List of Formulas of Complex Number

Complex Numbers: Definition

A complex number is expressed as \( z = a + bi \), where:

• \(a = R(z)\) → Real Part
• \(b = Im(z)\) → Imaginary Part
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• \(i^{4n} = 1\) where \(n=1,2,3,...\)

Basic Properties

• Conjugate: \( \overline{z} = a - bi \)
• Modulus: \( |z| = \sqrt{a^2+b^2} \)
• Product with conjugate: \( z\overline{z} = |z|^2 \)
• Argument: \(\arg z = \theta = tan^{-1}|\frac{Im}{R}|\)
  Note: (1) If \(\theta \leqslant 90^{\circ} \) then \(\arg z = \theta \)
  (2) If \(90^{\circ} < \theta \leqslant 180^{\circ} \) then \(\arg z = \pi - \theta \)
  (3) If \(180^{\circ} < \theta \leqslant 270^{\circ} \) then \(\arg z = \pi + \theta \)
  (4) If \(\theta \leqslant 360^{\circ} \) then \(\arg z = -\theta \)

Algebra of Complex Numbers

For \( z_1 = a+bi,\; z_2 = c+di \):

• Addition: \( z_1+z_2=(a+c) + i(b+d) \)
• Subtraction: \( z_1-z_2=(a-c) + i(b-d) \)
• Multiplication: \( z_1 z_2 = (ac-bd) + i(ad+bc) \)
• Division: \( \dfrac{z_1}{z_2} = \dfrac{(ac+bd) + i(bc-ad)}{c^2+d^2} \)

Polar / Trigonometric Form

\( z = r(\cos\theta + i\sin\theta) \), where \( r=|z| \) and \( \theta=\arg z \).

Exponential Form (Euler’s Formula)

\( e^{i\theta} = \cos\theta + i\sin\theta \)

Thus, \( z = re^{i\theta} \)

Multiplication & Division (Polar Form)

• \( z_1 z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)} \)
• \( \dfrac{z_1}{z_2} = \dfrac{r_1}{r_2} e^{i(\theta_1-\theta_2)} \)

De Moivre’s Theorem

\( (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \)

General: \( (re^{i\theta})^n = r^n e^{in\theta} \)

Complex Roots

The nth roots of \( z = re^{i\theta} \) are:

\( w_k = r^{1/n} e^{i(\theta + 2\pi k)/n},\ k=0,1,\dots,n-1 \)

Important Identities

• \( \overline{Z_1+Z_2} = \overline{Z_1} + \overline{Z_2} \)
• \( |Z_1 Z_2| = |Z_1|\,|Z_2| \)
• \( R(Z) = \dfrac{Z+\overline{Z}}{2} \)
• \( Im(Z) = \dfrac{Z-\overline{Z}}{2i} \)
• \( |\frac{Z_1}{Z_2}|=\frac{|Z_1|}{|Z_2|} \)
• \( |Z^n|=|Z|^n \)

Inequalities

• Triangle inequality: \( |z_1+z_2| \le |z_1|+|z_2| \)
• Reverse triangle inequality: \( \big||z_1|-|z_2|\big| \le |z_1-z_2| \)

Complex Logarithm & Powers

\( \log z = \ln|z| + i(\arg z + 2\pi k),\ k\in\mathbb{Z} \)

\( z^w = e^{w\log z} \)

Geometrical Interpretation

• Distance between two complex numbers: \( |z_1 - z_2| \)
• Roots of unity are equally spaced on the unit circle.

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🔑 Conclusion

Complex numbers extend the real number system and provide a powerful tool in algebra, geometry, and calculus. These formulas are highly useful for Class 11–12 mathematics, engineering, and competitive exams like JEE and GATE.