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# Annotation of /trunk/NationalProblemLibrary/Rochester/setComplexNumbers/ur_cn_1_20.pg

 1 : jj 143 ##DESCRIPTION 2 : ## Some simple complex number algebra 3 : ##ENDDESCRIPTION 4 : 5 : 6 : DOCUMENT(); # This should be the first executable line in the problem. 7 : 8 : loadMacros( 9 : "PG.pl", 10 : "PGbasicmacros.pl", 11 : "PGchoicemacros.pl", 12 : "PGanswermacros.pl", 13 : "PGauxiliaryFunctions.pl", 14 : "PGcomplexmacros.pl" 15 : ); 16 : 17 : TEXT(beginproblem()); 18 : $showPartialCorrectAnswers = 1; 19 : 20 :$a = new Complex( 0, $PI ); 21 :$b = new Complex( -2*sqrt($one = random(2, 3, 1 )), -($two = non_zero_random( 1, 5, 1 )) ); 22 : $c = (new Complex( 1, -1 ) ) * ( new Complex( -sqrt($three=random(2,3,1)), 1 ) ); 23 : $d = (new Complex( sqrt($four=random(2,3,1)), ($five = random (-5, -1, 1 )) ) )**2; 24 :$e = ( new Complex( ($six = random(-5, -1, 1 )), sqrt($seven = random(2,3, 1))))/( new Complex( 25 : ($eight = random(1, 5, 1)), ($nine = random( 1, 5, 1 ) ))); 26 : $f = -sqrt(7) * ( new Complex( 1, 1 ))/( new Complex( sqrt($ten = random( 2, 3, 1)), 1 ) ); 27 : 28 : 29 : BEGIN_TEXT 30 : 31 : Write the following numbers in the polar form $$re^{i \theta}$$, $$-\pi < \theta \le \pi$$: $PAR 32 : (a) $$\pi i$$$PAR 33 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 34 : (b) $$-2\sqrt{one}\ -\ two i$$$PAR 35 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 36 : (c) $$( 1\ -\ i)( - \sqrt{three}\ +\ i)$$$PAR 37 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 38 : (d) $$(\sqrt{four}\ five i)^2$$$PAR 39 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 40 : (e) $$\displaystyle \frac{six\ +\ \sqrt{seven}i}{eight\ +\ nine i}$$$PAR 41 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 42 : (f) $$\displaystyle \frac{-\sqrt{7}(1\ +\ i)}{\sqrt{ten}\ +\ i}$$$PAR 43 : $$r =$$ \{ans_rule(15)\}, $$\theta =$$ \{ans_rule(15)\}, $PAR 44 : 45 :$PAR $BR 46 : END_TEXT 47 : 48 : 49 : ANS(num_cmp($a -> rho )); 50 : ANS(num_cmp( $a -> theta)); 51 : ANS(num_cmp($b -> rho )); 52 : ANS(num_cmp( $b -> theta)); 53 : ANS(num_cmp($c -> rho )); 54 : ANS(num_cmp( $c -> theta)); 55 : ANS(num_cmp($d -> rho )); 56 : ANS(num_cmp( $d -> theta)); 57 : ANS(num_cmp($e -> rho )); 58 : ANS(num_cmp( $e -> theta)); 59 : ANS(num_cmp($f -> rho )); 60 : ANS(num_cmp( \$f -> theta)); 61 : 62 : 63 : 64 : ENDDOCUMENT(); # This should be the last executable line in the problem. 65 :