordered pairs of real numbers which can be added and multiplied such that<br></br>  <br></br> $(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 +y_2)$<br></br>  <br></br> $(x_1, y_1)\times (x_2, y_2) = (x_1x_2 -y_1y_2, x_1y_2+y_1x_2)$<br></br> <br></br>  <br></br> <span style="font-weight: bold;">Exhibit C</span><br></br>  <br></br> A set defined recursively such that<br></br>  <br></br> $+_k(1) = +_1(k)$<br></br>  <br></br> $+_k(+_1(n)) = +_1(+_k(n))$<br></br>  <br></br> $\times_k(1) = k$<br></br>  <br></br> $\times_k(+_1(n)) = +_k(\times_k(n))$<br></br>  <br></br> In these rules, $k$ and $n$ are allowed to be any natural numbers<br></br>  <br></br>  <br></br> Once you have figured out what these structures represent ask yourself this: Are these good representations? What benefits can you see to such a representation? How might familiar properties from the structures be represented in these ways? <br></br>  <br></br>  <br></br>  <br></br>  </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p><strong>Exhibit A</strong></p> <p>The condition $$(Na, Nb) \equiv (a, b) \mbox{ for all } N\neq 0$$ ought to give it away: $$(a,b) \iff \frac{a}{b}$$ This statement simply says that if the numerator and denominator of a fraction share a common factor, they can be cancelled down. </p> <p> </p> <p><strong>Exhibit B</strong></p> <p>If we represent a complex number a+bi by the ordered pair (a,b), we get the required properties:</p> <p>$$(a+bi) + (c+di) = (a+c) + (b+d)i \iff (a,b) + (c,d) = (a+c, b+d)$$</p> <p>$$(a+bi) \times (c+di) = (ac-bd) + (ad+bc)i \iff (a,b) \times (c,d) = (ac-bd, ad+bc)$$</p> <p> </p> <p><strong>Exhibit C</strong></p> <p>These formally define addition and multiplication over the natural numbers. Can you see how the familiar properties we&#39;re used to follow from them?</p> <p>The first implies $k+1 = 1+k$, i.e. addition is commutative. </p> <p>The second implies $k+(1+n) = 1+(k+n)$, i.e. addition is associative.</p> <p>The third implies $k\times 1 = k$, i.e. 1 is the multiplicative identity.</p> <p>The fourth implies $k\times(1+n) = k+(k\times n)$, that mulitplication is distributative over addition.</p> <p>This is a rigorous treatment of a very familiar concept. For more information on this subject, you could start by reading this <a href="http://en.wikipedia.org/wiki/Peano_axioms">Wikipedia article</a>. </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3>Why do this problem</h3> This problem gives a taster of abstract representations framed around familiar mathematical concepts. It is useful to prepare the way for students to start thinking about abstract mathematics such as group theory, as concepts such as Identity, Inverse, Equivalence and Closure will emerge during the task. The latter parts of the task are good fun to have ongoing over the course of a week or term.<br></br> <br></br> <h3>Possible approach</h3> <div>This problem would be a difficult challenge for keen students to consider individually, perhaps as an extended homework or holiday task.</div> <div> </div> <div>In lesson time it is suited to a group or class discussion where the problem is gradually solved together -- focus initially only on exhibit A and the related questions, saving exhibits B and C for high-fliers or open problems. All students will know the mathematics for Exhibits A and C, but perhaps skip B (complex numbers) if students are not in their final year.</div> <div> </div> <div>Start with Exhibit A and write up the rules. Say something like &quot;I've got a set containing pairs of integers, such as (1, 22) and (-34, 8). I have a rule which allows me to combine pairs to give another pair in my set. </div> <div> </div> <div>For example $(1, 2) + (2, 3) = (7, 6)$ and $(7, 2) + (3, 5) = (41, 10)$</div> <div> </div> <div>Oh, and we also have rules such as $(14, 21) \equiv (2, 3)$&quot;</div> <div> </div> <div>This is a good place to discuss abstract rules and to introduce the concept of equivalence if the class has not met this before.</div> <div> </div> <div>Allow the class to discuss what might be going on before writing down the general rule</div> <div> </div> <div>$(a, b) + (c, d) = (ad+bc, db)$</div> <div> </div> <div>Suggest that small groups attempt to work out what is happening by testing out various numerical examples. When a group feels that they have worked out that the structure is 'addition of fractions' they will then need to convince the rest of the class of their reasoning. </div> <div> </div> <div>You can then allow the class to move onto consideration of the other Exhibits if a longer task is desired.</div> <br></br> <h3>Key questions</h3> <div>Have you tried exploring with small numbers, both positive and negative to get a feel for the structure?</div> <div>Using our rules, can two pairs be combined to give $(0, N)$?</div> <div>What happens if you combine two of the same pair together?</div> <div>Using our rules, which pairs can be combined to give $(N, 0)?$</div> <h3>Possible extension</h3> <div>Exhibits B and C are likely to offer sufficient extension. If more exploration is desired, students can attempt to alter the combination rules and see if any structures emerge.</div> <div> </div> <div>You could also ask students to devise exhibits of other structures, such as vectors and matrices.</div> <div> </div> <h3>Possible support</h3> <div>You can greatly simplify this tasks by asking 'This is similar to addition of fractions! Can anyone see why?' or 'This is addition of complex numbers' or 'These are the rules of arithmetic'.</div> <br></br>  <br></br>  <br></br>  <br></br>  <br></br>  <br></br>  <br></br>  <br></br>  <br></br>  <br></br></mdoxml> 2 5 0 0 0 0 1 Can you work out what simple structures have been dressed up in these advanced mathematical representations? Advanced Algebra Groups Advanced Algebra Complex numbers Advanced Algebra Isomorphism Advanced Algebra Relations Secondary processes PM - Representing