Formally, a complex number is an expression \(a + bi\), with \(a\) and \(b\) real numbers and \(i\) an entity satisfying \(i^{2} = -1\). The square of an arbitrary real number is non-negative, so whatever \(i\) may be, it is not a real number.
Complex Addition and Multiplication
Suppose \(z = 3 + 4i\) and \(w = -5 + 2i\) are complex numbers. How might we add them? How might we multiply them? In each case, the goal is represent an expression, \(z + w\) or \(zw\), in the form \(a + bi\) for suitable real numbers \(a\) and \(b\). In an exploratory sense, it's reasonable to assume complex numbers satisfy the commutative, associative, and distributive properties, and to gather like terms: \[ z + w = (3 + 4i) + (-5 + 2i) = (3 - 5) + (4i + 2i) = -2 + 6i. \] Similarly, multiplying out (the distributive property), rearranging (associative and commutative properties), and collecting like terms gives \begin{align*} zw &= (3 + 4i)(-5 + 2i) \\ &= (3)(-5) + 3(2i) + (4i)(-5) + (4i)(2i) \\ &= (-15) + 6i + (-20i) + 8i^{2} \\ &= (-15) + (6 - 20)i - 8 \\ &= -23 - 14i. \end{align*}
Generally, we might write \(z = a + bi\) and \(w = c + di\), and calculate \begin{align*} z + w &= (a + bi) + (c + di) \\ &= (a + c) + (bi + di) \\ &= (a + c) + (b + d)i. \end{align*} Since \(a + c\) and \(b + d\) are real numbers (being sums of real numbers), the preceding formula defines the sum of complex numbers on the left-most side.
Similarly, \begin{align*} zw &= (a + bi)(c + di) \\ &= (ac) + a(di) + (bi)c + (bi)(di) \\ &= (ac) + (ad + bc)i + (bd)i^{2} \\ &= (ac - bd) + (ad + bc)i. \end{align*}
To emphasize, we have defined \begin{align*} (a + bi) + (c + di) &= (a + c) + (b + d)i, \\ (a + bi)(c + di) &= (ac - bd) + (cb + ad)i. \end{align*} These formulas obviate logical concerns about the formal manipulations that led us to them. But are these definitions useful?
It turns out these definitions obey the associative and commutative properties, and the distributive property, on the set of complex numbers. This justifies our definitions: We have defined a new, logically-consistent set of complex numbers, including the operations of addition and multiplication.
The complex number \(a + bi\) is said to have real part \(a\) and imaginary part \(b\). (Note carefully: The “imaginary part” is real!) A complex number of the form \(a = a + 0i\) “behaves just like” an ordinary real number. Practically, we may view real numbers as special complex numbers, those whose imaginary part is \(0\).
Complex Division
A complex number can be divided by a non-zero complex number. This fact is striking if we try naively to make sense of an expression such as \((3 + 4i)/(-5 + 2i)\). On the other hand, there is a sneaky algebra trick that lets us find the reciprocal of a non-zero complex number. Division by a non-zero complex number is multiplication by the reciprocal.
Note first that if we multiply \(-5 + 2i\) by \(-5 - 2i\), the number obtained by “reversing” the imaginary part, we get \[ (-5 + 2i)(-5 - 2i) = (-5)^{2} - (2i)^{2} = 25 - (-4) = 25 + 4 = 29, \] a positive real number! (We used the “difference of squares” identity. The definition of multiplication gives the same result.)
Now, although we do not yet know how to divide arbitrary complex numbers, we can perhaps agree that dividing a non-zero complex number by itself gives a quotient of \(1\). Thus, \begin{align*} \frac{1}{-5 + 2i} &= \frac{1}{-5 + 2i} \cdot 1 &&\text{multiplying by \(1\) does not change a number} \\ &= \frac{1}{-5 + 2i} \cdot \frac{-5 - 2i}{-5 - 2i} &&\text{rewriting \(1\) using the conjugate} \\ &= \frac{-5 - 2i}{(-5 + 2i)(-5 - 2i)} &&\text{multiplying numerators and denominators} \\ &= \frac{-5 - 2i}{29} &&\text{prior calculation} \\ &= -\frac{5}{29} - \frac{2}{29}i &&\text{a complex number!} \end{align*}
Generally, if \(z = a + bi\) is complex, we define its complex conjugate to be \(\bar{z} = a - bi\). By the definition of multiplication, the product of a complex number and its conjugate is \begin{align*} z \bar{z} &= (a + bi)(a - bi) \\ &= \bigl(a^{2} - b(-b)\bigr) + \bigl(a(-b) + ba\bigr)i \\ &= a^{2} + b^{2} + 0i. \end{align*} This is a sum of real squares, and is positive unless \(a = b = 0\), i.e., unless \(z = 0 + 0i\).
By the reasoning above (with a few steps elided), if \(z = a + bi\) is non-zero as a complex number (at least one of \(a\) and \(b\) is a non-zero real number), then \[ \frac{1}{z} = \frac{1}{a + bi} = \frac{1}{a + bi} \cdot \frac{a - bi}{a - bi} = \frac{a - bi}{a^{2} + b^{2}} \] as complex numbers.
Now assume \(z = a + bi\) and \(w = c + di\) are complex, and \(w\) is non-zero. As just noted we have \[ \frac{1}{w} = \frac{1}{c + di} = \frac{c - di}{c^{2} + d^{2}}. \] Consequently, \[ \frac{z}{w} = z \cdot \frac{1}{w} = (a + bi) \cdot \frac{c - di}{c^{2} + d^{2}}, \] which we can multiply out to obtain a formula if desired. In practice the process is faster than the formula. For example, \[ \frac{3 + 4i}{-5 + 2i} = \frac{(3 + 4i)(-5 - 2i)}{29} = \frac{-15 + 8 - (20 + 6)i}{29} = \frac{-7 - 26i}{29}. \]
One philosophical take-away is that complex numbers may be viewed abstractly as a “number system”: They are a well-defined set of mathematical entities that can be added, multiplied, and divided just like real numbers. The next paragraph can be skipped on first reading if this imprecision doesn't bother you.
More carefully, addition and multiplication are associative and commutative, and multiplication distributes over addition. Further, there is a complex number \(0 = 0 + 0i\) satisfying \(0 + z = z\) for all complex \(z\), and every complex number \(z = a + bi\) has a negative \(-z = -a - bi\) satisfying \(z + (-z) = 0\). Similarly, there is a complex number \(1 = 1 + 0i\) satisfying \(1 \cdot z = z\) for all complex \(z\), and every non-zero complex number \(z = a + bi\) has a reciprocal \(1/z = (a - bi)/(a^{2} + b^{2})\) satisfying \(z \cdot (1/z) = 1\).
Geometry of Complex Operations
A complex number \(a + bi\) amounts to an ordered pair \((a, b)\) of real numbers. We may view the set of complex numbers as the Cartesian plane: The Euclidean plane with a distinguished origin and coordinate axes. The real axis is horizontal, and comprises complex numbers \(a + 0i\) with imaginary part \(0\). The imaginary axis is vertical, and comprises complex numbers \(0 + bi\) with real part \(0\).
Addition and multiplication have pleasant interpretations in Euclidean geometry. We won't prove these geometric assertions, but will note some useful, intuitive consequences.
Addition translates the plane. Specifically, if \(\alpha\) (alpha, the first letter of the Greek alphabet) is a fixed complex number, there is a unique translation of the plane carrying \(0\) to \(\alpha\). This is the mapping that sends each complex number \(z\) to \(z + \alpha\). Particularly, adding \(1\) shifts the plane one unit to the right, adding \(i\) shifts the plane one unit upward.
Multiplication rotates and scales the plane about the origin \(0\). Specifically, if \(\alpha\) is a fixed complex number, there is a unique way to rotate-and-scale the plane keeping \(0\) fixed and carrying \(1\) to \(\alpha\). This is the mapping that sends each complex number \(z\) to \(\alpha z\). Particularly, multiplying by \(1\) is the identity mapping, while multiplying by \(-1\) is a half-turn about the origin.
Multiplying by \(i\) has a concrete meaning as a one-quarter turn counterclockwise about the origin. Performing two quarter turns, namely squaring \(i\), gives a half-turn, multiplication by \(-1\)! To say “imaginary numbers don't exist” amounts to denying existence of rotations.
Although complex conjugation is not an arithmetic operation, it too has a useful geometric interpretation: The mapping that sends each complex number \(\alpha = a + bi\) to \(\bar{\alpha} = a - bi\) is reflection across the real axis. Although reflection is a Euclidean motion (preserves distances), it does not preserve the sense of angles, but instead swaps clockwise and counterclockwise.
Magnitude and the Unit Circle
Every non-zero complex number lies on a unique circle centered at \(0\). If we write \(\alpha = a + bi\) with \(a\) and \(b\) real, then the distance from \(0\) to \(\alpha\) is \(\sqrt{a^{2} + b^{2}}\) by the Pythagorean theorem applied to the right triangle with corners \(0\), \(a\), and \(\alpha\). This radius is useful enough to earn a name: We call \[ \sqrt{a^{2} + b^{2}} = \sqrt{\alpha\bar{\alpha}} \] the magnitude of \(\alpha\), and denote it with absolute value symbols, \(|\alpha| = \sqrt{a^{2} + b^{2}}\). It can be checked (by somewhat laborious algebra) that if \(\alpha\) and \(\beta\) (beta) are complex, then \[ |\alpha\beta| = |\alpha| \cdot |\beta|. \] 
In words, the magnitude of a product is the product of the magnitudes. Particularly, if \(\alpha\) is non-zero, we may multiply and divide by \(|\alpha|\), obtaining \[ \alpha = |\alpha| \cdot \frac{\alpha}{|\alpha|}. \] The first factor \(|\alpha|\) is a positive real number. Multiplication by \(|\alpha|\) geometrically effects scaling (zooming in or out) about the origin. The second factor \(\alpha/|\alpha|\) is a point of the unit circle, a complex number of magnitude \(1\). Multiplying by a point of the unit circle is rotation about the origin.
Complex multiplication plays a starring role at the Math Art Shop, and the unit circle is a particular favorite. An earlier post on circles and trigonometry noted that every point of the unit circle in the Cartesian plane has coordinates \((\cos\theta, \sin\theta)\) for some real number \(\theta\). If we identify each real ordered pair \((a, b)\) with the complex number \(a + bi\), we are led to describe points of the unit circle in the form \[ \cos\theta + i\sin\theta,\qquad\text{\(\theta\) real.} \] A remarkable identity, often called de Moivre's identity or Euler's formula, says \[ e^{i\theta} = \cos\theta + i\sin\theta,\qquad\text{\(\theta\) real.} \] Even to define the terms involved, especially the left-hand side, requires calculus. One approach, often seen in the second semester of a careful study of real numbers and functions, uses “power series” to place this formula on a solid footing.
The particular case \(\theta = \pi\) reads \[ e^{i\pi} = \cos\pi + i\sin\pi = -1, \] or \(e^{i\pi} + 1 = 0\). Perhaps more interestingly, the law of exponents holds for complex exponents. Multiplying the equations \begin{align*} e^{i\theta} = \cos\theta + i\sin\theta, \\ e^{i\phi} = \cos\phi + i\sin\phi, \end{align*} gives a formula for \(e^{i(\theta + \phi)} = \cos(\theta + \phi) + i\sin(\theta + \phi)\) that contains two hard-to-remember trig identities, the sum formulas for cosine and sine.