View of /trunk/NationalProblemLibrary/Rochester/setComplexNumbers/ur_cn_1_20.pg

    1 ##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