What Are Complex Numbers?

This article is a series of emails and posts I made concerning Steven Strogatz’s 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 Strogatz’s column to read all of this.  Enjoy.

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. “What is the square root of i?”   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. “Why don't we go on and make a number system out of 3-dimensional space or 4-dimensional space?”   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

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

_______________________________________________________

Hello,

I watched your [Mr. X’s 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 “imaginary” and “chasm” 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 “rational” and numbers like sqrt(2) to be frightening and bizarre, hence “irrational.”  These words survive, but that doesn’t mean we today think that sqrt(2) is bizarre or senseless. Quite the contrary.  Similarly, there is nothing “imaginary” 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 “>” 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 “first”, “second”, etc.]

Hi,

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

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. Hardy’s “Apology”: “[mathematical reality] lies outside us, ... our function is to discover or observe it.”  (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 “note”, not an explanation) that confirmed the non-ordering.  I don’t know why this situation exists, but it’s 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 it’s used in electrical engineering to represent physical quantities.  There’s 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 doesn’t 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 you’re 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 doesn’t 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, you’ve stated that both Strogatz and I have the definition of i wrong: it’s point (0,1) in the plane, not sqrt(-1).  But the plane definition is both impossible and unnecessary without the prior sqrt definition.  You’re again assuming that a later development erases prior developments.  It’s 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