Imaginary unit

The imaginary unit is the symbol $i$ (or $j$ in electrical engineering) defined by

$i^2=-1.$

That is the whole definition. No need to ask what $i$ “really is” any more than we ask what $5$ really is. Manipulate it with the ordinary algebraic laws plus the rule $i^2=-1$, and the system does useful work.

Historical motivation

In the Renaissance, Italian algebraists solving the cubic $x^3+px+q=0$ found that even when all three roots were real, their formulas required taking $\sqrt{-1}$ in an intermediate step. Refuse the maneuver and you can’t get the answer; allow it and the imaginary parts cancel at the end, leaving the correct real roots. The mathematics was telling them you sometimes have to pass through an imaginary region to reach a real answer.

Euler later gave the symbol $i$ for “imaginary,” and Gauss provided the geometric picture (the complex plane).

i vs j

Mathematicians and physicists use $i$. Electrical engineers use $j$, because $i$ is already taken as the symbol for current. Same object, different notation. Vector Calculus and Complex Analysis uses $j$ in Chapters 1–3 (the phasor/circuit chapters) and switches to $i$ from Chapter 4 onward, where complex analysis proper starts and the unit vector $\mathbf{j}$ also appears.

Combining with reals

Multiplied by a real number, $i$ gives a pure imaginary number like $3i$ or $-i$. Added to a real number, it gives a complex number $x+iy$. The full algebra is generated from the reals and $i$ by closing under addition and multiplication, with $i^2=-1$.

Powers of i

The powers cycle with period 4:

$i^0=1,i^1=i,i^2=-1,i^3=-i,i^4=1,\dots$

So $i^n$ depends only on $n\mathrm{mod}4$. This is the first hint of the periodicity that becomes Euler’s formula and the multivaluedness of the complex logarithm.