UNIVERSITY OF NEBRASKA–LINCOLN
COLLEGE OF EDUCATION AND HUMAN SCIENCES
TEAC851P – MATH TEACHING METHODS I
TERM REPORT
i AND COMPLEX NUMBERS
Prepared for
Dr. David Fowler
Department of Teaching, Learning and Teacher Education
University of Nebraska–Lincoln
by
Brian Hirt
April 20, 2006
Introduction
The question “What is i?” has a simple response: “That’s very poor grammar! The
correct way to say it is, ‘What am I?’”1 Clearly, one must depart the world of everyday
language to even entertain this as a question of mathematics rather than as possible
philosophical musing of Aristotle or Sartre.
This paper will investigate i (written in lower case and in italic script) not as a pronoun
but as a number, one with an interesting history, novel properties, and of many uses
despite the intuition of many people that i does not have any meaning in the physical
world. In the process, the paper will delve into a discussion of abstraction in
mathematics and numbers, touching on the acceptance of different kinds of numbers
throughout history.
This paper is written for an audience of adults that has general knowledge of mathematics
through college (up to but not including calculus), but with a complete ignorance of
imaginary numbers.
The Unsolvable Equation
Knowledge of high school-level algebra gives an adult the tools to solve the following
equation:
(1)
x2 = 9
There are two solutions: x is equal to 3 or –3, as one recognizes that both 3 2 = 9 and
(− 3)2 = 9 . Generally, the properties of multiplication describe that the product of two
positive numbers is positive, and that the product of two negative numbers is also
positive. So any non-zero number multiplied by itself can only result in a positive
product.
In contrast, high school algebra shows the next equation to be intractable:
(2)
x 2 = −9
Clearly, no such number can exist while satisfying the multiplication rules stated above.
The answer to an equation such as this is typically given as “no solutions exist,” and this
is arguably a satisfactory way to handle the dilemma. Without any numbers on the real
number line to satisfy this equation, there is little motivation to go looking further. One
might (as this paper later does) develop a novel number system to solve equation (2), but
there would be little motivation for a high school student to do so.
1
The author credits and thanks his brother, Kenneth Gerber, Ph.D. of Mathematics, for this rejoinder.
2
Indeed, historically the quandary of x 2 = −k (for some positive number k) for centuries
did not provoke mathematicians to seek out new ways to think about math in order to
solve them. The earliest known appearance of such a problem is presented in Nahin (pp.
3-4), where the story of the engineer Heron of Alexandria of the first century AD is
presented. Heron used the following equation to find the height of a truncated square
pyramid based on several of its measurable parameters:
(3)
h = c2 − 2
where:
a−b
2
2
h is the height of the frustum,
c is the slant edge length of the frustum,
a is the side length of the bottom base, and
b is the side length of the top base.
Heron applied this problem to a theoretical frustum of slant edge c = 15 units, and side
lengths a = 28 and b = 4. Using equation (3):
28 − 4
h = 15 − 2
2
2
2
h = 225 − 2(12 )
2
h = 225 − 288
h = − 63
Heron was at an impasse. Since the number − 63 was meaningless to him given the
rules stated above, he completed the solution by removing the “ –” sign and concluded
(incorrectly) that the answer was 63 . A more revealing approach would have been a
diagram (as created in Figure 1) showing that the pyramid in question was an impossible
one. Some simple algebra and geometry is suppressed in developing Figure 1 below:
3
2 2
A
4
15
h
h
B
C
28
12 2
14 2
Figure 1. Heron’ s impossible frustum (not drawn to scale for obvious reasons)
As shown in Figure 1, the parameters given require a right triangle ABC with hypotenuse
of length 15 and with one leg of length 12 2 . Since 12 2 > 15 and the hypotenuse
must be the longest side of a right triangle, the parameters are physically meaningless,
and it not unexpected that equation (3) fails to produce a meaningful answer. In fact,
rather than “ fudging” his arithmetic (Nahin, p. 4), Heron might have instead concluded
that the nonsensical answer of − 63 in fact supported that this was a nonsensical
problem. Returning, then, to equation (2), the conclusion that x 2 = −9 has “ no solutions”
in fact may be a useful insight depending on the context of the question.
Inventing a Number
However, it is possible to invent a number outside of what are considered the real
numbers that makes solving equation (2) possible. To this end, a wholly new kind of
number, called i, can be used, where i has the property:
(4)
i 2 ≡ −1
All of the existing operational rules of algebra that apply to real numbers, such as
distribution and commutivity, are allowed to be applicable to i as well. Further, since the
existence of i does not contradict the use of any operators in the pre-existing
mathematical system (Asimov, p. 103) of real numbers, it may be considered a number
that simply extends the existing mathematical system. Solving equation (2) now
becomes possible as follows:
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x 2 = −9
x 2 = 9 ⋅ −1
x2 = 9 ⋅ i2
x = ±3i
The number i was first used conceptually by Girolamo Cardano in the first half of the
sixteenth century (Reid, p. 108) precisely for solving abstract polynomial equations such
as the one above, although he did not use today’ s notation. The letter “ i” “ ...was first
used by Leonhard Euler in a memoir presented in 1777” (Miller) and was selected to
reflect the idea that the number represented something that Cardano had called
“ imaginary” (Reid, p. 108) since it does not represent any quantity in the real world.
Expanding beyond this one imaginary number i, it is possible to create an entire number
line of numbers that behave like i (i.e., which when squared yield negative numbers).
Such an imaginary number line may be drawn in a manner analogous to the real number
line. Where on the real number line one may find 5, –2.7, or π, on the imaginary number
line one may similarly find 5i, –2.7i, or πi. i itself is the number analogous to 1 on the
real number line, and it is called the imaginary unit. This is illustrated in Figure 2.
2.7
0
1
π
5
Real
2.7i
0
i
πi
5i
Imaginary
Figure 2. The real and imaginary number lines
The question remains why such an undertaking—the development of this new kind of
number—is even desirable. To find the answer, one may look first to another class of
numbers that was once considered outside the realm of understanding: the negative
numbers. The Greeks at the time of Pythagoras, the 6th century BCE, did not accept
numbers smaller than zero (Reid, p. 104, Ifrah p. 596). To them, equation (2) would be
no less solvable than:
(5)
x+7 = 4
Clearly, no positive number could be added to 7 to get a smaller result. From the
standpoint of geometry, a number like –3 (which is today considered the solution to
equation (5)) was meaningless, since polygons and polyhedra never have side lengths of
–3 units.
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However, negative numbers, if abstract, can be extraordinarily useful in real-world
applications. For the purposes of accounting, surpluses may be thought of as positive
numbers, and shortfalls may be thought of as negative numbers. So even if a debtor
never physically carries bills totaling –$54 in his pocket, the number is a useful
representation of the fact that he owes his creditors $54 dollars.
The parallel may be drawn back to imaginary numbers. Indeed, nobody carries $54i
dollars in his or her pocket either, yet there are instances where i may have enormous
practical applications. From a historical perspective, it is unsurprising that it was through
pursuit of making practical math problems easier that i was fully developed and accepted
as a useful and necessary number.
Making Use of i
Credit for pioneering imaginary numbers beyond the wholly abstract studies of Cardano
and Euler is credited to Caspar Wessel, a Norwegian surveyor (Nahin, ch. 3). In 1797 he
presented a paper in which he sought to explain how two-dimensional vectors (arrows
drawn on a coordinate plane with their tails on the origin) may be easily manipulated,
such as multiplied or taken to powers. Such tasks were essential to his profession of
surveying and were considered to be very cumbersome using the traditional mathematics
of his day. For his purposes, Wessel found it useful to draw his two-dimensional grid and
to represent one of the directional axes (such as the east-west axis) with the real number
line, but the other (north-south) with the imaginary number line.
Wessel could draw any vector as an arrow whose tail was at zero (the intersection of the
axes) and whose head was at some point that had a real component and/or an imaginary
component. Such a point is called a complex number. This is shown in Figure 3.
Imaginary
+
a + bi
b
r
N
–
θ
0
a
Real
+
–
Figure 3. Wessel’ s geometry of complex numbers
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Using algebra and trigonometry, the general complex number a + bi shown above may be
equivalently expressed as r∠θ ,
r = a 2 + b 2 , the magnitude of the vector in Figure 3, and
b
θ = arctan
, the angle between the vector and the positive real axis.
a
where:
a + bi is called the binary form of a complex number, and r∠θ is called the polar form.
The payoff behind this elaborate setup is found in the uncomplicated arithmetical
properties that result when one attempts to raise a vector to a power or to multiply or
divide it by another vector. Given complex numbers z1 = r1∠θ1 and z 2 = r2 ∠θ 2 (which
can represent vectors drawn on a real-world north-east-south-west plane as in Figure 3):
(6a)
z1 = r1 ∠nθ 1
(6b)
z1 ⋅ z 2 = (r1r2 )∠(θ1 + θ 2 )
(6c)
z1 z 2 = (r1 r2 )∠(θ1 − θ 2 )
n
n
These three formulae are elegant and easy to use, yet they require considerable algebra
and trigonometry to develop from the rectangular (a + bi) form of each complex number.
The derivations of these formulae are not presented here, but may be found in detail in
Usiskin (pp. 56-59). Usiskin shows that these properties are developed given the critical
property that i 2 = −1 . In this manner, the number i, whether existent or not, has proved
itself to be extremely useful.
Other examples of the uses of complex numbers abound in the sciences and engineering.
One such is in the area of AC electrical circuits (Hambley, pp. 188-190). When these
circuits involve capacitors and inductors, and when voltage is given as a sinusoidal
function of time, finding a simple function for circuit current requires combining multiple
sine expressions. Using only real number-based mathematics, intensive use of
trigonometric identities is required to achieve the desired result.
However, representing the voltage as a polar complex number whose angle θ is a
function of time makes the problem a far easier one. In this manner, the function for
current may be quickly found using only arithmetic on complex numbers. This is called
the phasor method for circuit analysis, and it is a key tool in modern electrical
engineering. Given the alternative, no engineering student would deny the importance of
i.
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Conclusion: “Does or does not i exist?”
Invariably, though, the question returns to whether i and the family of complex numbers
actually exist. The answer may lie in the idea of context, and whether any number can be
said to exist.
Consider a foot race. Certainly, negative numbers, though populating the real number
line, do not have any meaning and could be said to be nonexistent in the context of timing
runners in a 50-meter dash (who can run it in –10 seconds?). Moreover, non-integer
rational numbers, such as 3/2 (equivalently 1 1/2), do not have any meaning and are
nonexistent in the context of counting votes on the floor of the U.S. Senate (how can a
senator cast half a vote?).
One might even go so far as to say that the most basic kind of number, the counting
number, is merely an abstraction that exists outside the realm of the physical world.
While the reader may have one heart and two lungs, what are the numbers 1 and 2 by
themselves? What is the physical manifestation of the counting numbers under
circumstances where they aren’ t actually counting anything?
From this perspective, one is hard-pressed to back up the assertion that the number 1
exists any more or less than the number i does. For its key role in solving difficult math
problems, an argument for the acceptance of i seems well justified, even if its existence
remains a debatable issue. In this light, the question “ What is i?” — like the question
“ What am I?” — is perhaps best left to the likes of Sartre as well.
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Works Cited
Asimov, Isaac. Asimov on Numbers. Garden City: Doubleday, 1977.
Hambley, Allan R. Electrical Engineering. 2nd ed. Upper Saddle River: Prentice Hall,
2002.
Ifrah, Georges. The Universal History of Numbers. New York: Wiley, 2000.
Miller, Jeff. “ Earliest Uses of Symbols for Constants.” 29 Dec. 2001. Gulf High School,
New Port Richey, Florida. 2 Apr. 2006
<http://hometown.aol.com/jeff570/mathsym.html>.
Nahin, Paul J. An Imaginary Tale: The Story of –1. Princeton: Princeton University
Press, 1998.
Reid, Constance. A Long Way from Euclid. New York: Crowell, 1963.
Usiskin, Zalman, et. al. Mathematics for High School Teachers. Upper Saddle River:
Pearson Education, 2003.
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