TIME 2012
Technology and its Integration in Mathematics Education
10th Conference for CAS in Education & Research
July 10-14, Tartu, Estonia
Working Environment:
Real or Complex?
TI-Nspire CAS CX and V200 offer the
user the choice.
Which one should we select?
What kind of maths do you want to do?
Who are you teaching to?
Who is using this machine?
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Working Environment:
Real or Complex?
If you are teaching at university level,
there is no doubt about it:
the complex format gives you the
opportunity to do MORE mathematics;
moreover, the complex format allows you
to also recall simple formulae used by
high school students.
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Working Environment:
Real or Complex?
Using Nspire CAS, we will now give
examples of some computations,
switching from the real branch to the
complex branch.
Most examples will be done using the
handheld instead of the software version.
The price we have to pay, when using the
complex branch, is relatively small
compared to the benefits.
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Overview
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Examples (simple ones).
Complex Menu of Nspire CAS.
Solving Equations.
Improper Integrals.
Antiderivative of 1/x.
Another Problem with log(x).
Conclusion.
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Examples (simple ones)
What is the “cubic root” of 8?
Benefit: let’s factor, (c)solve an
equation, recall remarquable formula.
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Examples (simple ones)
Sometimes,
looks “strange”:
Benefit: how can I use an integer? For a
CAS, what does mean?
Examples (simple ones)
Does
always simplify to
?
Benefit: let us use a complex number z
(defining z as x + iy) and “play” with it.
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Complex Menu of Nspire CAS
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Solving Equations and
Intermediate Complex Values
Sometimes, the solver finds a (real)
solution … but graphs don’t show it!
Let’s try with the following functions:
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Solving Equations and
Intermediate Complex Values
What is the domain of the following
function?
What happens when you solve the
equation dj(x) = c for a constant c?
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Solving Equations and
Intermediate Complex Values
Sometimes, the solver needs a “csolve”
command in order to find a ... real
solution!
Let’s try solving 3x a(n) + a(n + 1) =0
with
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Solving Equations and
Intermediate Complex Values
Sometimes, the ODE solver needs
complex arithmetic… Let’s try with the
following ODE, using the real or
complex branch:
Benefit: where the solution you are
looking for should be defined?
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Improper Integrals
(infinite integrands)
Here is an improper integral that first
year calculus students learn to compute:
The handheld confirms the result, using
the REAL branch:
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Improper Integrals
(infinite integrands)
Using the COMPLEX branch displays an
answer that is correct according to …
complex analysis! But this is not what
we want here.
So, when teaching this subject, you must
tell the students to select the REAL
branch!
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The Problem with the Antiderivative of 1/x
Calculus textbooks use
Major CAS are using ln(x);
Derive
Maple > int(1/x, x); ln(x)
Mathematica Integrate[1/x,x] = Log[x]
wxMaxima (%i2) integrate(1/x, x)
(%o2) log(x)
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The Problem with the Antiderivative of 1/x
Nspire CAS and V200 offer the user the
choice, by selecting the appropriate
branch.
What is the problem with the absolute
value? Here is an answer:
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The Problem with the Antiderivative of 1/x
If f is a continuous function defined over
an interval I, the every antiderivative of f
on I differs by a constant.
“continuous over an interval” is
important: for example, over , the
unit-step function and the signum
function both have 0 as derivative but
they don’t differ by a constant.
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The Problem with the Antiderivative of 1/x
The function 1/x is continuous on every
interval that does NOT contain the origin.
So, when integrating 1/x, it would be
better to write this:
(but probably it would take up “too much
space” in a textbook...).
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The Problem with the Antiderivative of 1/x
If we use
instead of
,
then we must pay attention to
(intermediate) complex values: we should
tell students that, for x > 0,
ln(x) = ln(x) + p i.
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The Problem with the Antiderivative of 1/x
But we can use the fact that ln(x) is also
an antiderivative of 1/x… and this is a
good choice when x < 0.
Let us give a concrete example, using
complex values in the first case and using
only real numbers in the second case.
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The Problem with the Antiderivative of 1/x
For example,
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The Problem with the Antiderivative of 1/x
The absolute value:
1. unnecessarily makes the antiderivative incorrect for non-real endpoints
(and some people will eventually use non-real endpoints not realizing the
incorrectness),
2. completely hides the non-integrable singularity if someone integrates
from a negative to a positive value. (With ln(x) you get a some warning
from the i p term.)
3. makes it very difficult to simplify subsequent expressions that use the
antiderivative -- for examples, mixtures of ln(x) and ln(abs(x)). As
another example, iterated integrals become very difficult if an inner integral
generates a ln(abs(x)).
With all of its extra closure, the complex domain is so much easier than the
real domain. I don't know why we put these extra hurdle constraints on
students who have been exposed to complex numbers in earlier courses.
(David Stoutemyer)
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Another Problem with the
Absolute Value in the log
Partial fraction expansion can be hidden
when you integrate a rational function,
using the real branch:
Moreover, the “temptation” to simplify
ln(ab) into ln(a) + ln(b) is a bad habit
students should lose when using a CAS.
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Another Problem with the
Absolute Value in the log
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Conclusion
We believe that the complex branch is the
one we should select when students have
been introduced to (elementary) complex
numbers.
This should ALSO be the case in a single
variable calculus course where students
are using a CAS calculator in the
classroom.
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Conclusion
First year calculus students don’t
compute line integrals in the complex
plane … so, for improper integrals with
infinite integrands, the REAL branch
must be the selected one.
(especially for rational powers having
odd denominator!)
For all other computations, we
recommend the COMPLEX branch.
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Conclusion
First year calculus students do need
implicit plotting (when they study
implicit differentiation).
In some cases, there is a possibility to
“fake” implicit plot with Nspire CAS,
using the “zeros” command … but the
complex branch should be selected if all
parts of the curve have to be shown. For
example;
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Conclusion
Here is the graph of this curve, using
Derive:
Nspire CAS can also plot the curve:
(Real branch)
(Complex branch)
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Conclusion
The subject of Taylor series is studied by
first year calculus students.
A graphical approach is often used to try
to find the interval of convergence.
Without complex numbers, it can be quite
difficult to find it exactly.
Here is a very nice example provided by
Bill Bauldry.
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Conclusion
Consider the function tan(sin(x)) whose
domain consists of all real numbers.
What is the domain of convergence of its
Taylor expansion?
For real x, sin(x) is bounded by 1, so
tan(sin(x)) has no real singularities but
for some complex numbers x, we might
have sin(x) = p /2.
Here are a graph of tan(sin(x)) along with
2 of its Taylor polynomials about 0.
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Conclusion
The intervalle of convergence will be
found, using complex numbers. This can
be done with Nspire CAS:
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Conclusion
Usually, first year calculus students
will follow an ODE course at the
beginning of their second
year/semester.
In this course, the Laplace transform
package requires the Complex
branch:
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Conclusion
For engineering students – not only those
in the electrical department – complex
numbers should be a natural tool: it can
help to simplify some computations or to
obtain/prove some identities.
For example, using Euler’s formula and
taking real and imaginary parts is a good
way to prove the following identity:
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Conclusion
The fact that a CAS system doesn’t
simplify an expression when the domain
is unknown can become a great
advantage for the teacher.
1) We have the opportunity to recall
some properties of logarithms.
2) And we can show students two
properties that are valid
EVERYWHERE
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Conclusion
In fact, no matter which branch you
select, the CAS system won’t simplify an
expression if some additional information
is needed.
So, at college/university level, we should
(start to) add the domain of validity of the
formula we are using.
Opting for the complex branch is a good
way to think of this.
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