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L3-01 · Complex numbers and phase

Calculate a complex magnitude and explain the difference between magnitude and phase.

Level 3 icon Level 3 · High-school bridge
Objective. Calculate a complex magnitude and explain the difference between magnitude and phase.
Before this lesson. L2-01
Hands-on lab. Interference

Learn

A complex number has a real part and an imaginary part. Write it as z = a + ib. The numbers a and b are real. The symbol i satisfies i squared = -1. Plot a on the horizontal axis and b on the vertical axis.

The complex conjugate changes the sign of the imaginary part. For z = a + ib, its conjugate is a - ib. Multiply the two numbers. The imaginary terms cancel. The result is a squared plus b squared. Its square root is the magnitude.

Phase is the angle of the point from the positive real axis. Two numbers can have equal magnitude and different phases. Quantum amplitudes use both properties. Probability depends on squared magnitude, but a later operation can turn a phase difference into a probability difference.

Practice

Solution

The squared magnitude is 1² + (-2)² = 5. The two points lie on opposite sides of the real axis, so their phases differ.

Check your understanding

Which operation produces the squared magnitude of z?

Show the answer

B. Multiply z by its conjugate

The product with the conjugate is a² + b². It is real and cannot be negative.

Sources

This lesson draws on Illowsky & Dean, 2023Watrous, 2018.

Next: L3-02 · Vectors and inner products

References
  1. Illowsky, B., & Dean, S. (2023). Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
  2. Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/