On the interpretation of the number i, the imaginary

Question

Numbers can express how much there is of a certain object. The objects can exist in the mind or in reality. You can have 1 marble, 35 marbles, and even wish for an infinite amount of them. You can have 33,3452 marbles and you can have $\sqrt 2$ marbles. You can even have a minus number of marbles. A bit more abstract but I can imagine that should Ì get them in reality I have to give them to someone else.The negative number is always related to the real positive number. A negative charge exists in comparison to the positive charge only. Likewise, a negative temperature can only be defined wrt the real positive temperature. So negative numbers are numbers that can only be defined in the context of poitive quantities. On their own they have no meaning. They are the result from taking the opposite of real positive numbers. Positive numbers are real in the sense that they represent a true quantity. You can have one apple for real but minus one only in the mind. A charge of 5 exists as well as a charge of -5 but these numbers don't point to a really existing thing which you can have a number of. The plus and minus are merely used to point at opposite effects they can have. A velocity can be plus or minus but you can't have -6 of them. So a negative number (if used when expressing how much of a thing you actually can have) is imaginary too.

So this means you can have -5 marbĺes but this merely means that you must give them away when you should get a hold on 5 real marbles.

But what about i marbles. What mental operation do we have to perform on the marbles to say you have 5i of them? If -5 apples just is the operation of taking the negative (meaning you owe) to what corresponds multiplicating by i (instead of -1)? If I have -5 marbles I think of giving back 5 marbles as soon I get them. But what should I think when I have 5i marbles?
Will the marbles themselves change (contrary to taking the negative)? Is there something I have to do with them? Insread of giving them away, maybe rotate them? So if I have -5i (sqrt-25) marbles I have to rotate all five and then give them away? Or what? Why is it called imaginary?

Answer 1

Numbers are used for more than just counting things. You are thinking of cardinal numbers, which represent the size of a set; in your case, if you have 5 marbles then your set of marbles has cardinality 5. Numbers have other uses, such as ordinal numbers which represent the position of something in a sequence; imagine putting your marbles in a row, then marble number 5 would be the last one.

So the number 5 has different interpretations in different contexts, depending on what you use it for. Note that it is our choice how to interpret a number in a given context, and we choose to interpret numbers in ways that reflect how we use those numbers; the fact that some numbers (e.g. -5 or 0.5) cannot be interpreted in certain ways just reflects the fact that those numbers cannot be used for those purposes. Case in point, you cannot interpret i as a cardinal number because you cannot have a set of size i.

Generally, we use i purely formally as a number whose square is -1, which exists only in our imaginations (much like other numbers do in the abstract). So you can interpret it as purely a formalism, and that is how mathematicians tend to interpret it.

Answer 2

In electronics, we use imaginary numbers to represent the reactance of certain components (capacitors and inductors). The overall impedance of a circuit is then a combination of the various components' resistance and reactance, expressed as a complex number.

This reactance causes alternating voltage and current to lose phase (in power engineering this is called the power factor). Across a pure reactance they are 90 deg out of phase, with the voltage represented as a real number and the current as an imaginary one.

I once built a resonant oscillator with a period of around 1 second, and attached moving-needle voltage and current (ampere) meters. The needles oscillated back and forth with the signal, the current visibly lagging the voltage by 90 deg. Before my eyes I watched a real number changing on the voltmeter, an imaginary number changing on the ammeter.

Einstein's theory of Relativity similarly places Time in the imaginary domain (but I did not have the components for a Time oscillator to hand - you need a small bench-top wormhole through which to thread your closed timelike loops).

The moral of this tale is that numbers can be used for far more things than merely counting marbles and marking out speedometers. i is simply √-1, the square root of minus one, and if the word "imaginary" troubles you then it may be safely ignored.

So what are numbers? It turns out there are many kinds. Some people like to treat a number as the set of sets containing that many objects. Others recursively as the set of smaller numbers. These are I think the cardinals and ordinals, I forget exactly. Neither makes much sense of i, or even irrationals like π for that matter. Intriguingly, Wikipedia says that a number is a "mathematical object", which it in turn says is "an abstract concept which has been formally defined", yet it offers no formal definition of number. Dictionary definitions as an "amount" come closest, but then we get things like imaginary, complex and quaternionic numberings which do not really fit that. We just have to accept that quantities can be less intuitive things than we might have supposed.

Answer 3

The key here is to notice that i is a square root, which means that — geometrically — it is a 2-dimensional concept; it relates to areas. In one dimension (as the question notes) we can have positive numbers (assets) or negative numbers (deficits). Thus I could say I have three pineapples (+3) but I absolutely need four lemons that I do not have (-4), and it makes sense to most people.

We can do a similar thing with areas, but it becomes a bit more complicated. Say that I'm a pineapple farmer (don't ask why I'm hung op on pineapples this morning) and I have four acres of pineapple grove: a square roughly 416 feet on a side. That is an 'asset' area: something I have at my disposal. Now say that I want to expand into lemon production, and I need another acre to do that. Now I have a 'deficit' area: a square roughly 208 feet on a side that I currently do not have at my disposal. These are both positive areas, of course: 416 × 416 = 173,056; −208 × −208 = 43,264. But now we have a third (mathematical) possibility, where I can have an asset in one dimension and a deficit in the other. It's difficult to visualize, though you might think about having a 208 foot mirror reflecting the land behind you; it looks like an (asset) acre, but it has a problematic existence.

This is where imaginary numbers enter the picture. If we want to talk about the area of the land in the mirror, we'd have to say that it is 208 × −208 = −43,264. But if we take that area and try to get the side-length of its square (its square root) we get √(−43,264) or 208 × i. 'i' is a recognition that we know one dimension is an asset and one dimension is a deficit, but we don't really know which dimension is which. So we mark them as having indeterminate evaluation, knowing that it will all come out in the wash once we square things again.

Answer 4

It doesn't make much sense to have an imaginary amount of marbles. It might make a little bit of sense to have an imaginary number of meters. Suppose that you owe someone 100 m^2 of land. This might be represented as you having -100 m^2 of land, perhaps arranged in a square that is 10i meters on each side. Or perhaps a hundred squares that are each i meters on a side.

Ultimately i is defined by the property that, when you square it, the result is -1. This is mostly a mathematical formalism that makes certain equations come out nicely; for instance, see the fundamental theorem of algebra which says that all polynomials with complex coefficients have at least one complex root. In other words, there is always a solution to polynomial equations in complex variables. This is not true for the real numbers; polynomials with real coefficients do not always have at least one real root.

Answer 5

Many excellent answers here. I will add a slightly different perspective:

The square root of minus one (call it "i") suggests itself in the solution of certain mathematical equations (for example, x^2 + 1 = 0), which has a solution x = sqrt(-1). Regarding its relationship with the real numbers, once you understand that the negative reals start at zero on the number line and extend in a direction opposite to the positive reals, then it is possible to geometrically interpret imaginary numbers ( i, 2i, 3i, 4i, ...) as being on a line that is perpendicular to the real number line, intersecting it at the zero origin.

Paul Nahin's book The Square Root Of Minus One: An Imaginary Tale contains an excellent exposition of this; I highly recommend it.

Answer 6

Generally, when we get beyond counting... we apply mathematics to the real world according to its usefulness. The real world is not dictated by the mathematics we're using to model it. So I think it's mistaken to search for an interpretation of "5i" apples. The right way is to look at how we use apples in the real world, and see if there's any way to apply complex numbers to it.

So generally (maybe not always) the real world comes first... then we create equations to model the real world. If we get solutions that don't make sense in the world we're modeling than we throw them out.

If we have a vector in the 2d plane... like 2d velocity... we can model this using complex numbers. For example the real axis can represent east/west, the imaginary axis can represent north/south. So we're taking a real world situation and modeling it with complex numbers, rather than taking a complex number and trying to find an interpretation for it.

Similarly since we know electrons come in discrete units, it would be a mistake to look for an interpretation of "6.23 electrons".