Complex Numbers - Comprehensive Notes (JEE Style)

Introduction to Complex Numbers

Complex number definition: z is a complex number if it can be written as
z = a + ib, where a,b \in \mathbb{R} and the imaginary unit satisfies i^2 = -1.
Real and imaginary parts: \Re(z) = a, \quad \Im(z) = b. A complex number can also be viewed as the ordered pair z \equiv (a,b) in the plane.
Notation: conjugate of z is denoted by \overline{z} = a - ib. The real part and imaginary part can be recovered by
\Re(z) = \frac{z + \overline{z}}{2}, \qquad \Im(z) = \frac{z - \overline{z}}{2i}.
Importance: Extension of real numbers to include imaginary unit allows solving equations that have no real solutions (e.g., x^2 = -1).
Algebraic operations with complex numbers
- Addition: (a+ib) + (c+id) = (a+c) + i(b+d).
- Subtraction: (a+ib) - (c+id) = (a-c) + i(b-d).
- Multiplication: (a+ib)(c+id) = (ac - bd) + i(ad + bc).
- Division: \frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{(c+id)(c-id)} = \frac{(a+ib)(c-id)}{c^2 + d^2}.
- Equality: two complex numbers are equal iff their real parts and imaginary parts are equal, i.e., a=c\text{ and }b=d.
Example-style illustrations (conceptual): Complex numbers enable straightforward manipulation via real and imaginary components, and via conjugation to rationalize denominators in division.

Representation in the Argand Plane

Argand plane: complex numbers correspond to points in the plane with horizontal real axis and vertical imaginary axis.
The complex plane is denoted as the Argand plane. A point P = (a,b) represents z = a + ib with coordinates (a,b).
The origin O corresponds to z = 0 (which is both real and purely imaginary).
Distance from origin (modulus) will be defined next; the geometric view underpins modulus, argument, and polar form.

Modulus of a Complex Number

For z = a + ib, the modulus is
|z| = \sqrt{a^2 + b^2}.
Interpretation: the distance of the point z from the origin in the Argand plane.
Key properties
- Non-negativity: |z| \ge 0, with equality iff z=0.
- Multiplicativity: |zw| = |z|\,|w|.
- Subadditivity (triangle inequality): |z + w| \le |z| + |w|.
- For real numbers, modulus behaves like the usual absolute value: if z = x \in \mathbb{R} then |x| = |x|.
Unimodular numbers
- A complex number is unimodular if |z| = 1. Such numbers can be written as
  z = \cos\theta + i\sin\theta = e^{i\theta}
  for some real \theta (modulus 1).
Relationship to polar form: modulus is the radial coordinate in polar representation; the direction is given by the argument (angle).

Argument of a Complex Number

The argument of a non-zero complex number z is the angle θ formed with the positive real axis, defined by
z = |z|\, (\cos\theta + i\sin\theta).
Principal value: called the principal argument, usually denoted as \text{Arg}(z) and lies in (-\pi, \pi] (context can require otherwise).
Quadrant-based intuition: depending on which quadrant z lies in, θ takes an appropriate value in the corresponding interval.
Polar (or exponential) form (summary):
z = r(\cos\theta + i\sin\theta) = r e^{i\theta}, \quad r = |z| \ge 0, \theta = \text{Arg}(z) \in \mathbb{R}.
Important properties of arguments
- Arg identities for multiplication and division (up to addition/subtraction of multiples of 2\pi).
- Conjugation symmetry: \text{Arg}(\overline{z}) = -\text{Arg}(z).
Relationship with real and imaginary parts
- If z = x + iy, then x = Re(z) and y = Im(z); the angle θ satisfies \tan\theta = \frac{y}{x} in the appropriate quadrant.

Polar Representation and Euler’s Formula

Trigonometric (polar) form: for z with modulus r and argument θ,
z = r\, (\cos\theta + i\sin\theta).
Euler’s formula: e^{i\theta} = \cos\theta + i\sin\theta.
Therefore, polar representation can be written as
z = r\, e^{i\theta}.
Applications
- Foundational basis for De Moivre’s theorem and roots of unity.

De Moivre’s Theorem and Roots of Unity

De Moivre’s theorem: for integer n,
(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta).
For a complex number in polar form, this yields
z^n = r^n (\cos(n\theta) + i\sin(n\theta)) = r^n e^{i n \theta}.
Roots of unity (example): cube roots of unity are the solutions to z^3 = 1 given by
z_k = e^{2\pi i k/3} = \cos\left(\frac{2\pi k}{3}\right) + i\sin\left(\frac{2\pi k}{3}\right), \quad k = 0,1,2.
Hence the set is {1, \omega, \omega^2} with (\omega = -\tfrac{1}{2} + i\tfrac{\sqrt{3}}{2}).
Real and imaginary components of powers: useful in evaluating powers and roots; concept extends to general z^n via polar form.

Conjugation and Algebraic Identities

Conjugate of z: \overline{z} = a - ib.
Geometric interpretation: reflection of z across the real axis.
Basic identities
- z + \overline{z} = 2\Re(z)
- z - \overline{z} = 2i\Im(z)
- z\overline{z} = |z|^2
- \Re(z) = \frac{z + \overline{z}}{2}, \quad \Im(z) = \frac{z - \overline{z}}{2i}.
Conjugation across sums and products
- \overline{(z1 + z2)} = \overline{z1} + \overline{z2}, \qquad \overline{(z1 z2)} = \overline{z1}\ \overline{z2}.
Useful corollaries
- If z is unimodular, then \overline{z} = 1/z. (since |z|=1 implies z\overline{z}=1.)

Representation and Geometry: Lines, Circles, and Loci in the Argand Plane

Cartesian viewpoint: write z = x + iy with x = Re(z) and y = Im(z).
Line through two points z1 and z2
- The ordinary slope (complex slope) is
  m = \frac{\Im(z2) - \Im(z1)}{\Re(z2) - \Re(z1)}.
- A point z lies on the line through z1 and z2 iff the ratio
  \frac{z - z1}{z - z2}
  is a real number. This is a commonly used complex-plane criterion for collinearity.
General straight line form in complex notation
- In Cartesian terms, the line is y = mx + c with z = x + iy.
- In complex form, lines can be represented by appropriate linear relations among z and \overline{z} (e.g., α z + \overline{α} \overline{z} + β = 0 with α ∈ C, β ∈ R).
Circle in the complex plane
- The locus of points z such that |z - z0| = R is a circle with center z0 and radius R.
- General circle equation in complex form can be written via equations in z and \overline{z}.
Curves of common loci
- Circle: |z - z0| = R.
- Line: (z - z1)/(z - z2) ∈ R (or equivalently a linear relation in z and \overline{z}).
- Concyclicity: four points z1, z2, z3, z4 lie on a common circle if a certain cross-ratio or determinant condition is satisfied (standard tests exist in complex geometry).
Geometric interpretations of modulus and argument
- The modulus gives radial distance; the argument gives angular position about the origin.
- Polar coordinates link to Cartesian via
  z = x + iy = |z|\, (\cos\theta + i\sin\theta),
  with x = |z|\cos\theta, \quad y = |z|\sin\theta.

Complex Slope, Parallelism, and Perpendicularity (Summary)

Two lines with slopes m1 and m2 are parallel iff m1 = m2.
They are perpendicular iff m1 m2 = -1.
Real slope of a line in the Argand plane corresponds to the standard cartesian slope when the line is viewed as a locus in the plane.
Perpendicular distance from a point z to a line AB is a standard geometry formula (adapted to complex form via coordinates x,y).

Algebra of Complex Numbers: Quick Reference

Basic products and conjugates
- For z = a + ib and w = c + id,
- Sum: z + w = (a+c) + i(b+d).
- Difference: z - w = (a-c) + i(b-d).
- Product: zw = (ac - bd) + i(ad + bc).
- Quotient: \frac{z}{w} = \frac{z \overline{w}}{|w|^2}.
Conjugation properties
- \overline{z+w} = \overline{z} + \overline{w}, \quad \overline{zw} = \overline{z} \ \overline{w}.
- Re(z) and Im(z) as above; useful identities: Re(z) = (z+\overline{z})/2, Im(z) = (z-\overline{z})/(2i).
Polar form and conjugation
- If z = r(\cos\theta + i\sin\theta) , then
  \overline{z} = r(\cos\theta - i\sin\theta) = r e^{-i\theta}. .
Unimodular and representation on the unit circle
- If |z| = 1, then z = e^{i\theta} = \cos\theta + i\sin\theta for some real \theta.
Logarithm of a complex number (brief):
- For z = re^{i\theta} with r > 0 , a branch of the logarithm is
  \log z = \ln r + i\theta,
  where \theta = \text{Arg}(z) (principal value may be chosen).

Applications and Examples (Illustrative Problems)

Example 1: Convert to polar form
- Given z = 3 + 4i, compute modulus and argument:
- |z| = \sqrt{3^2 + 4^2} = 5.
- \theta = \arctan\left(\frac{4}{3}\right) lying in the first quadrant.
- Polar form: z = 5\, (\cos\theta + i\sin\theta) = 5 e^{i\theta}.
Example 2: De Moivre application
- Compute (\cos\tfrac{\pi}{4} + i\sin\tfrac{\pi}{4})^3
- Using De Moivre: \cos\tfrac{3\pi}{4} + i\sin\tfrac{3\pi}{4}.
Example 3: Roots of unity and conjugates
- The three cube roots of unity are 1, \omega, \omega^2 with \omega = e^{2\pi i/3} = -\tfrac{1}{2} + i\tfrac{\sqrt{3}}{2}.
Example 4: Conjugate identities
- If z = a + ib, then z \overline{z} = a^2 + b^2 = |z|^2.
Example 5: Internal division (section formula in the plane)
- If a point P divides AB internally in the ratio m:n, then in Cartesian coordinates
  P = \left( \frac{m xB + n xA}{m+n}, \frac{m yB + n yA}{m+n} \right).
Example 6: Complex slope and line through two points
- Points z1 = x1 + i y1 and z2 = x2 + i y2 determine slope
  m = \frac{y2 - y1}{x2 - x1},
  and the line is described by y - y1 = m (x - x1) or equivalently by the complex condition
  \frac{z - z1}{z - z2} \in \mathbb{R}.

Logarithms, Conics, and Advanced Loci (Overview)

Logarithm of a complex number connects modulus and argument: if z = r e^{i\theta}, then
\log z = \ln r + i\theta.
Loci in the complex plane include circles, lines, parabolas, ellipses, and hyperbolas depending on distance relations and sums/differences of distances to fixed points.
Concyclicity: four points are concyclic if there exists a circle passing through all four; in the complex plane, several algebraic tests exist (e.g., cross ratios real, determinant criteria).

Practice and Practice-Set Themes (reflected in the transcript)

Common problem types appearing in JEE Main style practice include:
- Modulus and argument manipulations, polar forms, and De Moivre’s theorem.
- Operations with conjugates and their geometric interpretations.
- Root of unity identities and their algebraic consequences.
- Straight line and circle equations in the Argand plane, including locus problems.
- Concyclicity and locus problems involving multiple complex numbers.
- Several questions on mapping geometric shapes to algebraic equations in z and \overline{z}.
Note: A large set of Multiple Choice Questions (MCQs) with an answer key is provided to reinforce these concepts; the key includes numerous numeric results and symbolic relations (e.g., results for minimum/maximum values, Arg values, etc.). This notes pack emphasizes understanding the concepts, with practice questions serving to test application.

Quick Reference: Key Formulas to Remember

Complex number and parts
- z = a + ib, \quad a = \Re(z), \quad b = \Im(z).
- \Re(z) = \frac{z + \overline{z}}{2}, \quad \Im(z) = \frac{z - \overline{z}}{2i}.
Modulus and conjugate
- |z| = \sqrt{a^2 + b^2}, \quad z\overline{z} = |z|^2.
- \overline{z} = a - ib.
Multiplication and division
- (a+ib)(c+id) = (ac - bd) + i(ad + bc).
- \frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{c^2 + d^2}.
Polar form and De Moivre
- z = r(\cos\theta + i\sin\theta) = r e^{i\theta}, \quad r \ge 0.
- e^{i\theta} = \cos\theta + i\sin\theta.
- De Moivre: (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta).
Argument and principal value
- z = |z| (\cos\theta + i\sin\theta) \Rightarrow \text{Arg}(z) = \theta.
Conjugate and geometric interpretation
- Conjugation reflects across the real axis; \overline{z} has the same real part and opposite imaginary part.
Roots of unity (example)
- 1, \omega, \omega^2 \quad (\omega = e^{2\pi i / 3}).
Loci and geometry in the plane
- Circle: |z - z_0| = R.
- Line: condition involving z and \overline{z} or the ratio (z - z1)/(z - z2) being real.
Note on exam practice material
- A substantial set of MCQs with answers covers the breadth of complex-number geometry: modulus, arguments, conjugates, De Moivre, roots of unity, perpendicularity/parallelism of lines, loci of points (circles, lines, ellipses, hyperbolas), and concyclicity. Use the formulas above to set up and solve these problems efficiently.