Complex countdown

6441 /www/nrich/html/content/id/6441/ 2 2011-06-14T11:25:37 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Complex countdown is like <a href="http://nrich.maths.org/6499">normal countdown</a> except that in addition to the usual arithmetical rules of add, subtract, multiply, divide, you are allowed to use the operations of: Mod(z), Re(z), Im(z), Arg(z) and CC(z) These stand for: Modulus, Real Part, Imaginary Part, Argument (measures in degrees) and Complex Conjugate. Oh, and it also uses complex numbers. Have fun! <mdo:flash height="150" id="/content/id/6441/countdown.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/6441/countdown.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash> We can think of various scoring systems for this game; we'll leave it up to you to devise your own. You might well wish to play this collaboratively with friends and take you time to try to hit the targets exactly. Extension: If you wish to develop your IT skills, why not try implementing a version of this game on a spreadsheet or in code? As an extra challenge you might try to implement a version in which you know that there is always an exact solution for every game played. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why play this game?</h3> Confidence with basic manipulation of complex numbers is very important in school courses in complex numbers and critical in university mathematics, physical sciences and engineering courses. This game is a fun (to maths folk, at the very least) way to practice these skills in a way which will increase fluency and understanding far more than simply working through a bank of exercises: striving to hit a target requires the players to consider all aspects of the complex numbers on offer; as there is no 'recipe' players must actively engage with the mathematical properties of each complex number leading to a far richer learning experience. <h3>Possible approach</h3> Complex countdown is initially best played at a slow and thoughtful pace, where students collaborate to try to hit the targets. It is possible that certain games might seem difficult, but (in our experience) it is possible to hit most targets exactly. In <a href="http://nrich.maths.org/6441?part=clue">the hint</a> to the problem we give a few worked solutions so that you can get an idea for the possibilities. Once students are more fluent you might consider timed competitions. Extension ideas: The scoring system provides an interesting mathematical question: how can we determine whether two complex numbers are 'close'. Do not be tempted to provide up-front the default answer that any mathematician would come up with (look at the modulus of the difference) - this would be a very good opportunity for students to come to this realisation themselves (perhaps as a homework) that 'closeness' can be summarised by distance in the complex plane. Some different scoring systems might be offered, such as "Look at the difference between the modulus of the target and the modulus of the answer", "Look at the differences between the real and imaginary parts separately and add them" or "Hitting either the real or imaginary part exactly always beats a score which hits neither" or simply voting on the "nicest" solution. As with any of our countdown games, a nice extension is for students to create their own set of cards and targets. Some motivation might be to deduce a set of impossible targets for a given set of cards. Can the impossibility be proved or strongly justified? Impossible solutions: In the event that a game seems to provide a set of cards and a target which appears to be impossible you can easily pose the very difficult challenge 'Can you prove that there is no solution to this particular game?'. Take care to record the values of the card before moving on! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Here are a couple of games of complex countdown in action to give you some ideas of how to get started. We use square brackets [ ... ] to show that we are using up one of the original cards. GAME 1 <mdo:image src="Countdown1.png"></mdo:image> One possible solution here is [8+9i] + [-2 - 9i] + [0-7i] + ([8i]/[4-0i]) GAME 2 <mdo:image src="Countdown2.png"></mdo:image> One possible solution is [-1 - 4i] - [-8i]/([8]/[-2]) GAME 3 <mdo:image src="Countdown3.png"></mdo:image> Arg[-4i] - (Re[7+1i] x Im[-8i]) + [-1] </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Hello Hello Hello Hello </mdoxml> 2 4 0 0 0 0 1 Complex countdown Play a more cerebral countdown using complex numbers. Advanced Algebra Complex numbers Advanced Algebra Modulus function Information and Communications Technology Interactivities