Josh: Let us revel in the joy of mathematics, Mick, for mathematics carries great joy in its learning.
Mick: How do you mean, Josh?
Josh: Mick, let me explain. Tell me, do you listen to music?
Mick: Why, yes, Josh; the works of Mozart belong amongst the highest music of the spheres.
Josh: What do you believe makes this music so virtuous?
Mick: As with all classical music, the virtue lies in the movement of tension to resolution.
Josh: As is the same with mathematics. Don’t you believe that a state of unknowing is a state of tension?
Mick: I suppose I do, Josh.
Josh: Don’t you also believe that a state of knowing, proceeding a state of unknowing, is itself state of resolution?
Mick: By Zeus, you are right.
Josh: Thus, Mick, it must be that learning mathematics, just as music, is a wholly virtuous activity.
Mick: I am in agreement, but I do not yet see the true beauty described.
Josh: Let me explain. Do you know what a complex number is?
Mick: I have never encountered such a concept in my studies.
Josh: So, Mick, you would say that you are in a state of tension, through your unknowing?
Mick: Yes, Josh, for I am not only ignorant, but also intimidated by this concept.
Josh: I will now attempt to resolve this tension. What do you consider to be a number?
Mick: In my own words, a number is a (possibly infinite) decimal, such as, 4, 15.6, 0, or the square root of two = 1.41421…
Josh: Good! Let us situate ourselves in sixteenth-century Italy.
Mick: I am situated. For what reason?
Josh: The mathematicians of this time, Mick, had a very similar notion of number that you do today.
Mick: They, too, had no idea of what a complex number is?
Josh: They will be the ones to invent the word in the first place! Surely, they have not already heard of it.
Mick: What were the events of this invention, Josh?
Josh: Precisely the right question. The Italians of this time were very interested in solving polynomial equations.
Mick: Such as (draw x^2 + x = 0), or (draw x^2 + 1 = 0)?
Josh: Precisely, Mick. I see that Math 111 has trained you well. Do both of these equations have solutions, in numbers you are familiar with?
Mick: Not both! The first equation is solved at x = 0, or x = -1, while the second equation has no solutions.
Josh: Right you are. Wouldn’t it, however, be convenient to act as if the second equation also had two solutions?
Mick: Perhaps it would, Josh; if the second equation had two solutions, then I could solve many equations that I have been challenged with in the past.
Josh: Further than that, you could solve any polynomial equation.
Mick: It seems as if we wish for there to be a number that solves this second equation.
Josh: To that end, let us define the imaginary unit i: i is the unique number such that i^2 = -1.
Mick: Indeed, this imaginary unit solves our troublesome equation. This is well and good, Josh, but have we solved any problem? For we have not found a number that solves our equation, but have only named a solution that we assume to exist.
Josh: This must be why you are named among the most astute at Reed College, Mick. To answer your question, we must first ask what a number is in the first place.
Mick: Let me posit, then, that a number is a quantity.
Josh: When you describe a quantity, you refer to a quantity in the physical world, correct?
Mick: Correct, Josh.
Josh: Then tell me, Mick: is -3 a quantity?
Mick: I don’t see why not.
Josh: Can I have a negative amount of apples? Not in terms of debt, but purely in terms of quantity of possession?
Mick: No, Josh, for that does not make physical sense.
Josh: But you agree that -3 is a number?
Mick: I am fooled!
Josh: This shows us, Mick, that physical intuition is not an appropriate way to define numbers. This phenomenon goes back to Ancient Greece, where geometry ruled the definition of number. Indeed, Pythagoras refused to believe that the square root of two was irrational; he believed that this number was finite in its decimal representation.
Mick: How ridiculous this notion sounds today!
Josh: And yet it was his genuine mathematical belief. The Italian mathematicians, while not as zealous as Pythagoras, too questioned the validity of the imaginary unit. This is the origin of the terms “imaginary” and “real” in mathematics: the imaginary unit was not seen as a “real” number, and thus other numbers were seen as more “real”.
Mick: If I am correct, Josh, you are implying that there should be no ontological distinction between a real number and an imaginary number.
Josh: Exactly. And thus we are led to the complex number: a number of the form a+b*i, where a and b are real numbers, and i is the imaginary unit.
Mick: I understand.
Josh: Furthermore, this “imaginary” unit does have physical and geometrical significance! Do you remember what a number line is, Mick?
Mick: You mean the system of visualization where real numbers are plotted on a line, corresponding numbers to lengths?
Josh: The very same. Such a system exists for complex numbers, as well:
Mick: Ah! So, I can think of complex numbers as just directions vertically on the plane, just as real numbers are directions horizontally?
Josh: Exactly, Mick. There, too, is a way to correspond complex numbers to lengths, just as we can do with real numbers. Tell me, what real numbers correspond to a length of one?
Mick: I suppose that would be the two real numbers +1 and -1.
Josh: Indeed. Now, what complex numbers correspond to a length of one?
Mick: Would those complex numbers form a circle on the complex plane?
Josh: Precisely; there are in fact infinitely many complex numbers that correspond to a unit length.
Mick: Wait! Looking at the unit circle reminds me of trigonometry I learned many months ago. Are these two concepts related?
Josh: That, Mick, is a conversation for another day.