Real & Imaginary Numbers

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Summary

Complex numbers are of the form \(a + bi\), where \(a\) and \(b\) are real and \(i\) is the unit with \(i^2 = −1\). We’ll use magnitude and phase a lot in quantum: \(z = re^{i\theta}\).

  • Magnitude: $$|z| = \sqrt{a^2+b^2}$$
  • Angle: $$\theta = atan2(b, a)$$
  • Euler: $$e^{i\theta} = cos(\theta) + i sin(\theta)$$

Practice

1) Convert \(z = 1 + i\) to polar form \((r, θ)\).

$$r = \sqrt{2}, \theta = \pi/4$$

2) Compute \((1 + 2i)(2 − i)\).

\((1·2 − 2·(−1)) + (1·(−1) + 2·2)i = 4 + 3i\)

3) What is \((a+bi)(a−bi)\)?

\(a^2 + b^2\) (always real and non‑negative)