#A09 · Algebra & Number Theory · Lv.3View in 3D Mind-Map

Complex Numbers & Polar Plane

Bombelli, Euler & Gauss · 1799 CE

z = a + bi = r(\cos\theta + i\sin\theta),\quad i^2 = -1

r = |z| = \sqrt{a^2 + b^2},\quad \theta = \operatorname{atan2}(b, a)

Extends real numbers into the 2D complex plane via i² = -1, completing the Fundamental Theorem of Algebra and turning multiplication into 2D rotation and scaling.

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Real part a

Imag part b

z = 3 + (4)i \implies |z| = 5.00, \; \theta = 53.1^\circ

절댓값 r = |z|

5.000

편각 θ (deg)

53.1°

제곱 회전 z²

-7.0 + (24.0)i

Prerequisite Path (8 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 6#A04Quadratic Formula & Discriminant

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

STEP 7#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 8#A09Complex Numbers & Polar Plane

z = a + bi = r(\cos\theta + i\sin\theta)

🚀 Next Unlocks (Derived Formulas) (2)

#C07Euler's Formula & Euler's Identity #L10Quaternions & 3D Spherical Interpolation (SLERP)

Geometric & Intuitive Insight

실수선에서 -1을 곱하는 것이 180° 반전이라면, i를 곱하는 것은 정확히 90° 반시계 방향 회전입니다! 두 번 90° 회전하면 180° 반전이 되므로 i × i = -1 이 기하학적으로 당연해집니다.

Step-by-Step Derivation

Step 1.1단계: 판별식 D < 0 인 이차방정식 x² + 1 = 0 의 해를 i = √(-1) 로 정의합니다.
Step 2.2단계: 복소수 z = a + bi 를 좌표평면의 점 (a, b) 로 대응시키면 원점 거리는 피타고라스 정리에 의해 r = √(a² + b²)
Step 3.3단계: 삼각함수로 치환하면 a = r cos θ, b = r sin θ 이므로 z = r(cos θ + i sin θ) 극형식이 얻어집니다.

AI & Coding Real-World Use

FFT audio/RF signal processing, quantum computing qubit amplitudes, and fractal rendering

#Quantum Computing#RF Signal DSP#2D Rotation

const mag = Math.hypot(re, im); const phase = Math.atan2(im, re);

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