Why Try to Avoid Using Complex Numbers?

TIME 2012 Technology and its Integration in Mathematics Education

10th Conference for CAS in Education & ResearchJuly 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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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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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. 4

OverviewOverview

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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Sometimes, looks “strange”:

Benefit: how can I use an integer? For a CAS, what does mean?

( 1)n

cz



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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2a a

Complex Menu of Nspire CAS 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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31( ) 1, 2( ) 4f x x f x x



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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( ) 1 1dj x x x



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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2

( 1)( ) .

1

n

a nn



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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2

4, (0) 5.

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dy yy

dx x

Improper Integrals (infinite integrands) 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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8

32 3

0

13 1 7

( 1)dx

x

Improper Integrals (infinite integrands) 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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8

32 3

0

13 1 7

( 1)dx

x

The Problem with the Antiderivative of 1/x The Problem with the Antiderivative of 1/x

Calculus textbooks use

Major CAS are using ln(x); DeriveMaple > 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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1lndx x C

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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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 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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ln( ) ( 0)1

ln( ) ( 0)

x C xdx

x D xx


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) + i. 20

1lndx x C

x

1ln( )dx x C

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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For example,

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22

33

1 2ln( ) ln(2) ln(3) ln

3dx x

x

22

33

1ln( ) ln( 2) ln( 3)

ln(2) ln(3)

2ln(2) ln(3) ln .

3

dx xx

i i

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 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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3

33

1ln(3) ln( ) ln(3) ln(1) ln(3) ;

2 2

1ln ln 3 ln ln(3) ln(1) ln(3).

i

ii

idz i i

z

dz z iz

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11

1ln ln(1) ln 1 0 0 0...dx x

x

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20 2

1

16

x

dy dxy

sign( ) 1ln

1ln

d xx x

dx x x

dz z

dz z


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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22 1 11 1 1

( 1)( 4) 4( 4) 1 4

x

x x x x x x



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ConclusionConclusion

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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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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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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3 2 2 33 16 0x y x y


Here is the graph of this curve, using Derive:

Nspire CAS can also plot the curve: (Real branch) (Complex branch)

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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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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) = 2.

Here are a graph of tan(sin(x)) along with 2 of its Taylor polynomials about 0.

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The intervalle of convergence will be found, using complex numbers. This can be done with Nspire CAS:

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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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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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0

sin ( 1) 2sin( ) sin 2

sin 2

n

k

nk n


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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ln

,

0,

z w z w

w z w

e e e z w

e z z w

ln( ) ln( ) ln( ) ( or 0)zw z w z w


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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Why Try to Avoid Using Complex Numbers?