Algebra - Complex Plane

Complex Plane

Graphical Understanding of Complex Numbers

The point M(a, b) represents the complex number \( a + bi \) on the complex plane, where:

  • \( a \): real part (x-axis), \( b \): imaginary part (y-axis)
  • \( r = \sqrt{a^2 + b^2} \): modulus of the complex number
  • \( \varphi \): argument (angle with real axis)
Complex Plane Diagram

Trigonometric Form of Complex Number

\[ a + bi = r(\cos \varphi + i \sin \varphi) \]

\[ [r(\cos \varphi + i \sin \varphi)]^n = r^n (\cos n\varphi + i \sin n\varphi) \]

Angle and Trig Ratios

\[ \tan \varphi = \frac{b}{a}, \quad \cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}, \quad \sin \varphi = \frac{b}{\sqrt{a^2 + b^2}} \]

Applications of Complex Numbers

  • Used in AC circuit analysis in electrical engineering.
  • Modeling oscillations and waveforms in physics.
  • Fractal geometry and computer graphics.
  • Solving polynomial equations and transformations.

Why It Matters

Complex numbers extend the idea of one-dimensional numbers to two dimensions. They help interpret mathematical and physical phenomena where direction and magnitude are involved.

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