The Case of
The Imaginary Numbers
|describes a further extension of number to the really useful notion of imaginary numbers.|
One of the most useful extensions of straightforward multiplication is the notion of squaring a number or multiplying it by itself. All sorts of geometric relationships have a natural expression in terms of squares and square roots. However, when we do algebra with squares, we find ourselves again with insoluble equations. For example, the question "for which values of x does x*x=-1?" has no solution among the combination of positive and negative numbers.
Fairly early in the history of algebra (in 1545 at least, by the mathematician Cardan), mathematicians began to talk about solutions to equations like "x*x=-1" by introducing the sqrt(-1) as a term in their equations and solutions. In 1777, the mathematician Euler started referring to this with the symbol i from the German word for "imaginary," indicating their difference from the supposedly "real" positive and negative numbers.
For the same reason that negative numbers "realized" algebraically had to be distinct from the positive numbers, imaginary numbers have to be different from the real numbers. However, addition and multiplication of real and imaginary numbers with each other could not systematically produce either a real or imaginary number. This required the definition of yet a third sort of number, the complex number, which had both real and imaginary parts. A complex number was written like 3+5i, where "3" was the regular (real) component and 5*sqrt(-1) was the imaginary component. In fact, technically, both regular and imaginary numbers are complex numbers: 3 is just 3+0i and 5*sqrt(-1) is just 0+5i.
As with negative numbers, the introduction of complex numbers had to be systematic and the rules for adding complex numbers had to be the same as for adding real numbers. This has two important advantages. The first is that the systematicity of reference is unaffected. If we have certain descriptions of real numbers, they are not affected by the addition of complex numbers to the system we are describing.
The second fruit of systematicity is that the patterns of access remain the same. The rules we have for manipulating numbers --- arithmetic, algebra, calculus --- look pretty much the same when we are manipulating variables and functions of complex numbers. This is an instance of the "five pounds in a ten pound bag" phenomenon of Chapter 5: we can handle much more complex problems without drastically changing either what we already "know" or what we already know how to do.
Are imaginary and complex numbers "real objects"? This is the wrong question. Imaginary numbers, like negative numbers, make certain kinds of problems easier to solve. One way they can do this is by describing intermediate results, as in the example with negative intermediate results. Many equations are easier to solve if we permit imaginary intermediate results.
Complex numbers can also be used to describe physical phenomena by combining two separate parameters into a single complex quantity. For instance, Maxwell's laws for electromagnetism describe the interaction of magnetic and electric fields. A field is a function which assigns a value to every point in space and the use of complex numbers in describing Maxwell's equations allows two fields to be replaced by one complex-valued one. The complex value a+bi assigned to a point in space by this field describes both the point's magnetic field (b) and its electrical field (a). It is not that electrical fields are particularly real or that magnetic fields are particularly imaginary. The two could have easily been switched, but the independence and inseparability of real and complex components mirrors the independence and inseparability of the electrical and magnetic fields.
Another way in which complex numbers can "be real" is best introduced by looking at complex numbers as points on a plane. This "model of the model" reveals structure in the complex numbers which enables their application to many other phenomena.
The first steps in making complex numbers less "imaginary" were taken by the mathematician Wessel in 1797. Wessel proposed that complex numbers be thought of as points on a plane, where one axis represented the real component and the other represented the imaginary component. Complex numbers, rather than being mysterious half imaginary entities, were suddenly quite simple points on the plane, as in these complex numbers represented as points:
What is the point of this visual model? One advantage is that it allows mathematicians to visualize the solutions to equations on the two dimensional plane. Though mathematics seems to be about the transformation of equations, visualizing solutions is vitally important to mathematical thought. So, the visualization of complex numbers --- a model of a model --- makes it easier to work within the model being redescribed. These sorts of chained models are especially common in mathematics, where the purposes are sufficiently similar that the links between models lose nothing of importance to the model's purpose.
But the graphical model also makes it easy to see some properties of complex numbers which we might not have identified otherwise. In a slightly different model, a complex number can be seen as a arrow or vector (the term I'll use) reaching from the origin (the center) to a point on the plane:
When it is described in this way, arithmetic on complex numbers has geometric consequences. Adding two complex numbers "chains" together the vectors representing the numbers, attaching the tail of one to the head of the other, as in:
Multiplying a complex number by a real number changes the length of the vector, as in:
while multiplying the complex number by an imaginary number changes the direction of the vector:
without neccessarily changing the length.
These geometric descriptions demonstrate a property of complex numbers which we might not have noticed without them. Rotations repeat, returning to the same place after some time, and so do complex multiplications. Because of this, it makes sense to use complex numbers to describe repeating phenomena such as radio or sound waves. The real part of the complex number (shown here on the vertical axis) can be tied to the "amplitude" of a wave:
while the imaginary part describes the progress of the wave by rotation of the vector around the origin. When a wave signal is described in this way, the effects of certain materials and components are just particular sorts of complex multiplications or additions applied to the signal. This is what makes this model so useful for describing oscillating phenomena.
It is important to be clear here about what is not happening. First, this model is not using the "imaginary" component to directly represent time. When engineers reason about waves using this model, time is represented separately. Second, this model is not using the "imaginary" component to represent something independent but inseparable, as in the merger of electrical and magnetic fields. Instead, the imaginary component is being used to separate the points in the wave with the same value but different directions.
We can illustrate this with a charming story about the mathematician Norbert Wiener.
Wiener met McCulloch in the hallway of Building 20 and they spent some minutes discussing some minor matter. After finishing the discussion, but before parting, Wiener asked "When we met was I going this way (pointing up the hallway) or this way (pointing down the hallway). McCulloch pointed down the hallway. "This way," he replied. "Ah," said Wiener, "then I was going to lunch."
Like Wiener in his daily lunchtime excursion, an oscillating wave regularly reaches the same place with great regularity, but the position alone does not tell the direction. In many electrical components and circuits, the response to a given level of an oscillating signal is very different if the level has been dropping than if it has been rising. The imaginary component indicates "what just happened" and "what will happen next" (assuming an oscillating signal) which makes it easier to describe the individual components and their combination into circuits.
This is worth repeating. The "imaginary" quantity in the mathematical description of circuits does not correspond to any mysterious property carried along with the measurable energy flowing through the circuit. Instead, it arises from the assumption that the energy is travelling in oscillating cycles with predictable (though variable) properties. Looked at across time, much as we could see the motion of plants in time-lapse photography, we can see the changes which the imaginary quantity supposedly describes. However, if the signal to the system stopped obeying the regular oscillation which the model assumes, the quanitites we measured would cease to behave like the imaginary and complex numbers we were using to describe them. Just as the imaginary islands for Micronesian sailors don't represent actual islands but stand for progress in their journey, imaginary components in the mathematical circuit model do not really correspond to the changes in energy, they merely track the expectations for the oscillating voltage and current.
This chapter has looked at how a number of different models make inferences. These inferences give the models the completeness which makes them so valuable. Because models "complete" partial descriptions, they are typically postulating things which have not actually been encountered. In many cases, the model relies on things which do not actually "exist" in order to make a more complete description. But the model may later change or evolve to "find" things which match these imaginary constructions.
Central to all these acts of inferential imagination was the systematicity of whatever was constructed. Whether they were imaginary islands, negative numbers, or complex numbers, the constructions each had to be consistent with the real islands, the positive numbers, or the real numbers. And because of this, they were not arbitrary but were tightly constrained.
This requirement of systematicity returns us to Coleridge's striking distinction between fancy and imagination. Fancy involved the mere juxtaposition of familiar components but not their systematic combination, while Imagination honors the logic of our models while still putting things together in unexpected ways.