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Complex Number formulas

Master Complex Number through 22 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Complex Number, every formula

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

Complex number
$z = x + i y, i = - 1$
Q1MCQComplex number
The value of $i^{2}$ is:
- A
  $-1$
- B
  $1$
- C
  $i$
- D
  $-i$
Modulus
$∣ z ∣ = x^{2} + y^{2}$
Argument
$ar g z = tan^{- 1} \frac{y}{x}$
Q1MCQArgument
The argument of $z=1+i$ is:
- A
  $\dfrac{\pi}{4}$
- B
  $\dfrac{\pi}{2}$
- C
  $0$
- D
  $\pi$
Conjugate
$\overset{z}{ˉ} = x - i y$
Q1MCQConjugate
The conjugate of $z=2-3i$ is:
- A
  $2+3i$
- B
  $-2+3i$
- C
  $2-3i$
- D
  $-2-3i$
Modulus–conjugate
$z \overset{z}{ˉ} = ∣ z ∣^{2}$
Q1MCQz·conj
$z\bar z$ equals:
- A
  $|z|^{2}$
- B
  $|z|$
- C
  $z^{2}$
- D
  $\bar z^{2}$
Polar form
$z = r (cos θ + i sin θ)$
Q1MCQPolar form
The polar form of a complex number is:
- A
  $r(\cos\theta+i\sin\theta)$
- B
  $r\cos\theta$
- C
  $r+i\theta$
- D
  $re^{\theta}$
Euler form
$z = r e^{i θ}$
Q1MCQEuler form
The Euler form of a complex number is:
- A
  $re^{i\theta}$
- B
  $re^{\theta}$
- C
  $e^{ir\theta}$
- D
  $r+i\theta$
Product (polar)
$z_{1} z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$
Q1MCQProduct polar
The product $z_1 z_2$ in polar form is:
- A
  $r_1 r_2 e^{i(\theta_1+\theta_2)}$
- B
  $r_1 r_2 e^{i(\theta_1-\theta_2)}$
- C
  $\dfrac{r_1}{r_2}e^{i\theta_1}$
- D
  $r_1+r_2$
Quotient (polar)
$\frac{z _{1}}{z _{2}} = \frac{r _{1}}{r _{2}} e^{i (θ_{1} - θ_{2})}$
Q1MCQQuotient polar
The quotient $\dfrac{z_1}{z_2}$ in polar form is:
- A
  $\dfrac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$
- B
  $r_1 r_2 e^{i(\theta_1+\theta_2)}$
- C
  $\dfrac{r_1}{r_2}e^{i(\theta_1+\theta_2)}$
- D
  $\dfrac{r_2}{r_1}$
De Moivre's theorem
$(cos θ + i sin θ)^{n} = cos n θ + i sin n θ$
Q1MCQDe Moivre
De Moivre's theorem states $(\cos\theta+i\sin\theta)^{n}$ equals:
- A
  $\cos n\theta+i\sin n\theta$
- B
  $\cos\theta^{n}+i\sin\theta^{n}$
- C
  $n(\cos\theta+i\sin\theta)$
- D
  $\cos\theta+i\sin n\theta$
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
$1, ω, ω^{2}; 1 + ω + ω^{2} = 0, ω^{3} = 1$
Q1MCQCube roots
For cube roots of unity, $1+\omega+\omega^{2}$ equals:
- A
  $0$
- B
  $1$
- C
  $3$
- D
  $\omega$
Triangle inequality
$∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣$
Q1MCQTriangle inequality
$|z_1+z_2|$ is:
- A
  $\le|z_1|+|z_2|$
- B
  $\ge|z_1|+|z_2|$
- C
  $=|z_1|+|z_2|$
- D
  $=|z_1||z_2|$
Modulus of a product
$∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣$
|z1+z2|²
$∣ z_{1} + z_{2} ∣^{2} = ∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2} + 2 Re (z_{1} \overset{z}{ˉ}_{2})$
Q1MCQ|z1+z2|²
$|z_1+z_2|^{2}$ equals:
- A
  $|z_1|^{2}+|z_2|^{2}+2\,\text{Re}(z_1\bar z_2)$
- B
  $|z_1|^{2}+|z_2|^{2}$
- C
  $(|z_1|+|z_2|)^{2}$
- D
  $|z_1|^{2}-|z_2|^{2}$
Square root of a complex number
$z = \pm (\frac{∣ z ∣ + x}{2} + i sgn (y) \frac{∣ z ∣ - x}{2})$
Q1MCQSquare root
The square root of a complex number gives:
- A
  two values
- B
  one value
- C
  infinitely many
- D
  no value
Rotation of a vector
$z^{'} = z e^{i θ}$
Q1MCQRotation
Multiplying $z$ by $e^{i\theta}$ rotates it by:
- A
  angle $\theta$
- B
  angle $-\theta$
- C
  $2\theta$
- D
  $0$
Distance between points
$∣ z_{1} - z_{2} ∣$
Equation of a circle
$∣ z - z_{0} ∣ = r$
Q1MCQCircle
The equation $|z-z_0|=r$ represents:
- A
  a circle of radius $r$ centred at $z_0$
- B
  a line
- C
  an ellipse
- D
  a point
Real and imaginary parts
$Re (z) = \frac{z + z ˉ}{2}, Im (z) = \frac{z - z ˉ}{2 i}$
Q1MCQReal part
$\text{Re}(z)$ equals:
- A
  $\dfrac{z+\bar z}{2}$
- B
  $\dfrac{z-\bar z}{2}$
- C
  $\dfrac{z-\bar z}{2i}$
- D
  $z\bar z$
Argument of a product
$ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}$
Q1MCQArgument of product
$\arg(z_1 z_2)$ equals:
- A
  $\arg z_1+\arg z_2$
- B
  $\arg z_1-\arg z_2$
- C
  $\arg z_1\cdot\arg z_2$
- D
  $0$

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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

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

Join QuestPix, Today!