Explosive Multiplication
What if you want to calculate \((1 + i)^{10}\)? Using FOIL would take forever. De Moivre's Theorem gives us a shortcut for raising complex numbers to any power \(n\).
If \(z = r(\cos\theta + i \sin\theta)\), then: \[z^n = r^n(\cos(n\theta) + i \sin(n\theta))\]
The Rule: Raise the radius to the power, and multiply the angle by the power.
Worked Examples
Example 1: Basic Power
Calculate \((1 + i)^4\).
- Convert \(1+i\) to polar: \(r = \sqrt{2}\), \(\theta = 45^\circ\).
- Apply Theorem: \((\sqrt{2})^4 \text{ cis }(4 \cdot 45^\circ) = 4 \text{ cis }(180^\circ)\).
- Convert back: \(4(\cos 180 + i \sin 180) = 4(-1 + 0i) = -4\).
- Result: -4
Example 2: Higher Powers
Calculate \([2(\cos 10^\circ + i \sin 10^\circ)]^6\).
- \(2^6 = 64\).
- Angle: \(6 \cdot 10 = 60^\circ\).
- Result: \(64 \text{ cis } 60^\circ = 64(0.5 + 0.866i) = 32 + 55.4i\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the energy of a particle is often found in the "Exponent" of its wavefunction. When we look at how a state changes over a long time \(t\), we are effectively raising the wavefunction to a very high power. De Moivre's Theorem is how we predict the "Long-Term Evolution" of a quantum system. It explains why a wave that starts small can grow or oscillate into a complex pattern over time. This theorem is the bridge between a single measurement and the lifetime of a particle.