Some equations have no real solution. The simplest is $x^2 = -1$, because the square of any real number is never negative. To repair this gap we introduce a single new symbol, the imaginary unit $i$, defined by the one rule that does all the work:
A complex number is any expression of the form $z = a + ib$, where $a$ and $b$ are real numbers. Here $a$ is the real part, written $\operatorname{Re}(z)$, and $b$ is the imaginary part, written $\operatorname{Im}(z)$. Notice that $\operatorname{Im}(z) = b$ is the coefficient of $i$, not $ib$ itself. When $b = 0$ the number is purely real, and when $a = 0$ (with $b \ne 0$) it is purely imaginary.
Equality: two complex numbers are equal exactly when their real parts match and their imaginary parts match — a single equation in complex numbers is really two equations in disguise.
Powers of $i$ cycle every four steps. From $i^2 = -1$ we get $i^3 = i^2 \cdot i = -i$ and $i^4 = (i^2)^2 = 1$. After that the pattern repeats, so to find any power of $i$ you only need the remainder of the exponent on division by $4$.
| Power | Value | Power | Value |
|---|---|---|---|
| $i^1$ | $i$ | $i^5$ | $i$ |
| $i^2$ | $-1$ | $i^6$ | $-1$ |
| $i^3$ | $-i$ | $i^7$ | $-i$ |
| $i^4$ | $1$ | $i^8$ | $1$ |
Arithmetic. Addition and subtraction act part by part. Multiplication uses ordinary distribution, after which every $i^2$ is replaced by $-1$ and like terms are gathered:
Division is the one operation that needs a trick: multiply the top and bottom by the conjugate of the denominator, $c - id$, which turns the denominator into the real number $c^2 + d^2$.
A caution with square roots of negatives: the schoolbook rule $\sqrt{x}\,\sqrt{y} = \sqrt{xy}$ fails when both numbers are negative. Always convert first: $\sqrt{-4} = 2i$, so $\sqrt{-4}\,\sqrt{-9} = (2i)(3i) = 6i^2 = -6$, never $\sqrt{36} = 6$.
Deeper Insight — one symbol, and the whole number system closes up: The entire chapter rests on a single act of definition: declare $i^2 = -1$ and agree that $i$ obeys the ordinary laws of algebra. Everything else is forced. Once you accept $a + ib$, addition and multiplication are not new rules to memorise but the familiar distributive law applied while remembering to swap $i^2$ for $-1$. The deeper pay-off is closure: in the real numbers, polynomial equations could run off the edge of the system and have no answer, but the moment $i$ exists, every polynomial equation has a full set of roots — this is the Fundamental Theorem of Algebra, and it is why complex numbers feel less like an invention and more like a discovery. Treat $i$ as an ordinary quantity you can add and multiply, never as something mystical, and the algebra behaves exactly as you expect.