You have probably come across equations such as $x^2 + 4 = 0$, and written "no real solutions". Well, some mathematicians back in the 16th century essentially said "nah" to that, and came up with a new imaginary unit, defined as:
$$i^2 = -1$$
These, despite being called "imaginary", are actual mathematical objects you can do operations on.
Complex numbers are expressions of the form $a + bi$, where:
The real and imaginary parts can be any real number, including zero. When $b = 0$, we have a real number, and when $a = 0$, we have a purely imaginary number.
Let's multiply $(2 + 3i)(1 - 2i)$:
For any complex number $z = a + bi$, its conjugate is defined as $\bar{z} = a - bi$.
Adding a complex number to its complex conjugate, or multiplying a complex number by its own complex conjugate, always yields a real number.
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