Complex numbers have an intuitive way of thinking about them, they’re just not frequently given sufficient exposition for clarity. The fundamental concept to internalize for complex numbers is the square root of a negative number. In the real world, calculations involving square roots of negative numbers are practical. If all you have are the real numbers, you cannot resolve equations involving the square root of a negative number, like x^2 = -5. This isn't arbitrary - it becomes very handy when you're dealing with problems that can run into complex numbers. For example, we can use complex numbers to model phase and magnitude in physics and electrical engineering.

Complex numbers resolve this problem by segregating the discrepancy and reducing it to an imaginary unit i, where i^2 = -1. If you can agree to the definition of i, you can usefully model any square root of a negative number; if you recall that 1 is the multiplicative identity (1x = x), -1 can be considered to be something like the complex multiplicative identity. In other words, definining it axiomatically allows you to extrapolate the square root of arbitrary negative numbers using real numbers and i. In this way, the complex numbers extend the real numbers and encompass them.

If you’re with me so far, this next part might improve your intuition. A complex number is typically written z = a + bi, where a and b are real numbers and i is the imaginary square root of -1. This is presumably what you’re familiar with. For the complex number z, we have a real part, a, and the imaginary part, b. More formally, Re(z) = a and Im(z) = b. Instead of reasoning about complex numbers using this form, you can instead represent them as ordered pairs of the form (a,b), where a and b are real numbers and the same real and imaginary parts of z, respectively. This form doesn’t represent i explicitly, which might be more straightforward to understand. Moreover, complex numbers of the form (a,b) can be modeled as points in a plane, because they are coordinates.

Recall that a plane is a two dimensional coordinate space; a line is a one-dimensional space, and you can generalize this conceptually to k dimensions, represented by R^k. An ordered pair is a 2-tuple, which generalizes to a k-tuple, such that the points in a k-dimensional space R^k are k-tuples, (x1, ..., xk). Therefore, when we’re talking about ordered pairs, we’re talking about planes, and each element in an ordered pair is a dimension of the plane. For any given real number a, the real number’s ordered pair is (a,0). i = (0,1); for any x = (a,b) and y = (c,d), xy = (ac - bd, ad + bc). This defines multiplication on the set of all complex numbers (recall, then, that any set defined with operations for addition and multiplication becomes a field, so this along with addition establishes the complex field, extending the real field).

Because any complex or real number can represented as an ordered pair of real numbers (a,b), we use R^2 to represent the complex plane, where R traditionally denotes the set of all real numbers. Finally we get to our result: by modeling the already intuitive real numbers and the complex numbers as points in a plane, we can practically model and reason about real world 2-dimensional problems using geometric and topological methods that would be unresolvable without a square root of negative numbers. Naturally, this features heavily in mathematical analysis :)

Now that we've gotten this far, we can look at a basic, practical example. Let's say we want to model a voltage or current on a two dimensional plane. Let z be a complex number representing the voltage or current on the complex plane R^2. Further let the vertical axis be Im(z) and the horizontal axis Re(z). Then we have line segments x and y representing the real and imaginary parts of z respectively, and drawn from z to the vertical Im(z) and horizontal Re(z) axes, respectively. The phase is initially represented by 0 (the origin), and the magnitude is a line segment drawn from the origin to the point of z, where the real and imaginary parts of z intersect on the plane. You could use trigonometry to reason about phase differences in a current, but it is much simpler to instead represent them as linear equations involving complex numbers.

I hope that helps a bit to build your intuition. If you can sit through it and work through the proofs, I have found that Rudin’s Principles of Mathematical Analysis is really excellent for building up the number systems from first principles in the first chapter. It’s dry and might take you 10 minutes per page, but it’s self-contained and will really enhance your understanding of why we have systems of numbers from a unifying theoretical perspective.

For example, you can (hopefully) see how complex numbers are really just a resolution to the problem of square roots of negative numbers, and that they have a practical application for modeling problems in physical space. You can think of irrational numbers in a very similar way: just as complex numbers extend real numbers with this resolution, irrational numbers extend the rational numbers with a resolution to the problem of square roots that have an infinite decimal expansion. We can look at the square root of 2: there is no rational p to satisfy p^2 = 2.

The intuitive theory of irrational numbers is not that they're "infinite decimals" (which leads to the question, "What are infinite decimals and why do we care about them?"). The intuitive theory of irrational numbers is that they are a set of numbers which resolve equations that we cannot resolve using only rational numbers. Similarly, the intuitive theory of complex number is not that they're numbers with some arbitrary "imaginary" part embedded in them, they're a system of numbers that express ideas we cannot reason about using only the reals.

Putting all of this together, we have:

1. The set of all natural numbers N: {0, 1, 2, 3, ...}. These are useful for counting.

2. The set of all integers, Z: {..., -2, -1, 0, 1, 2, ...} ("Z" stands for "Zahlen").

3. The set of all rational numbers, Q, where Q stands for quotient.

4. The set of all irrational numbers, sometimes denoted by I.

5. The set of all real numbers, R, which contains both the rational and irrational numbers.

6. The set of all complex numbers, sometimes denoted by C, which contains both the real and complex numbers.

Each number system is an extension of the number system preceding it, and expands the system in a way that allows new axiomatic operations to be defined such that we can resolve problems observed (and unsolvable) in the previous number systems.