1. The Imaginary Unit
The real number system has a massive structural limitation: it cannot calculate the square root of a negative value because squaring any real number always yields a positive result. To overcome this, we define the imaginary unit i such that i² = -1.
2. The Complex Plane and Polar Form
Complex numbers are plotted on a 2D coordinate space where the horizontal axis tracks the real part and the vertical axis tracks the imaginary component. A complex coordinates point can be represented as z = x + iy, or converted into trigonometric polar form z = r(cos θ + i sin θ).
3. Rigorous Proof of Euler's Identity
Euler's Identity seamlessly connects five fundamental mathematical constants: e, i, π, 1, and 0. Let us prove it using Taylor Series expansions.
Theorem: e^(iθ) = cos θ + i sin θ
Step 1: Look at the infinite series expansion for e^x:
e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ...
Step 2: Substitute x = iθ into the series:
e^(iθ) = 1 + (iθ) + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + ...
Step 3: Evaluate powers of i knowing i²=-1, i³=-i, i⁴=1:
e^(iθ) = 1 + iθ - θ²/2! - iθ³/3! + θ⁴/4! + iθ⁵/5! - ...
Step 4: Regroup into real and imaginary components:
e^(iθ) = (1 - θ²/2! + θ⁴/4! - ...) + i(θ - θ³/3! + θ⁵/5! - ...)
Step 5: Identify the matching trigonometric Maclaurin series expressions:
The real part is exactly cos θ. The imaginary scalar is exactly sin θ.
Therefore: e^(iθ) = cos θ + i sin θ
Step 6: Let θ = π:
e^(iπ) = cos(π) + i sin(π) -> e^(iπ) = -1 + i(0) -> e^(iπ) + 1 = 0