What Are
Complex Numbers?

This article is a
series of emails and posts I made concerning Steven StrogatzÕs NY Times article
on complex numbers, which appeared in early March 2010. I made one post, then a few more in response
to some I saw there. Then one
particular person, who I will call Mr. X, began a heated exchange with me about
a month later. He also produced a
video about complex numbers, the point of which is basically that
mathematicians are deceiving the general public about complex numbers.

It is not necessary to read StrogatzÕs
column to read all of this. Enjoy.

© Robert H. Lewis

Fordham University

My first post,
#19, March 7:

This is
a good article and I'm glad to see it appear in this fine series about mathematics.

I would have been happier if the
old-fashioned word "imaginary" had appeared always in quotes. Indeed, as he points out, complex
numbers are no more imaginary than is 2/3, -7, or the square root of 2.

I do take exception to his saying that
"i is defined to be the square root of -1." No. i is
defined to be the point (0,1) in the plane. Under an elegant and natural definition of multiplication of
such points, it is true that i^2 =
-1. After all, is 3 defined to be
the fourth root of 81?
Surely you remember in kindergarten, you sat around a circle and learned
to count: 1, 2, fourth root of 81, ...

It is quite ironic that this column
appears the same day (or maybe one later?) than the Op-Ed piece by the English
graduate student on Lewis Carroll and Alice in Wonderland. Reading that, one is left with the
impression that something is somehow "wrong" with complex
numbers. I'm glad this piece has
appeared so soon to set the record straight.

Prof.
Robert Lewis

Mathematics

Fordham
University

My
second post:

Poster
#49 asks two good questions:

1.
ÒWhat is the square root of i?Ó
Since squaring a number doubles the angle it makes with the positive
x-axis, we look for the square root of i on the line that makes a 45-degree angle
with the positive x-axis. Indeed, it is the point ( sqrt(2)/2, sqrt(2)/2), also
written as sqrt(2)/2 + sqrt(2)/2 i.
Multiply it out and see!
(Of course there is a second square root 180 degrees away from this
one.)

2. ÒWhy
don't we go on and make a number system out of 3-dimensional space or
4-dimensional space?Ó Great
question! The short answer is that
you can't! Now, yes, one can
define products of vectors in 3-space like dot product or cross product, and
yes, one can make 4-space into a sort of set of numbers with what are called
quaternions (mentioned by earlier posters), but these don't do the job. By "set of numbers" you would
want what mathematicians call a field:
you want to be able to add, subtract, multiply, and divide, according to
the intuitive rules that work for ordinary real (and complex!) numbers. This cannot be done in any dimension
higher than 2. That is not easy to
prove, but has been proven, rather recently.

This is a big subject. There are other
fields besides the reals and the complexes, in particular finite fields. Many of them contain square roots of
-1!

Prof.
Robert Lewis

Fordham
University

My
third post:

I feel
the need to respond to poster #..
[Mr. X]: Quoting:

"but the fact remains that there is
a fundamental DISCONTINUITY

between the reals and the imaginaries.
Despite widespread aversion

among mathematicians, the word
"imaginary" is perfectly apt.

The intriguing question for me is why
Strogatz and his fellow

mathematicians persistently mislead us
on this topic. Is it due

to professional myopia? Is the truth
too untidy? Too untidy for

what? Suggestions are welcome."

No, the
word "imaginary" is not apt.
There is nothing "imaginary" about complex numbers, nor is
there a huge leap or "discontinuity" between the set of real numbers
and the set of complex numbers.
(Note another old relic, the term "real number".)

There
is really no time to go into this wonderful subject here, but in the
progression of number systems, each included in the following:

positive
integers, all integers, rational numbers (fractions), real numbers, complex
numbers,

The
biggest and most difficult jump is that from rationals to reals, as Pythagoras
and his followers understood.

Prof.
Robert Lewis

Mathematics
Department

Fordham
University

[next day, personal
email from Mr. X]:

Dear Dr. Lewis:

In your second NYT
post re. the Strogatz column you refer to the progression in number systems and
state that, contrary to my post, continuity exists and the word "imaginary"
is not apt. This fails to address the point I raised: all numbers prior to the
imaginaries are necessary to reflect the physical world, whereas the
imaginaries were developed to create algebraic closure.

If you have time,
could you address this now?

[my response]:

Hi,

Historically,
sqrt(-1) was investigated and worked with so that various algebraic techniques
could be done, for example, finding roots of cubics. As you probably know, the
cubic formula was worked out in the Italian renaissance. Many people don't
realize that to solve the most interesting case (all three roots are real) with
that formula, it is necessary to manipulate complex numbers. That's one of the
reasons we don't routinely teach it.

Let Q = the
rationals, R = the reals, C = the complexes.

I'm not sure just
what you mean by "continuity". I think you are saying that the
transition from R to C needs some sort of extraordinary leap. I think that is
in the "eye of the beholder." As I tried to point out on the Times
website, the transition from Q to R is actually much more sophisticated. You
need Dedekind cuts or equivalence classes of Cauchy sequences. For R to C, all
you have to do is ask the very appealing and intuitive question, can we make
the plane R^2 into a field? (Of course one has to have the experience of, say,
a good 11th grader to find such a question natural and compelling.) One does
not need to ever mention solving the equation x^2 + 1 = 0 in this process.
Also, and this is more sophisticated, once one studies Algebra (i.e. abstract
algebra) at about the college junior level, one sees a purely algebraic and
very natural way to construct C from R by taking the polynomial ring R[x] and
modding out by the prime ideal <x^2 + 1>. BTW there is then a purely
algebraic* proof of the Fundamental Theorem of Algebra.

About physical
reality, I think that was gone into on the Times discussion. However, you say
that "all numbers prior to the imaginaries are necessary to reflect the
physical world". Not really. R is not necessary to any engineer. All they
need is arbitrarily precise decimal numbers. Those are all rational, i.e., in
Q.

Secondly, I'm not
sure anyone pointed out that in special relativity, the difference (actually
ratio) between space and time is i. So if you think sqrt(-1) doesn't really exist,
then you think that either space or time doesn't really exist.

Regards,

Robert H. Lewis

http://fordham.academia.edu/RobertLewis

*One needs to know
first that all odd degree polynomials in R have at least one root in R.

_______________________________________________________

Hello,

I watched your [Mr. XÕs video]. I have to tell you it is misguided.

I do recall the exchange of emails we
had in mid March. I am pasting below
the comments from Strogatz's column that I felt were noteworthy and saved, including
yours and mine. I hope you will
take the time and effort to carefully reread them.

Concerning your video, you are making a
profound error, as I think you were doing in March. You are saying that to define the complex numbers we essentially
go through three steps:

1. we define i = sqrt(-1).

2.
later, we place i on the plane at the point (0,1).

3.
then we think of multiplying by i as a rotation by 90 degrees.

You have it backwards. First, one must
understand the concept of field.
Acting on a quite natural desire to see if R^2 can be a field, one
defines a multiplication on R^2 via rotations and stretches. One checks that this makes R^2 a
field. We call R^2, with this
multiplication and vector addition, the field of complex numbers, C. One observes that R is naturally
imbedded in C as the x-axis.
"Natural", a very important concept in mathematics, means here
that all the arithmetic one is used to in R is exactly the same, whether we
think of the points on the x-axis as in R or C.

Then one observes that (0,1)^2 = -1,
i.e., (0,1)^2 = (-1,0). Since
(0,1) is a natural choice for the other basis element of the vector space R^2,
it is worth giving it a name. We
call it i. Thus, rewritten, we
have i^2 = -1.

We do not define i to be the square
root of -1 any more than we define 3 to be the fourth root of 81.

So, to summarize, your points 1, 2, 3
above are backwards. I suspect this is the heart of your confusion. By continually using words like
ÒimaginaryÓ and ÒchasmÓ you reveal your inability, or unwillingness, to learn
conceptually. But the essence of
mathematics is understanding, and understanding means to build concepts upon
concepts. Instead, you are
wallowing in your immaturity. That
is why you think that the complex numbers are a kludge, instead of a profound
insight.

It may be that you learned the points
1, 2, 3 in this order when you
were in high school. Could be, but
that only shows the poor quality of many American high schools. Furthermore, no doubt 500 years ago
some people thought of it that way (indeed, 3 was not thought of until much
later). That is of historical interest,
but we need not be a slave to 500 year old notions. Think Copernicus.

By analogy, as I wrote in March, many
years ago the Greeks considered the fractions to be ÒrationalÓ and numbers like
sqrt(2) to be frightening and bizarre, hence Òirrational.Ó These words survive, but that doesnÕt
mean we today think that sqrt(2) is bizarre or senseless. Quite the
contrary. Similarly, there is nothing
ÒimaginaryÓ about the complex numbers.
We need not be enslaved by 2000 year old notions.

A few more comments on your video. You point our correctly that there is
no natural ordering Ò>Ó on C.
That is true. It is true of
many other very interesting and very useful fields, such as finite fields. If you think C is strange, you ought to
read about the field of p-adics, and their completion. Or how about the hyperreals?

A
true student listens, learns, works, and thinks. A true student enters a subject with a desire to know the
truth, not a desire to cease learning at a certain level, or to be a slave to
misconceptions and compromises that were foisted on him, perhaps by less than
ideal teachers. Nor should
such a "drop out" then think of himself as qualified to teach others.

I hope you will take this to heart and
endeavor to rise above your misconceptions.

[response
to his next email, parts of which are quoted as ÒfirstÓ, ÒsecondÓ, etc.]

Hi,

First: " One mathematician who was
challenged on this in an interview retorted, ÒHave you ever seen a negative
number of sheep?Ó The interviewer
laughed, and the remark probably plays well in the classroom, but itÕs absurd. " The implication is of course that negative numbers have no
correspondence to physical reality, so theyÕre just as ÒimaginaryÓ as the
imaginary numbers.Ó

OK, I see both sides of this. The original comment [negative number
of sheep] was meant to be an analogy.
At some point in one's life (sixth grade?) one learns about negative
numbers, and the response that they make no sense is pretty common (especially
among parents trying to help with homework). So the original speaker was saying in essence, "remember
when you were 11 years old and thought negative numbers sounded weird. Later you learned that it makes perfect
sense: there are many situations where you have to set a zero point on a scale,
and then the values on one side become negative. It's very natural. Learning complex numbers is like that
too." Your response, since
you aren't 11 years old, is "that's silly, negative numbers make a lot of
sense, but complex don't."
It's just a matter of where you are on the intellectual growth scale.

Second: "the strange relationship mathematicians appear to have
with the physical world. There are
many examples, but one of the more disturbing is found in G.H. HardyÕs
ÒApologyÓ: Ò[mathematical reality] lies outside us, ... our function is to
discover or observe it.Ó (p.
123) To me this is delusional:
mathematics is transparently a human creation."

This
is an old discussion, and I see both sides of it. Hardy's book is very good. He may be taken as representing
the strong pure mathematician viewpoint, that mathematics is something we
discover. On the one hand,
obviously many aspects of mathematics are manmade. Our definition of limit, for instance. And yet, I too have felt often what
Hardy is saying: it's like
astronomy. We choose words like
"planet" and "star", we make up units of measure like
"year" and "meter". We view them through telescopes. We take photographs, sometimes computer
enhanced, sometimes with false colors. So it's all man-made? No, Jupiter really exists, by that or any other name. It really is eleven times the diameter
of the earth.

Mathematics is like that. There is something there. I've felt it.

Third:
"widespread confusion about the ordering or non-ordering of the imaginary
number line. In trying to sort
this out I consulted numerous sources, none of which clarified the matter to my
satisfaction. I finally went to my
university library and found a brief statement in an algebra book (a three-line
ÒnoteÓ, not an explanation) that confirmed the non-ordering. I donÕt know why this situation exists,
but itÕs certainly convenient for those wanting to misrepresent the
plane."

The
imaginary axis, which is just the y-axis in the usual Cartesian coordinate
system, can certainly be ordered, just by up-and-down, in the obvious way. The point, I think, is that there is no
natural (there's that word again) ordering ">" on all of C (i.e.,
one that has familiar properties, like:
if a > b, then for any c, a+c > b+c). Depending on the book you found, I can see that it would be
given just a passing reference. It's really not hard to prove. Advanced math books, written for
graduate students at least, always leave a lot unwritten. I'm guessing you can find it on
line. Otherwise, I can write one
out later. I've got a final exam
tomorrow! I don't understand your
comment "those wanting to misrepresent the plane." Who would that be?

Fourth: "Further: the imaginary axis is
non-quantitative, but itÕs used in electrical engineering to represent physical
quantities. ThereÕs nothing inherently
wrong with this - if engineers find the plane handy, let them use it. However, this usefulness is then cited
as proof that imaginary numbers are just as real as the real numbers. In other words, a non-quantitative
construct is applied quantitatively - that is, its formal properties are
ignored - and this application demonstrates that the quantitative and
non-quantitative are essentially the same. This represents such a massive intellectual cheat that I
feel I must be making an error somewhere."

"non-quantitative?" I don't understand. Do you mean you don't need it to do
mathematics up through, say, college freshman level? Do you mean an accountant doesn't need it? That's probably true, but nor does an
accountant need irrational numbers, or finite fields. (BTW finite fields are crucial to the RSA cryptosystem,
something VERY practical. Google it.)

"just
as real as the real numbers."
Hmm, that's a tough one. In
some sense they are not. Is the addition of rational numbers (fractions) that
we learned in fifth grade

"real"
to someone in, say, Africa struggling to survive with his goat herd? I think reality is in the eye of
the beholder. Of course, someone
may respond, if that goatherd became educated to, say, an eighth grade level,
he would be able to understand better agricultural and husbandry
practices. Suddenly a silly abstraction
becomes real!

Fifth: R^2 is our shorthand for RxR, the
Cartesian plane, the plane with two coordinate axes that we graph on and plot
functions on. (The little ^ means
raise to the power.) It's the set
of all ordered pairs (x,y) where x and y are real numbers. It really is like multiplying the
entire set R by itself.
Example: roll a pair of
dice. The first has set of outcomes
{1,2,3,4,5,6}, which I'll call D.
The second one has the same set of outcomes. The list of all possible
outcomes for the experiment of rolling 2 dice is DxD, or D^2 = { (1,1), (1,2),
.... (6,5), (6,6) }. So there are
36 outcomes. 36 = 6^2.

Sixth: "What this demonstrates to me is
that, AT THIS LEVEL OF ABSTRACTION, R and C are conceptually homogeneous. What it doesnÕt demonstrate is that
this is true at lower levels of abstraction." I doubt they are! Why should they be?
The essence of mathematics is to grow intellectually by layering concept
upon concept.

Seventh: "What youÕre doing with the
field-based story, in my view, is abstracting away from the underlying
messiness of complex numbers. But
turning your conceptual eye away from this messiness doesnÕt make it
disappear." But it's
not messy!! It's very
beautiful! Maybe this is a bad
analogy, but the first time I ever saw a painting by van Gogh, when I was
perhaps 12, I thought it was childish and, yes, messy. But when I was about 45 an exhibition
of van Gogh's works came to the Metropolitan Museum in New York. I encountered
his painting "the night cafe."
I was mesmerized. I was
truly awestruck, "blown away."
The difference in my reaction between ages 12 and 45 was not because I
willed the messiness to disappear.

Eighth:
"By the way, youÕve stated that both Strogatz and I have the definition of
i wrong: itÕs point (0,1) in the plane, not sqrt(-1). But the plane definition is both impossible and unnecessary
without the prior sqrt definition.
YouÕre again assuming that a later development erases prior
developments. ItÕs not true."

I
am certainly not erasing the idea of sqrt, if that's what you mean. What I said about the definition of C
is correct. Strogatz knows it, but he couldn't go into that in the space limit
he was working under. You are
utterly wrong in saying that one first has to use sqrt – of anything
– to define C.

I'm
going to try to explain in the brief time and space I've got here. I first learned this when I was about
15 from a really nice little book by Irving Adler, The New Mathematics. (I just
googled him, and I am amazed to see he is apparently still alive.) If you can find this little book on
Amazon, buy it! Really, I think it
would help you a lot.

From
ages about 4 to 14 we expand our concept of "number" from positive
whole numbers, to 0, to fractions, to negatives, to irrationals. At each point we do not lose what we
previously had. Changing the order
slightly (at least from what I learned) we have

N
< Z < Q < R

where
by "<" I mean contained in.
N = the set of positive whole numbers and 0, Z = all whole numbers,
integers, Q = all fractions (including whole numbers n as n/1), R is Q plus
irrationals. Q and R are
fields (def. above). Z is what we
call a ring: you have + and x but you can't divide just anything by anything,
as in a field. R is naturally
identified with the continuous, complete line.

Big
question: can we make R^2, the plane, into a field. Answer: yes.
I don't have time to do
this justice here. short
answer: if (a,b) is one point in
the plane and (c,d) is another, write each also in polar coordinates. Let r = length of radius vector of
(a,b), s = that of (c,d). Let
theta and gamma be their angles.
(Hope you know polar coordinates).
The product of (a,b) and (c,d) is defined to be the point at angle theta
+ gamma with radius rs (r times s).
That's it!! It is indeed a
field - that takes some checking.
See, I never said "sqrt". BTW, the sum (a,b) + (c,d) is of course just the
vector sum (a+c, b+d). With C so
defined we don't lose R and we now have

N
< Z < Q < R < C

Each
inclusion is natural - i.e., the arithmetic is the same; for example, 2 + 3 in
Z is the same as 2/1 + 3/1 in Q.

One
truly beautiful fact is that we can go no further. That's it.
There is no way to make R^3 (3 space) into a field, or any R^n larger
than 2. (but Google "the
quaternions".)

I've
really got to wrap this up. Maybe
I'll write a book someday.

Robert
H. Lewis

Fordham
University