My Calculator Is Broken; It Says the Log of (—1) Is …Author(s): Jeremy A. Kahan and Glen W. RichgelsSource: The Mathematics Teacher, Vol. 96, No. 2 (FEBRUARY 2003), pp. 108-111Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871253 .

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Jeremy A. Kahan and Glen W. Richgels

My Calculator Is Broken; It Says the Log of (-1 ) Is

R

The tables had been

turned

ecently, a student in the mathematical methods for

secondary teachers class that I (Richgels) teach stop ped me in the hall and asked a question: 'What does it mean when a calculator gives you a number for the

log of negative one?" Immediately, smiling inside be cause I thought that I had caught a rookie, I asked the student to show me the calculator. I was impressed because the student had a new TI-86. The student entered wra -1. When he pressed the key, I

expected an error message. Instead the calculator

displayed (0,1.36437635384). I did a double take and asked for the calculator. I had seen the student enter the correct expression but had to verify the result for myself. For almost thirty years, I had taught that the logarithm of a negative number is not defined. We used to enjoy asking students to use their

calculators to find the log of a negative number. When the calculator gave an error message, we could follow up with "Is your calculator broken?

Why is it doing that?" Students might then appeal to the graph of y = 10* and note the horizontal

asymptote. Or they might look at a table of values or simply argue that 10 raised to a power could never give an answer of -1. Or they might even look at their textbook's graph of y = log or the log table and note that no negative jc-values exist for which y-values occur. In any case, setting up a sur

prising experience with the calculator allowed stu dents to deepen their mathematical understanding.

EXPLORING MATHEMATICS The tables had been turned. I repeatedly entered B 3-1 and obtained the same result as the stu dent. Finally, I explained that something must be

wrong, and the student agreed because he knew that the log of any negative number is undefined. I

pursued the problem for several weeks, by dis

cussing it with the student and colleagues, by exchanging e-mails with Texas Instruments, by experimenting with the calculator to seek patterns, and by relating to what I already knew about loga rithms and exponents. Here is what I learned, as well as some of the important steps in the process.

One strategy was to test how log (-100), log (-10), and log (-1) are related. For 1,10, and 100, the log

arithms increase by 1 each time we multiply by 10, so would a similar pattern occur here? The calculator

reported (0,1.36437635384), (1,1.36437635384), and (2,1.36437635384) as the logarithms of-1, -10, and -100, respectively. At least this pattern of

magnitudes increasing by 1 as the input increased

by a factor of 10 was familiar. Another clue was that the TI-86 was the newer

version of the TI-85, which worked in complex mode all the time. The problem, a form of which can also occur when someone sets other calculators (for ex

ample, the TI-83, which is a bit less puzzling since it uses a + bi notation) to complex mode (and radian

mode), arose by default here. The ordered pair (a, b) on the TI-86 represents a + bi, so log (-1) =

(0,1.36437635384) meant log (-1) = 1.36437635384?. I knew that the common logarithm could be viewed as an exponent of 10, so it meant that 10L36437635384? =

-1. If that assumption was true, then

jq2+1.36437635384? _

^q2 ?

jq1.36437635384?

= 100 ? (-1) =

-100,

so that the change in the real part of the exponent fit well with the laws of exponents and the value for the common logarithm of-100 given by the calcula tor matched the exponent 2 + 1.36437635384/ to which 10 was just raised to obtain -100.

But what did raising 10 to an imaginary power mean? At this point in my thought process, I recalled the equation em = -1. Then, using a base e

logarithm, the logarithm is the exponent, which is to say, In (-1) = m. On a TI-86, it should look like

(0, 3.1415 ...), and it did. The rest followed more

easily.

Jeremy Kahan, kahan002@umn.edu, teaches mathematics education at the University of Minnesota, Minneapolis,

MN 55455. He is interested in using problems to help teachers reflect on important mathematics and its teach ing. Glen Richgels, grichgels@bemidjistate.edu, teaches

mathematics at Bemidji State University, Bemidji, MN 56601. His area of concentration is working with pre service and in-service mathematics teachers and their pro

fessional development.

108 MATHEMATICS TEACHER

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An important topic featured in many calculus textbooks (e.g., Repka [1994]) is the development of series. The following three series, the derivations of which we omit, are called MacLaurin series, and

they are crucial in understanding why em = -1.

They are

/yO vl v2 v3 v4 v5

"0! 1! 2! 3! 4! 5!

0! 2! 4! 6!

and

S1I13C=___ + ___+...,

Working with e", we obtain the following:

Bh _ (w? . ijtf . (jx? . iixf . (jxf . iixf . 0! 1! 2! 3! 4! 5!

vl 1! 2! 3! 4! 5!

f-t X X .

2T 4 ~6 : cos +1 sm

!1! 3! 5! 7!

Some readers call it eis for short, a notation whose appealing compactness we avoid here for the sake of clarity.

e111 = cos + i sin = -l + -?(0) = -1

In this way, Euler's formula ew = cos 0 + i sin ? can be the foundation for realizing negative num bers as positive numbers raised to some power, which is equivalent to finding logarithms of nega tive numbers. Euler's formula also connects imagi nary exponents and the sine and cosine functions, whose inputs are the real numbers?and in turn it can serve as the basis for extending sine and cosine to take complex inputs (see Beals [1973, p. 59]).

In addition to learning how the mathematics fits

together, I also echoed my students' question of, When are we ever going to use this?" I learned that this area of complex analysis arises in issues of wind or fluid flow, for example, when designing an

airplane wing. As a learner, I was pleased. As a teacher, my mind was dazzled by the oppor

tunities that this problem presented. Students could investigate series, complex numbers, approxi mations of periodic functions, and modes of opera tion on graphing calculators. This problem is a rich one. But how would we teach it?

EXPLORING TEACHING We would take the following approach, which builds on four conference sessions attended by sec

ondary teachers. The teachers enjoyed the combina

tion of graphs and symbols in our approach and believed that it would work with students. We first eliminated the direct, lecture approach, because the

approach should be a problem-solving journey like

mine, but it should be one that precalculus stu dents could take. We plan to make the students comfortable first and then guide them toward a solution.

So we would start with the graph of f(x) = log x. What is the domain? What is the range? We would let students graph f(x) to put them at ease.

The next step would be to help the students dis cover that a series could approximate a function. As an introduction, teachers could begin by sharing that in this lesson students will see how to approxi mate values of trigonometric functions in terms of more familiar polynomial functions. The following are the steps in that process (see fig. 1):

Graph Yl (x) = sin(x).

Graph Y2 (x) = x; notice that Y2 approximates Yl near the origin. Alternatively, students could use a calculator-based or printed trigonometry table, in radians, for sine, and the teacher could ask students what they notice. Both Yl and Y2 are odd functions; so as we extend to Y3 and beyond, we build with odd exponents only and thereby preserve the oddness of the approximating poly nomial functions.

Graph and examine the table values for the function

X x^ Y3(x) = Y|-g|;

what do you notice? Next graph and examine the table values for the function

Y4(r) = ?. _ ?? + . 1! 3! 5!'

what do you notice?

After these investigations, students have at hand a plausibility argument (rigorous proof needs to

The next

step would be to help the students

discover that a series could

approximate a function

Vol. 96, No. 2 ? February 2003 109

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Students should be

able to make the

connection

that e = cos +

i sinx

wait) that indicates why we can calculate (or some

times define) sine by its MacLaurin series. Next, the teacher can ask students to investigate the

MacLaurin series for the cosine and exponential functions; for the cosine series, students who are

reasoning by analogy with the sine series may even guess the series by themselves if they observe that cos is about 1 when is near 0 and that the cosine function is even. Next, the teacher can ask the students to evaluate elx, as above. Because of the investigations of the sine and cosine, students should be able to make the connection that elx =

cos + i sin .

Last, we need to look at examples and then check to see whether the log or natural log is what we

would expect. First, we check that what we already knew remains true, that the logarithm calculated

through the complex form agrees with the real one.

(I) 1 = cos 0 + i sin 0 =

ei0,

so extending the basic assumption that if a = eb, then In a = b, we obtain In 1 = ?0 = 0, as expected, on the basis of experience with natural logarithms.

Example (1) can be extended to any positive real

number, r. For example,

(2)

so

5 = 5 (cos 0 + i sin 0) =

5e'?,

ln(5e?0J = ln5 + ln (ei0) = In 5 + 0 = ln 5.

Thus, in treating logarithms in this way, we are

extending our earlier use of the logarithm function on positive real numbers. Students can also check that toggling calculator modes for positive real numbers does not affect the values reported for nat ural logarithms. Although this result might seem

obvious, toggling modes does change the value of the cube root of -1 on the TI-89.

Then we could extend to numbers whose loga rithms were previously undefined, starting with

negative numbers, proceeding to pure imaginaries, then to the complex. Each of these examples may be predicted first, then confirmed by the calculator; or if the teacher prefers, the examples can be done

by using the calculator first, then students can jus tify their results.

(3)

so

-1 = cos + i sin

e171,

In (-1) = 0 + .

(4)

so

(5)

so

(6)

= cosi||

+ ?sin(|

ln/ = 0 + S?

- = cosf-yj

+ i sinf-y

= e* ,

ln(-?) = 0 + ̂?.

V2 V2. -I - cos

-3 +1 sin

-3

:e<(--n

so

Ini ̂ V2. 4

As in the transition from example (1) to example (2), we can use the property for the logarithm of a

product?which we ought to show formally again for the complex logarithm?to extend from numbers of the form elB to those of the form rew,

(7) _i_/ = V2(cos^)

+ /sin^))

= V2 -3

so

ln(-l-i) = (lnV2) +Inferi = 0.34657359028 + ?. 4

The examples thus far illustrate the tidy rela

tionship In (e*) = . Nevertheless, the choice of

rather than

-

2

3tt 2

in example (5) may look strange to some readers.

Similarly strange is the following example:

(8) 1 = cos (2 ) + i sin (2 ) = em

but In 1 = 0, not 2 .

These examples lay the groundwork for stu

dents to learn that the logarithm of a complex number is not a function unless we restrict the

range of values of the imaginary part (Beals 1973,

110 MATHEMATICS TEACHER

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p. 61); students can relate this issue to their expe rience in defining inverse trigonometric functions, where arccosine (cos 2 ) 2 . Indeed, the issue is

fundamentally the same for defining the inverse of

any many-to-one function, that is, functions that fail a horizontal-line test, as students will have encountered in defining square roots of positive numbers to be positive only, with the upshot that

?**-3.

Teachers may wish to extend the method of

example (7) to representing nonzero complex num

bers in the form of - r (cos 0 + i sin = rel6, which then leads (with some care to the sets from which r and may be chosen) to In as (In r + ). Students may then visit or revisit r feos 0 + i sin 0) notation and investigate analogies between loga rithmic and trigonometric functions applied to

sums, differences, products, quotients, and pow ers. Another simpler challenge might be to find In (3?) in two different ways, in addition to using the calculator.

Finally, students need to return to the base 10 log arithm that introduced this investigation and devel

op a justification for it. At one time, students were

(perhaps not so unwisely) forced to memorize that

Ina loga=EI?

and that

1 In 10

is approximately 0.4343. Assuming that the change of-base formula still applies in the complex realm

(an assumption that readers may wish to verify) and using the result of example (3), we have

(9) k?M>.!?$

=ir?w-u ? 0.4343(0+ - (0 + 1.364/).

Contrasting this example with example (3), we see an answer to the question, What's so natural about natural logarithms?

This problem has given us an opportunity to

investigate functions, domain, range, approxima tion of functions with series, and representation of and operations on complex numbers. Readers who

attempted the exercise of finding In (3/) two differ ent ways may wish to check that their result is

ln(3) + 2?*

1.09861 + 1.5708/.

REFLECTIONS ON TECHNOLOGY The foregoing story exemplifies the ways that tech

nology will provide new opportunities and challenges

for teachers, whether we like it or not. Students with access to this technology will present teachers with mathematical challenges. For problems like the one with which we began, the TI-89 does not

cooperate when we say that no solution is possible. At least, it will tell us the piquant "nonreal result." When we set a TI-89 to complex rectangular format and exact mode, it yields In (-1) = m or

log(-l) = ??.?. We could not tell students that nothing was going on there, and we worry about the messages about mathematics and learning that a brush-off would send. We also believe that teachers who are pas sionate about their subject would want to know more when thus challenged. The technology prods teachers to develop a deeper knowledge of their

subject. At the same time, the technology challenges us

as teachers. We do not want to rush students into material for which they are not prepared; but in our example, technology pushes us to find a way to

explain complex analysis in a mathematically meaningful way that is accessible to precalculus students. The good news is that if we take the chal

lenge, or if we engineer it by design, technology offers us opportunities to guide students into a

mathematically rich terrain, perceiving beautiful connections within mathematics. In the process, we

deepen our understanding and help deepen theirs.

TEACHING WITH TECHNOLOGY: A NEW COMPLEXITY In this article, we have explored how the calcula tor's capability to work with complex numbers chal

lenged a teacher to deepen his mathematical under

standing of the logarithms of negative numbers, in addition to exploring the beautiful mathematical connections that this journey revealed. We have also seen how the practice of teaching has become more complex as teachers try to help students make sense of, and then build on, what the calcula tor is saying. Any trigonometry teacher or able stu dent should know that arccosine 2 is undefined; right? Some calculators confirm this claim with an error message. But others?for example, the TI

89?yield 1.3169579? in some modes. What are

they trying to tell us about taking the cosine of an

imaginary number, and how should we teach it to others?

Technology prods teachers to

develop a

deeper

knowledge of their subject

REFERENCES Beals, Richard. Advanced Mathematical Analysis. New York: Springer-Verlag, 1973.

Repka, Joe. Calculus and Analytic Geometry. Dubuque, Iowa: William C. Brown Publishers, 1994.

Mr

Vol. 96, No. 2 ? February 2003 111

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