The Ultimate Notation
Now that we have Euler's formula, we can rewrite any complex number in its most powerful form. This notation turns multiplication into addition and powers into simple multiplication.
\[z = r e^{i\theta}\]
Worked Examples
Example 1: Multiplication Revisited
Multiply \(z_1 = 5e^{i\pi/4}\) and \(z_2 = 2e^{i\pi/2}\).
- Numbers: \(5 \cdot 2 = 10\).
- Exponents (Add): \(\pi/4 + \pi/2 = 3\pi/4\).
- Result: \(10e^{i3\pi/4}\).
Example 2: Division
Divide \(z_1 / z_2\).
- Numbers: \(5 / 2 = 2.5\).
- Exponents (Subtract): \(\pi/4 - \pi/2 = -\pi/4\).
- Result: \(2.5e^{-i\pi/4}\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the total energy of a particle is represented by the frequency of this exponential: \(e^{-iEt/\hbar}\). If you have two different energy levels, their total wavefunction is the sum of two different exponentials: \(\psi = c_1 e^{-iE_1t/\hbar} + c_2 e^{-iE_2t/\hbar}\). By using the laws of exponents on these complex forms, we can predict exactly when a particle will "tunnel" or "decay." This notation isn't just a convenience; it's the only way to handle the time-dependent nature of reality.