The Imaginary Unit i

Complex numbers pervade quantum mechanics: state vectors, amplitudes, and time-evolution operators all live in the complex plane. Everything starts with one simple definition.

Defining i

The imaginary unit $i$ is defined by the single equation

$i^2 = -1.$

No real number satisfies this equation because squaring any real number gives a non-negative result. We introduce $i$ as a new symbol that obeys ordinary algebra together with this one rule. That is enough to build all of complex arithmetic.

Powers of i cycle with period 4

Repeatedly multiplying by $i$ produces a four-step cycle that never leaves the set $\{1, i, -1, -i\}$ :

i^0 = 1, \quad i^1 = i, \quad i^2 = -1, \quad i^3 = i^2 \cdot i = -i, \quad i^4 = i^3 \cdot i = -i \cdot i = -i^2 = 1.

After four steps we are back to 1, so the cycle repeats. For any integer $n$ , we reduce:

$i^n = i^{n \bmod 4},$

where the right-hand side uses the table above. For example, $i^{13} = i^{13 \bmod 4} = i^1 = i$ .

Why this matters for quantum mechanics

In quantum mechanics the state of a system is described by a state vector whose components are complex numbers called probability amplitudes. The time-evolution operator involves the factor $e^{-iEt/\hbar}$ , which oscillates in the complex plane because of $i$ . Without $i$ there would be no interference — the phenomenon that underlies every quantum algorithm. Learning to work fluently with powers and products of $i$ is therefore the very first step toward understanding quantum states.

Try it

This is a numerical exercise — your code should return a number. Compute the real part of $i^{22}$ .

Run your code to see the quantum state.