Complex Number formulas

Every formula JEE Advanced expects you to know for Complex Number, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 22
Subject: Mathematics
Exam: JEE Advanced
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Complex Number, every formula

22 formulas, typeset and free. Print it, or keep it open beside your practice.

Complex number
$z = x + i y, i = - 1$
Modulus
$∣ z ∣ = x^{2} + y^{2}$
Argument
$ar g z = tan^{- 1} \frac{y}{x}$
Conjugate
$\overset{z}{ˉ} = x - i y$
Modulus–conjugate
$z \overset{z}{ˉ} = ∣ z ∣^{2}$
Polar form
$z = r (cos θ + i sin θ)$
Euler form
$z = r e^{i θ}$
Product (polar)
$z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$
Quotient (polar)
$\frac{z _{1}}{z _{2}} = \frac{r _{1}}{r _{2}} e^{i (θ_{1} - θ_{2})}$
De Moivre's theorem
$(cos θ + i sin θ)^{n} = cos n θ + i sin n θ$
nth roots of unity
$z = e^{2 π ik / n}, k = 0, 1, \dots, n - 1$
Sum of nth roots of unity
$1 + ω + ω^{2} + \dots + ω^{n - 1} = 0$
Cube roots of unity
$1, ω, ω^{2}; 1 + ω + ω^{2} = 0, ω^{3} = 1$
Triangle inequality
$∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$
Modulus of a product
$∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$
|z1+z2|²
$∣ z_{1} + z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} + 2 Re (z_{1} \overset{z}{ˉ}_{2})$
Square root of a complex number
$z = \pm (\frac{∣ z ∣ + x}{2} + i sgn (y) \frac{∣ z ∣ - x}{2})$
Rotation of a vector
$z^{'} = z e^{i θ}$
Distance between points
$∣ z_{1} - z_{2} ∣$
Equation of a circle
$∣ z - z_{0} ∣ = r$
Real and imaginary parts
$Re (z) = \frac{z + z ˉ}{2}, Im (z) = \frac{z - z ˉ}{2 i}$
Argument of a product
$ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}$

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Complex Number formulas

Complex Number, every formula

Complex number

Modulus

Argument

Conjugate

Modulus–conjugate

Polar form

Euler form

Product (polar)

Quotient (polar)

De Moivre's theorem

nth roots of unity

Sum of nth roots of unity

Cube roots of unity

Triangle inequality

Modulus of a product

|z1+z2|²

Square root of a complex number

Rotation of a vector

Distance between points

Equation of a circle

Real and imaginary parts

Argument of a product

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