Complex Numbers Formulas

Definitions

A complex number is written as a + bi where a and b are real numbers an i, called the imaginary unit, has the property that i2 = -1.
The complex numbers a + bi and a – bi are called complex conjugate of each other.

Formulas

1) Equality of complex numbers

a + bi = c + di if and only if a = c and b = d

2) Addition of complex numbers

(a + bi) + (c + di) = (a + c) + (b + d)i

3) Subtraction of complex numbers

(a + bi) – (c + di) = (a – c) + (b – d)i

4) Multiplication of complex numbers

(a + bi)(c + di) = (ac – bd) + (ad + bc)i

5) Division of complex numbers

\( \frac{a + bi}{c + di}  = \frac{a + bi}{c + di} . \frac{c – di}{c  – di} = \frac{ac + bd}{c^2 + d^2} + \left( \frac{bc – ad}{c^2 + d^2} \right)i \)

6) Polar form of complex numbers

\( x + iy = r (\cos \theta + i \sin \theta  )  r – modulus, \theta – amplitude\)

7) Multiplication and division of complex numbers in polar form

\( [ r_1 ( \cos \theta_1 + \sin \theta_1)] . [r_2 (\cos \theta_2 + \sin \theta_2) ] = r_1r_2[\cos(\theta_1 + \theta_2) + \sin(\theta_1 + \theta_2)] \)

\( \frac{r_1(\cos \theta_1 + \sin \theta_1)}{r_2(\cos \theta_2 + \sin \theta_2)} = \frac{r_1}{r_2}[\cos (\theta_1 – \theta_2) + \sin (\theta_1 – \theta_2)] \)

8) De Moivre’s theorem

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

9) Roots of complex numbers

\([r(\cos \theta + \sin \theta)]^\frac{1}{n} = r^\frac{1}{n} \left( \cos \frac{\theta + 2k\pi}{n} + \sin \frac{\theta + 2k\pi}{n} \right) \)

From this the n nth roots can be obtained by putting k = 0, 1, 2, . . ., n – 1

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