How Imaginary Numbers Became “Real”

Last week we started a series on complex numbers, looking at how we introduce the concept. This time I want to look more at the actual history of the idea, leading to how mathematicians were able to define complex numbers without saying “Just suppose …”.

Why were they invented?

I’ll begin with a question from 2005:

History of Complex Numbers

I'm doing a project on imaginary numbers and I was stuck with the why.  If you could explain without getting to in depth about hypercomplex numbers etc., (I'm a sophomore) that would be great!

Why would mathematicians come up with this idea in the first place? I answered:

Hi, Audrey.

Here is a good place to start; I found this by searching our site for the words "imaginary number history":

  History of Imaginary Numbers
  http://mathforum.org/library/drmath/view/52584.html 

The first link given in the answer has some more details.  It mentions Cardan and Bombelli (in the 1500's), who were the first to work with complex numbers.

I will include part of that answer below.

What happened was that, in solving quadratic equations, it had previously been found that some equations could only be solved if you could take the square root of a negative number, which had been recognized to be impossible.

This is step 1: Imaginary solutions had no use; they were just a matter of curiosity, if that. No one pursued the idea, because mathematics was not yet regarded as an abstract game as it is now.

But then when these people tried solving cubic equations, they found that for some equations, if they went ahead and did their work as if you COULD take the square root of a negative number, those "imaginary" numbers would eventually cancel one another out and you would be left with a real solution that WORKED.  In other words, they made a trip through a world of numbers that presumably didn't exist, and came out at a proper destination in the real world, just as if those numbers they had used really meant something. In fact, they couldn't get to those solutions WITHOUT using imaginary numbers!  That gave them the beginning of a sense that imaginary numbers were worth thinking about.

This is the second step: Imaginary numbers used temporarily in the work led to real solutions to the whole problem. This made them useful.

So, over a long period of time, the rules for working with them, and appropriate notations, were developed, even though they were thought of as a sort of fiction that didn't feel quite valid.  Eventually more was discovered about them that made them seem more and more "real", until they were finally accepted as being no less "real" than the real numbers.

Like the Velveteen Rabbit, with age and experience, they became “real”. We’ll soon see how.

Who invented them?

Here is the 2001 question I referred to above:

History of Imaginary Numbers

I am a high school student Chicago, IL, and my class just started studying imaginary numbers. Someone asked my teacher who invented imaginary numbers, and my teacher had no idea, so she suggested we find out who did invent the theory. If you could please tell me or give me another place to look, I would be very thankful.

Doctor Rob looked it up in some of our favorite historical sites, which were linked in our FAQ:

Thanks for writing to Ask Dr. Math, Laurie.

This was an interesting question. After some research, I have found the following pertinent information.

At the MacTutor Math History archive in St. Andrews, I found

  The fundamental theorem of algebra
  http://www-history.mcs.st-and.ac.uk/history/HistTopics/Fund_theorem_of_algebra.html   

It says,

"Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation.  The formula when applied to the equation x^3 = 15*x + 4 gave an answer involving sqrt(-121) yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics."

Blind manipulations are not what we want students to do; but sometimes that is the only way you can start out! The page has a link to full information about Cardan (Girolamo Cardano, 1501 – 1576). It also says, though Doctor Rob didn’t quote this,

Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these ‘complex numbers’. Descartes in 1637 says that one can ‘imagine’ for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.

I recently found another nice summary of the entire history here.

After some other quotes, he concludes,

The conclusion is that probably Girolamo Cardano (or Cardan) (1501-1576) can be credited with the "discovery" of imaginary and complex numbers in the 16th century, but that the concept was not put on a firm footing until much later, especially in the work of Leonhard Euler (1707-1783) and Carl Friedrich Gauss (1777-1855).

Gauss once wrote,

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question."

It was Gauss who made the distinction between imaginary numbers a*i and complex numbers a + b*i (a and b real). Up until his work, both complex and imaginary numbers had been termed imaginary.

Can you see how his suggested terms (direct for positive, inverse for negative, and lateral for imaginary) make sense when you view the complex numbers as points on a plane as we did last week? Imaginary numbers move “laterally” (sideways), off the number line!

But what is this “firm footing” Gauss (and Hamilton) put it on? It was that same idea of points on a plane, as outlined in the next question.

How did they get past the fiction?

To take us there, here is a “question” from reader Howard Engel in 1997, that is really more answer than question:

Imaginary Numbers - History and Commentary

I have just discovered Dr. Math, as the result of a mention of the page in the Los Angeles Times this week.  I think it useful to youngsters through grade 12.  I have some comments to add to your presentation on imaginary numbers.

The ancient Greeks once believed that all numbers were rational numbers; that is, that every number could be expressed as the ratio of two integers, and they were very disturbed when it was demonstrated that the measure of the hypotenuse of an isosceles right triangle, having arms of unit measure, was not a rational number.  I omit the simple proof here.  The new numbers, of which I have given only one example, are now called irrational numbers to distinguish them from rational numbers. (Whether irrational numbers, or negative numbers, or the transcendental numbers yet to come were invented or discovered is a philosophical question I choose to avoid.)  The point I wish to make is that irrational numbers were a kind of number new to the experience of mathematicians. Prior to the proof of existence of irrational numbers, it was not necessary to distinguish between rational and irrational numbers; all numbers were expected to be rational.

The irrational numbers in a sense were already known (e.g. the length of that hypotenuse); they just weren’t yet known to be what they were (and were not used as numbers)! Other new kinds of numbers are different:

Mathematicians for a long time were unwilling to accept as solutions to equations numbers that were less than zero. Eventually numbers of this sort were accepted as solutions. Today we call them negative numbers, another kind of number once new to mathematicians, and requiring a revision of beliefs. Prior to the acceptance of negative numbers, it was not necessary to refer to positive and negative numbers; only positive numbers were believed to exist.

Here, as we saw last week, we have “numbers” that were thought of as fictional for a long time; their very existence was in question, and only gradually accepted because of their usefulness.

For centuries there were quadratic equations that were deemed not to have solutions. Equations like x^2 = -1 and x^2 - 2x + 2 = 0 have no solutions among the positive and negative numbers. The problem in seeking solutions to equations like these two is that the squares of positive and negative numbers are both positive.

Solutions for equations like these can be found, however, if we decide to invent a completely new number whose square is -1; of course, it is not a number that we have seen before. We name this number "i".  The square of -i is also -1.

As we’ve seen already, the concept started with this leap of imagination.

By multiplying i by positive and negative numbers (in other words, all the non-zero numbers we had before we added i) we can obtain a whole set of new numbers that have the property that their squares are negative numbers.  These new numbers, for better or worse, were called "imaginary" numbers, and the old positive and negative numbers (and zero) were called "real" numbers.

Still further, letting a and b be positive or negative real numbers, we can construct infinitely many numbers of the form a+ib. We then find that we can write the solutions to the equation x^2 = -1 as x = i or x = -i, and the solutions to the equation x^2 - 2x + 2 = 0 as x = 1+i or x = 1-i.

“Real number” was a retronym, like “analog clock”, “acoustic guitar”, and “manual transmission”; real numbers until then were just called “numbers”.

Unfortunately, because the word "imaginary" is associated with the make-believe, there has been a lot of confusion over the concept of this new number i.  The term "imaginary", when used to refer to multiples of i, is a technical term and because of its pervasive use amongst scientists and mathematicians, it helps to learn the term for the sake of communication.

Furthermore, numbers of the form a+ib, in which a and b are real numbers, were then called "complex" numbers.

“Complex” means “put together”, in this case from the two kinds of numbers obtained as square roots of real numbers.

Defining by construction

By the 1800’s. mathematicians had developed new, abstract ways of thinking about numbers, so it was time for a change:

If only mathematicians had waited a while before assigning these names! Hamilton, a few years later, found another way to express complex numbers where he never had to introduce the word "imaginary". Hamilton's solution was to expand the definition of number, just as other mathematicians in the past had expanded the definition of number, in the following way:

Hamilton decided that our ordinary "real" numbers are a subset of a larger set of numbers that are referred to as "ordered number pairs", and written (a,b), in which a and b are positive or negative numbers, including zero (in other words, in which a and b are real numbers).

The rules of arithmetic must be altered for ordered number pairs.  Letting letters represent real numbers, we have:

        (a,b) + (c,d) = (a+c,b+d)
        (a,b) - (c,d) = (a-c,b-d)
        (a,b) * (c,d) = (ac-bd,ad+bc)
        (a,b) / (c,d) = ((ac+bd)/(c^2+d^2),(bc-ad)/(c^2+d^2))

What he is doing is starting from scratch, defining numbers afresh, as ordered pairs with special definitions for the operations. These definitions are based on our experience with complex numbers as already understood:

$$(a+bi)+(c+di)=(a+c)+(b+d)i$$

$$(a+bi)-(c+di)=(a-c)+(b-d)i$$

$$(a+bi)\times(c+di)=ac+adi+bci+bdi^2=(ac-bd)+(ad+bc)i$$

$$(a+bi)\div(c+di)=\frac{a+bi}{c+di}\times\frac{c-di}{c-di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i$$

But they are definitions, not derivations; nothing here depends on assumptions about imaginary numbers.

These rules are considerably more complicated than those learned in elementary school for the elementary operations of arithmetic, but ordered number pairs continue to obey the laws of associativity, distributivity, and commutativity.

The ordered number pair (a,b) is equivalent to the complex number a+ib. That is, if b is zero, then (a,0) and a+i0 behave algebraically as the same real numbers.  If a is zero, then (0,b) and 0+ib behave algebraically as the same "imaginary" numbers.  Finally, if neither a nor b is zero, (a,b) and a+ib behave algebraically as the same complex numbers.

Having defined a new set called “complex numbers”, we now prove their properties, and in particular that the subset of numbers of the form \((a,0)\) behave exactly like the familiar real numbers. And we can use \(a+bi\) as a shorthand for \((a,b)\), which makes the rules feel natural.

By my argument and exposition, I do not mean to imply that ordered number pairs should be used to the exclusion of representations of the form a+ib. Once ordered number pairs and their algebra have been introduced, and used to express the roots of equations such as x^2 = -1 and x^2 - 2x + 2 = 0, the equivalent representation a+ib for (a,b) may be introduced, together with the simpler rules for manipulation, and it may be mentioned in passing that i may be treated as if it were a square root of the ordinary number -1 -- but do not dwell on the term "imaginary number".

Additional thoughts

Doctor Ceeks could only concur and comment:

Thank you very much for your thoughtful comments. I think you have some very good points.

One of the problems with the concept of "i" as a number is that most people associate the word "number" with the concept of a measure of the magnitude of some set... such as the number of people in a stadium. Since one cannot say there are "2+i" slices of bread in a loaf, people have a bad reaction to calling "i" some sort of number.

We have to change our whole idea of what a number is in order to accommodate imaginary numbers into our thought!

Mathematicians view the complex numbers as a construction which, as you point out, allows for the complete factorization of any polynomial with real (or complex) coefficients.  It's wonderful that it is possible to construct a system of numbers which contain a number whose square is -1, and deduce that such a system exists with many favorable properties!  For the sake of communication, mathematicians gave a name to some of the new objects relevant to the construction, and, unfortunately, the term "imaginary number" was introduced.

The reason this is unfortunate is because people have a natural tendency to want to reconcile the name with the old meanings of the words that make up the name. Since most people learn the words "imaginary" and "number" in a completely different context from that used by mathematicians, there is trouble.

But then this suggests that pedagogically, it helps if we can convince the student to accept the idea that there are new concepts and that it is misplaced to try to force the new concept into the mold of the old concept.

Doctor Pete expanded slightly on this idea (using ordered pairs to make imaginary numbers more concrete) in this question from later in 1997:

Defining Complex Numbers

Are imaginary numbers allowed as answers?

I’ll close with one more question about the “reality” of these numbers, this one from 2006:

Can an Imaginary Number Be a Valid Answer?

We know that the answer to the equation x^2 + 1 = 0 is root negative 1 and the only answers to that are +/- i.  But if "i" does not exist how can it be an answer?

I answered:

Hi, Greg.

Your question is just the one that bothered mathematicians when complex numbers were first discovered, and for centuries after.  But the same question was asked about negative solutions; only positive solutions to equations were considered meaningful, because it was assumed that the solution had to represent some measurable length.

As we’ve seen, the concept of “number” has expanded to include signs and fractions and irrationals, as the kinds of problems expanded (from counting whole objects, to measuring sizes, to locating coordinates, for example). Now we need to move beyond measurements, too.

Now, if your equation arises from a problem where only positive numbers make sense (such as a length), then the answer is that negative or imaginary solutions are invalid--but that's due to the problem, not the equation itself.  Likewise, there are problems in which only a real number makes sense (such as a coordinate in space); then positive and negative solutions are valid, but an imaginary one is not.

The domain of an applied problem consists only of numbers that are meaningful in that problem (only whole numbers, or only positive numbers, or only real numbers).

But an equation itself doesn't relate to any specific real-world problem; it only asks what numbers make it true, and complex numbers are numbers.  So we can ask "what are all the real solutions to this equation?", and the answer will be "there are none", only because the problem was stated in terms of real numbers.  Without that restriction, imaginary solutions are perfectly valid.

That is, an abstract question can have an abstract answer.

After this idea was accepted, that complex numbers really are numbers (and DO exist, in the sense that any numbers, though abstract, can be said to exist), it was found that there ARE real problems in which complex solutions make sense.  For example, in electronics there are problems in which the complex solutions of an equation tell you how a circuit will respond to certain inputs; imaginary numbers correspond to alternating current, and real numbers to direct current (more or less).  In this setting, complex solutions not only are valid, they are what you're looking for!

Most of these applications require too much background to be able to clearly show what they mean. But as we proceed later into the rotational aspects of complex numbers, we will be coming close to the ways in which complex numbers are used to represent waves in electronics or physics.