Real and Complex Domains in School Mathematics and in Computer Algebra Systems

1
Real and Complex Domains in School Mathematics
and in Computer Algebra Systems

• Eno Tõnisson
• University of Tartu
• Estonia

2
Plan

• Introduction
• School
• CASs
• Teacher
• Summary

3
Motivation Unexpected answers

• CASs
• are capable of solving many (school mathematics)
  problems
• mostly solve as used at school,
• but there are still answers more or less
  unexpected for school.
• Unexpected answers
• are not inevitably mathematically incorrect
• but may simply accord with another standard.
• Correctness, Completeness, Compactness
• Main goal is not only to find errors/dissimilariti
  es but to use them positively.

4
Calculation, simplification of expressions,
solving equations
Unexpected answers
Real and complex numbers (CADGME, today)
Branches (ICTMT8, in July 1)
Equivalence
Infinities and indeterminates
in CASs and school
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Questions for all areas

• What exact commands would be useful if we try to
  get more school-friendly answers? How much are
  the CASs adjustable? Are there any special
  packages?
• What do CASs need in order to give more
  school-friendly answers?
• Why do CASs solve the problems as they do? Are
  different standards used?
• Are these standards useful for the school? Would
  it be possible to integrate these approaches to
  school treatment? Would it be reasonable?

6
Real and Complex Domains
Complex
Real
Imaginary
Rather Border or bridge between R ? C Real and
imaginary
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School and CASs

• School (different countries, textbooks, teachers)
• Estonian, English, Norwegian, Russian
• Primary and secondary school Grades ??-12,
• University (teacher training)
• General (??), Specific
• CASs (different systems, versions)
• Derive 6, Maple 8, Mathematica 4.2, MuPAD 3.1,
  TI-92, TI-nspire (prototype) and WIRIS.
• General (??), Specific

8
Number Domains at School

• The available number domain gradually extends for
  the students during their school time.
• In many countries (incl. Estonia) N ? Q ? Q ? R
  (?C)
• Systematic
• Changeover may be complicated
• N ? Q discrete ? dense. Merenlouto. What is next?
• Students
• (probably?) work by default in their largest
  number domain
• 3(x-1)-(x5)2(x-4) ? 0 0
• The solution set of this equation is the entire
  set of numbers known to us, that is, the rational
  number set Q.
• usually do not think about number domain

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Domain is important

• The topic of number domains is certainly
  important
• there may be different transformation rules
  allowed or
• (x,y 0)
• (R/C, H. Aslaksen)
• the solution sets may differ in different number
  domains.
• It is not possible in real school to find
• Square root for a negative number
• Logarithm if argument or base are negative
• Arc sine and arc cosine if argument is less than
  -1 or greater than 1.
• Using complex domain allows these operations
• In case of square root is (probably) told that
  restriction will be removed later.

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Complex Numbers at schools

• The school curricula
• in many countries normally do not include complex
  numbers
• in other countries complex numbers are a part of
  the school curricula.
• Only some elementary properties and operations
  treated
• Introduction in secondary school ??? (if at all)
• College Algebra course
• Intermediate Algebra course
• Equality, Addition, Subtraction, Multiplication,
  (Division) (CA Barnett, Ziegler)
• Traditional university course of (Introduction
  to) Complex Analysis
• More thoroughly
• Imaginary unit occurs not only in case of square
  root but also in case of logarithms, inverse
  trigonometric functions, etc.??
• hopefully passed by math teachers

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CAS

• Use of a CAS in the learning process creates a
  necessity and provides a chance to treat real and
  complex number domains more thoroughly.
• Test problems that
• dont initially include imaginary numbers
• the solutions where CASs "cross the border" of
  real number domain.

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Visibility of domain C?

• Visible i or C
• The imaginary numbers may appear in solutions of
  equations (already in case of quadratic
  equation).
• solve(x2 -1) ? i, -i
• MuPAD solve(0x0,x) ?
• Invisible C answers
• CAS may provide a solution of equation that is
  real number but is not appropriate when operating
  with real numbers only.
• solve( ) ? -1
• Equivalence of expressions (Separate paper)
• Equivalences known in school may not hold in CAS
  because of use of complex numbers
  (Aslaksen)
• What is the least restrictive constraint to make
  a given expressions equivalent?

13
Expectedness

• There are examples that teachers (and students?)
• expect
• visible square root related (e. g quadratic
  equations)
• but some examples are less known (hardly
  expected)
• visible logarithm related ln(-1) ?
  piexponential equations ex 1 0
• trigonometry arcsin(2.0) ? 1.570796327-1.3169578
  97i trigonometric equations sin(x)2
• invisible C answers
• radical equations
• logarithm equations
• arcus equations arccos(2x)arccos(x2) solution 2

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Default (current) domain

• What is the default domain in CAS?
• User manual (not always very informative)
• By default
• Maple, Mathematica, MuPAD C
• Derive C/(R) (solve Complex/Real),
• TI-92, TI-nspire C/R (Complex Format
  Real/Rectangle/Polar, csolve)
• WIRIS R
• How complex?
• Test,
• may be more detailed

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Test problems
Square root Logarithm arcsin arccos
Calculation ln(-1) arcsin(2)
Equation (visible i) x2 -1 ex-1 sin(x)2
Equation (invisible C) arccos(2x)arccos(x2)
Equation 0x0 (visible C)
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Complex domain
x2 -1
All
ln(-1)
All except WIRIS
All except WIRIS and MuPAD
arccos(2x)arccos(x2)
All except WIRIS and TI-s
ex-1
All except WIRIS, Branches in Derive, MuPAD, TI-s
arcsin(2)
All except WIRIS, Numerically arcsin(2.0) in
Maple, Mathematica, MuPAD
All except WIRIS, Branches in Derive, MuPAD,
TI-s Numerically sin(x)2.0 in Maple,
Mathematica, MuPAD
sin(x)2
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Controllableness

• How could one set the domain (?R)?
• There are differences in the operation of
  different CASs
• in determination of domain
• of the calculation result,
• the variable value,
• the equation (inequality) solution
• the entire process.

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How to determine the
to R Derive Maple Math-ca MuPAD TI-92 TI-nspire WIRIS
calculation result Packace RealDomain Packace RealOnly Complex Format Real default
equation solution Solution Real Domain Packace RealDomain Packace RealOnly assume solve default
entire process Packace RealDomain Packace RealOnly default
Not complete Exceptions (e.g Maple logarithmic
equations)
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Technical approaches

• Special Commands (cSolve)
• Assumptions
• Menu -gtmode
• Menu-gt radio button (Derive, Solve)
• Packages

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Teacher actions

• Possible plan
• clarify how a particular CAS works on a
  particular problem
• In tables of this paper?
• Test (guide will be in paper)
• decide
• Avoid such problem in using CAS
• Adjust CAS (if possible)
• Add explanations (which?)
• Is explanation useful and meaningful for student?
  
• Will the topic be treated later?
• Dont explain
• ???

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Explanation? Too mathematical?
L. Euler 1746
Complex logarithm is multivalued.
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4x 64
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Summary

• School
• Merenluoto and Lehtinen little attention is
  paid to the underlying general principles of the
  different number domains in the traditional
  curriculum.
• School treats complex numbers slightly if at all
• Use of a CAS in the learning process
• creates a necessity and provides a chance to
  treat more thoroughly.
• CASs
• are different
• in default domain
• in determination of domain
• attempt to comply with pure mathematics rather
  than school mathematics
• relatively well-adjustable (Assumptions,
  RealDomain, RealOnly.)
• Teacher must
• know how particular a CAS works on a particular
  problem
• choose a proper action (avoid, adjust, explain,
  ??)

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Other areas
Unexpected answers
Real and complex numbers (CADGME, today)
Branches (ICTMT8, in July 1)
Restrictive constraints
Equivalence
Infinities and indeterminates
in CASs and school
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Future Work

• Systems and inequalities
• Other CASs, versions
• Related works?
• Suggestions?