In classical mathematics, we deal primarily with real numbers, which are points on a one-dimensional number line. In quantum mechanics and quantum computing, we require complex numbers, which extend this system to two dimensions.
A complex number z is written as: z = a + bi , where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1).
Geometric Representation
Complex numbers can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The magnitude or modulus of z is |z| = √(a² + b²) and the argument (angle) is θ = tan⁻¹(b/a). Using Euler’s formula, z can be represented as z = re^(iθ).
Example
Let z₁ = 1 + i and z₂ = 2 – i
Addition: z₁ + z₂ = 3
Multiplication: z₁z₂ = 3 + i
Magnitude of z₁: |z₁| = √2
Polar form: z₁ = √2 e^(iπ/4)
Quantum Connection
A qubit (quantum bit) is represented as |ψ⟩ = α|0⟩ + β|1⟩ where α, β ∈ ℂ and |α|² + |β|² = 1.
This normalization ensures the total probability equals 1.
