Real + Imaginary
A Complex Number (\(z\)) is a combination of a real number and an imaginary number. It is written in the form:
\[z = a + bi\]
- \(a\): The Real Part.
- \(b\): The Imaginary Part.
Arithmetic Rules
- Addition: Add the real parts and the imaginary parts separately.
- Multiplication: Use FOIL and remember that \(i^2 = -1\).
Worked Examples
Example 1: Addition
\((3 + 2i) + (4 - 5i)\)
- Real: \(3 + 4 = 7\). Imaginary: \(2 - 5 = -3\).
- Result: \(7 - 3i\).
Example 2: Multiplication
\((1 + i)(2 + 3i)\)
- First: \(1 \cdot 2 = 2\).
- Outer: \(3i\).
- Inner: \(2i\).
- Last: \(3i^2 = 3(-1) = -3\).
- Combine: \((2 - 3) + (3i + 2i) = -1 + 5i\).
- Result: \(-1 + 5i\).
The Bridge to Quantum Mechanics
When two quantum particles interact, we multiply their complex wavefunctions. This is exactly like the multiplication you just did. The real and imaginary parts of the wavefunction carry information about the particle's Energy and Momentum. If you can't add and multiply complex numbers, you won't be able to calculate how two atoms bond or how a photon is absorbed. This arithmetic is the "base-level" logic of every quantum calculation.