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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

• 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

Infinities and indeterminates

Real and complex numbers (CADGME, today)

Branches (ICTMT8, in July 1)

Equivalence

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

Real

Imaginary

Complex

Rather:Border or bridge between R CReal 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

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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

• 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

)( CRQZN

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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”.

yxyx 2z

ee z

012 x

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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

– – 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)

• “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?

C

12 xx

i 1

ze z )ln(

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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) πiexponential equations ex +1 = 0

• trigonometry: arcsin(2.0) 1.570796327-1.316957897i trigonometric equations sin(x)=2

• invisible C answers – radical equations – logarithm equations – arcus equations arccos(2x)=arccos(x+2) solution 2

)12log()log( xx

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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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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(x+2)

1

12 xx )12log()log( xx

Equation 0x=0 (visible C)

Test problems

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1

12 xx

)12log()log( xx

Complex domain

ln(-1)

arcsin(2)

x2 = -1

ex=-1

sin(x)=2

arccos(2x)=arccos(x+2)

All

All except WIRIS

All except WIRIS and MuPAD

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-sNumerically 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

• 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 ->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?

i )1ln(

...!

)(...

!3

)(

!2

)(

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32

n

xixixixie

nxi

L. Euler 1746

xixe xi sincos

x

i )1ln( kii 2)1ln(

Complex logarithm is multivalued.

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4x = 64

For example, Derive gives in the case of 644 x answers

x=)2(

33LN

i

v x=

)2(

23LN

i

v x=

)2(

23LN

i

v x=

)2(3LN

i

v x=

)2(3LN

i

v x=3;

MuPAD gives

kki

)2ln(

)2ln(3

.

Mathematica, TI-92+ and WIRIS gives 3. Maple gives )4ln(

)64ln(.

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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

Infinities and indeterminates

Real and complex numbers (CADGME, today)

Branches (ICTMT8, in July 1)

Equivalence

in CASs and school

i 1

C* Restrictive constraints

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Future Work

• Systems and inequalities

• Other CASs, versions

• …

• Related works?

• Suggestions?