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Real and Complex Domains in
School Mathematics and in
Computer Algebra Systems
Eno Tõnisson
University of Tartu
Estonia
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Plan
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Introduction
School
CASs
Teacher
Summary
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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/dissimilarities
but to use them positively.
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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?
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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)
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–
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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
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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 N  Z  Q  R ( C )
• 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)-(x+5)=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  x  y
z
(x,y ≥ 0)
(R/C, H. Aslaksen)
ez  e 2
– the solution sets may differ in different number domains.
• x2 1  0
• 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
– don’t 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
–
1  i
– The imaginary numbers may appear in solutions of equations (already in
case of quadratic equation).
• solve(x2 = -1)  i, -i
• MuPAD: solve(0*x=0,x)  C
• “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( x  2 x  1)  -1
– Equivalence of expressions (Separate paper)
• Equivalences known in school may not hold in CAS because of use of
complex numbers
ln( e z )  z
(Aslaksen)
• What is the least restrictive constraint to make a given expressions
equivalent?
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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)  πi
exponential equations ex +1 = 0
• trigonometry:
arcsin(2.0)  1.570796327-1.316957897i
trigonometric equations sin(x)=2
• invisible C answers
– radical equations
log( x)  log( 2 x  1)
– logarithm equations
– arcus equations arccos(2x)=arccos(x+2) 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
Calculation
1
Logarithm
ln(-1)
arcsin
arccos
arcsin(2)
Equation
x2 = -1
ex=-1
sin(x)=2
(visible i)
Equation
x  2 x  1 log( x)  log( 2 x  1) arccos(2x)=
arccos(x+2)
(invisible C)
Equation 0x=0 (visible C)
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Complex domain
1
x2 = -1
x  2x  1
ln(-1)
log( x)  log( 2 x  1)
All
All except WIRIS
All except WIRIS and MuPAD
arccos(2x)=arccos(x+2)
ex=-1
arcsin(2)
sin(x)=2
All except WIRIS and TI-s
All except WIRIS, Branches in Derive, MuPAD, TI-s
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
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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
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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
calculation
result
equation
solution
entire
process
Solution
Real
Domain
Maple
Math-ca MuPAD
TI-92+
WIRIS
TI-nspire
Packace
RealDomain
Packace
RealOnly
Complex
Format
Real
default
Packace
RealDomain
Packace
RealOnly
solve
default
Packace
RealDomain
Packace
RealOnly
assume
default
Not complete
Exceptions (e.g Maple logarithmic equations)
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Technical approaches
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Special Commands (cSolve)
Assumptions
Menu ->mode
Menu-> 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?
• Don’t explain
– ???
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Explanation?
Too mathematical?
L. Euler 1746
xi ( xi) 2 ( xi)3
( xi) n
e  1 

 ... 
 ...
1
2!
3!
n!
xi
e xi  cos x  i sin x
x 
ln( 1)  i
Complex logarithm is multivalued.
ln( 1)  i
ln( 1)  i  2ki
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4x = 64
For example, Derive gives in the case of 4 x  64 answers
2   i
2   i
 i
 i
3  i
x= 3 
v x= 3 
v x= 3 
v x= 3 
v x= 3 
v x=3;
LN (2)
LN (2)
LN (2)
LN (2)
LN (2)
 3  ln( 2)  ik

MuPAD gives 
k  
ln( 2)

.
Mathematica, TI-92+ and WIRIS gives 3. Maple gives
ln( 64)
.
ln( 4)
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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)
C
*
1  i
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?
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