Imaginary unit

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

$i^2=-1.$

That is the entire definition. We don’t ask what $i$ “really is” any more than we ask what $5$ really is; we manipulate it according to ordinary algebraic laws together with the rule $i^2=-1$, and we judge the system by whether it does useful work. It does.

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. The two are the same object — just 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 the course goes into complex analysis proper 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, subject to $i^2=-1$.

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.