I’m aware of the historical context of complex numbers, and I’m aware that the term “imaginary” is an ironic holdover from the days when the validity of complex numbers was in doubt. Now, I’ve worked with complex numbers, particularly with fractals, so I’m familiar with the basics of complex numbers and the complex number system. It’s simply been one of those things, like quantum superposition, that I’ve just had to take for granted and not think about too much. Well, I’m back to thinking about it, prompted in part by a question from a friend to whom I wanted to provide a better explanation that I received, which was, basically: there’s a number i that when squared is equal to 1. Believe it.
Well, it was a bit more expansive than that. A good summary of the argument I’m familiar with can be found here . The problem with this argument–for me–is that it uses a fourpart definition of a number system to show that complex numbers are a valid system, just like fractions, and therefore, really exist. Well, I don’t have a problem understanding they’re a valid number system and therefore exist as mathematical objects–but I’ve never been able to make the leap from “exists as a valid number system” to “exists in the real world.”
Another argument starts with the premise that all numbers, as we know them, are symbols, not real world objects. Thus i[sup]2[/sup] is no less valid than 2[sup]2[/sup] or even just 2, for that matter–they’re all just symbols. The number two doesn’t occur naturally, it doesn’t grow on trees–it’s a symbol. I can understand that. But the symbols I’m used to correspond to quantities which can found in the real world.
2: you can have two apples
3[sup]2[/sup]: you can have a quantity of three apples and multiply them by three
sqrt(4): you can have four apples and deduce the number of apples that, when multiplied by itself, will result in 4 apples
1/2: you can cut an apple in half; you can even imagine cutting it perfectly in half.
3/2: you can have three halves of apples
1.435435435…: you can have one and slightly less than one half of an apple such that it corresponds to the decimal fraction
Even negative numbers don’t require much of a conceptual leap. I can view a group of five apples next to a group of two apples and observe that the former has three more apples than the latter and the latter has three less than the former; we’re still talking about the number three, we’ve just invented negative numbers to obviate the need for the qualifiers “more than” or “less than.” So you can argue that negative numbers don’t really exist in the same way, but I could show you how real world relationships imply their existence. The best you could do is argue that negative numbers only exist when one is comparing quantities and therefore don’t really exist.
But I can’t show you i apples. I don’t know how I’d tell the difference between 2 apples and 2 + 2i apples. It’s easier for me to think of complex numbers as pairs of real numbers, using a separate dimension to track the complex part. But I still can’t visualize it.
Questions:

Can complex numbers be shown to represent realworld quantities or are they simply intellectual constructs? Note that by the latter, I’m not implying they’re not valid, just that there’s no way to quantify them.

Are all numbers really complex numbers? In other words, the natural number 2 is a complex number with complex part 0, correct? Once you’ve opened up the complex door, you’ve got a new system that includes all the familiar numbers already, right? Really, you can’t go back. You can, of course, go back to just dealing with the familiar numbers, but you’re simply ignoring the zero complex part.

Are complex numbers all there is? Essentially, when we expand our number system to include complex numbers, we’re essentially saying that every number has two parts–real and imaginary. Can we be certain we won’t someday discover a third dimension we’ve been overlooking all along?

How many more complex numbers are there than familiar numbers?

I understand that without complex numbers, much of our current technology would not have been possible, that their applications are ubiquitous in modern engineering and other fields. But I’ve also heard that everything that can be done with complex numbers can also be done with familiar numbers–it’s just much, much, much more difficult. Is that true–that complex numbers don’t actually bring anything new to the mathematical table, they essentially just greatly facilitate calculations which would otherwise be inordinately complex?