Are Imaginary Numbers as $Real$ as Real Numbers?

Question

In my Abstract Algebra book, ``A First Course in Abstract Algebra,'' by Fraleigh, the author seems to suggest that imaginary numbers are as $real$ as the real numbers, by asserting, for example, that generations of students have treated numbers which have a nonzero imaginary part with more skepticism than the real numbers (i.e., those number which have an imaginary part equal to zero).

I know this is a rather open-ended question, but can someone explain if one of these classes of numbers has more validity than the other?

Also, are there any other types of numbers (exempting Cantor's transfinite numbers) besides these complex numbers (i.e., numbers of the form $a + bi$, where $a, b \in \mathbb{R}$)?

Answer 1

These "number" things are only as valid as we make them. Historically, people used to reject the idea of negative numbers. Nowadays, negative numbers are almost second-nature. We've seen the same thing happen with irrational numbers: take the Pythagoreans for example. Complex numbers are the same. Here is what I mean.

How would we solve $x+4=0$? Well, we need to introduce a new class of numbers, the negatives. So we say $x=-4$.

How would we solve $x^2-2=0?$ Well, certainly no rational number will do the job, so we need to introduce the irrational number $\sqrt{2}$. Then we can say $x=\pm\sqrt{2}$.

Now, how would we solve $x^2+1=0$? Certainly no real number will do the trick, so we introduce complex numbers. Then we have $x=\pm i$.

We introduce new number systems because they are convenient and interesting. Just as fleablood said in the comments, what are numbers? This is something I leave you to ponder on your own.

Answer 2

I think it is common for students to find the complex numbers $\mathbb C$ to be strange when in reality they are entirely consistent with the lower rungs on the "ladder of numbers", as it were. Below is my impression of the construction of various number systems as described in chapter 1 of Analysis I by Amann and Escher.

(One of the key ideas here is each extension of our "number system" allows us to do things (such as solving certain equations) that we couldn't do previously, and each extension contains the previous system within it such that the previous system continues to function the way it did before.)

1. First assume that we are comfortable with basic set theory, functions, relations and operations. There are axioms called the Peano Axioms that we can use to build the natural numbers $\mathbb N = \{ 0, 1, 2, \dots \}$. Addition and multiplication work the way you expect.
2. We embed $\mathbb N$ into the so-called ring of integers $\mathbb Z$. This allows us to consider arbitrary differences $m - n$, which we couldn't do with $\mathbb N$ alone. (This is already a big leap in abstraction. If, 2000 years ago, I had asked you how many oranges I would have if I had three and gave you five, you would probably look at me like I was insane.)
3. We embed $\mathbb Z$ as a subring in the field of rationals $\mathbb Q$, which allows us to solve equations such as $2x = 1$. The natural numbers and integers continue to work normally within this larger structure $\mathbb Q$.
4. We notice that the equation $x^2 = a$ is in general not solvable in $\mathbb Q$, so once again we need to build a new structure containing what we've created so far. This is called the extension field $\mathbb R$ of $\mathbb Q$ and it allows us to solve equations like $x^2 = 2$. This is the "number line" that you've probably internalized so deeply that you're not even aware of the all the work that is required to rigorously define what it is.
5. We run into yet another obstacle, which is that the equation $x^2 = -1$ has no solution in $\mathbb R$. Notice that wanting to extend our number system in this case is entirely consistent with what we've done up to this point. Wanting to solve $x^2 = -1$ is no weirder than wanting to solve $3 - 5 = y$. We create an extension field $\mathbb C$ of $\mathbb R$ that allows us to solve $x^2 = -1$, and $\mathbb C$ is a big enough system to solve all algebraic equations.

On this ladder of numbers there are two more rungs: the quaternions and octonions. They are above my pay grade so I'm not even going to try to explain what they are. It is my understanding that nobody really understands how to use the octonions, for instance.

Answer 3

Complex numbers are definitely as valid as the reals - actually doing mathematics to them is exactly the same kind of activity of reasoning, arguments and proofs, it's just marginally more complicated if you're inexperienced because there are more definitions to deal with and complex numbers are further away from our experience so you have less helpful intuitions for how they should work. See the other answers for why mathematics with the complex numbers is interesting.

If you mean, are complex numbers "real" in that they describe phenomena in reality like the reals do? ...Well, arguably the reals don't either. The defining feature of the reals, what makes them distinct from the rationals, is that the rationals have "gaps" that are filled in by real numbers. These gaps are not very easy to visualize because you can chop up the space between rationals as small as you like while still talking about rationals, but you can sort of get the idea by thinking about how the $\sqrt{2}$ falls into a "gap" between $[1.4,1.5]$, and $[1.41,1.42]$ and $[1.414,1.415]$, and so on no matter how many digits you write out. You can make this interval as narrow as you like, but you can only get exactly the squareroot of $2$ by making it "infinitely" narrow - at which point you no longer have a rational number, because rationals can't have infinite denominators.

Quantities in the physical universe don't seem to work like that. Lengths, times, masses, energies and other similar measurable things all seem to be either integer multiples of some fundamental unit (e.g. all charges are multiples of the charge of certain fundamental particles) or are inherently "fuzzy", and make the universe behave strangely if you try to divide them up smaller than a certain scale of resolution. Even if you somehow get around that fuzziness, it's not clear how you would do this dividing up infinitely to get a true real within the finite lifespan of the universe. ($\sqrt{2}$ can be represented as a rational so long as you're happy rounding it to some, any, finite number of places)

So if we wave our hands, pretend the universe is Newtonian and space(time) is a genuinely smooth manifold and objects within it have genuinely real-valued positions, etc, can we use complex numbers to describe something "real?"

In short, yes, there are lots of completely sensible applications of complex numbers in physics and engineering. One example that doesn't seem to be mentioned in that thread is that quantum-mechanical wavefunctions are functions producing complex numbers, and these in turn determine the probabilities of outcomes of experiments. So in a quite fundamental sense, complex numbers and their structure underpin reality as we know it.

Since quantum theory was used to design the device you're reading this post on, yes absolutely, complex numbers are as "real" as the reals - if not more so - both in terms of their abstract properties as an axiomatically defined algebra system, and their applications to physical reality.

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As for "other types of numbers", that really depends what you want "numbers" to be able to do. There are countless structures that can do things that resemble "adding" or "multiplying" elements together. However, the complex numbers are relatively special in that they allow you to "divide" in a way that resembles division over reals, and in particular has no "zero divisors", i.e. a pair of numbers that multiply together to produce $0$. There's a certain sense you can "glue together" two copies of the complex numbers to get something called the quaternions, which are 4D and can be used describe rotations in 3D space, and glue together two copies of those to produce the 8D octonions. However, each time you do this you lose nice algebraic properties (like the complex numbers cannot be ordered like the reals can) and if you go any further than the octonions you get zero divisors, so your "division" isn't working very much like real division anymore.