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Transcript
Real and Complex Domains in
School Mathematics and in
Computer Algebra Systems
Eno Tõnisson
University of Tartu
Estonia
1
Plan
•
•
•
•
•
Introduction
School
CASs
Teacher
Summary
2
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.
3
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
4
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?
5
Real and Complex Domains
Complex
Real
Imaginary
Rather:
Border or bridge between R  C
Real and imaginary
6
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
7
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
8
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”.
9
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
10
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.
11
“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?
12
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
13
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
14
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)
15
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
16
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.
17
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)
18
Technical approaches
•
•
•
•
•
Special Commands (cSolve)
Assumptions
Menu ->mode
Menu-> radio button (Derive, Solve)
Packages
19
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
– ???
20
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
21
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)
22
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, ??)
23
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
24
Future Work
• Systems and inequalities
• Other CASs, versions
• …
• Related works?
• Suggestions?
25