Argand Diagram: Modulus and Argument

• The Argand diagram plots complex numbers
• It is a modified Cartesian plane representing complex numbers geometrically
• The horizontal axis shows the real part of the complex number, named the real axis
• The vertical axis shows the imaginary part of the complex number, named the imaginary axis
• A complex number z = a + bi is shown as a point (a, b)
• The position vector of the point (a, b) represents the complex number

Modulus

• The modulus of a complex number z = a + bi is written as |z|
• |z| represents the distance from the origin (0, 0) to the point (a, b)
• Calculated using the Pythagorean theorem: |z| = √(a² + b²)
• It is always a non-negative real number
• Geometrically, modulus represents magnitude or absolute value of the complex number
• Modulus is the length of the vector in the Argand diagram

Argument

• The argument of a complex number z = a + bi is written as arg(z)
• arg(z) represents the angle between the positive real axis and the line from the origin to the point (a, b)
• Can be measured in radians or degrees
• The principal argument, written Arg(z), lies in the interval (-π, π] or (-180°, 180°]
• Calculated using the arctangent function: arg(z) = tan⁻¹(b/a)
• Consider the quadrant of the complex number to find the correct argument:
  • If a > 0, arg(z) = tan⁻¹(b/a)
  • If a < 0 and b ≥ 0, arg(z) = tan⁻¹(b/a) + π
  • If a < 0 and b < 0, arg(z) = tan⁻¹(b/a) - π
  • If a = 0 and b > 0, arg(z) = π/2
  • If a = 0 and b < 0, arg(z) = -π/2
• Geometrically, the argument represents the direction of the vector representing the complex number

Complex Conjugate

• The complex conjugate of z = a + bi is written as z̄, defined as z̄ = a - bi
• To find it, change the sign of the imaginary part
• On the Argand diagram, it is a reflection of the original complex number across the real axis
• The modulus of a complex number and its conjugate are the same: |z| = |z̄|
• The argument of a complex conjugate is the negative of the original: arg(z̄) = -arg(z)
• z * z̄ = |z|²

Polar Form

• A complex number z = a + bi can be in polar form as z = r(cos θ + i sin θ)
• r is the modulus of z
• θ is the argument of z
• r = |z| = √(a² + b²)
• θ = arg(z)
• Polar form is used for multiplying and dividing complex numbers
• Euler's formula states e^(iθ) = cos θ + i sin θ
• Using Euler's formula, polar form can be written as z = re^(iθ)

Multiplication and Division in Polar Form

• If z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂):
  • z₁ * z₂ = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
  • z₁ / z₂ = (r₁/r₂)[cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
• When multiplying, multiply their moduli and add their arguments
• When dividing, divide their moduli and subtract their arguments

De Moivre's Theorem

• For any complex number z = r(cos θ + i sin θ) and any integer n:
  • zⁿ = rⁿ(cos(nθ) + i sin(nθ))
  • Using Euler's formula: (re^(iθ))ⁿ = rⁿe^(inθ)
• Useful for finding powers and roots of complex numbers

Roots of Complex Numbers

• To find the nth roots of a complex number z = r(cos θ + i sin θ):
  • z^(1/n) = r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k = 0, 1, 2, ..., n-1
  • Using Euler's formula: z^(1/n) = r^(1/n) * e^(i(θ + 2πk)/n)
• There are n distinct nth roots
• Roots are equally spaced around a circle with radius r^(1/n)

Geometric Transformations

• Multiplication by a complex number is a geometric transformation
• Multiplying by r(cos θ + i sin θ) scales by a factor of r and rotates by an angle of θ
• Conjugation is a reflection across the real axis
• Addition is a translation

Loci in the Argand Diagram

• A locus is a set of points satisfying a condition
• In the Argand diagram, loci are defined by equations using complex numbers
• |z - a| = r is a circle with center 'a' and radius 'r'
• arg(z - a) = θ is a half-line from point 'a' at an angle of θ
• |z - a| = |z - b| is the perpendicular bisector of the line segment joining 'a' and 'b'
• |z - a| + |z - b| = k (where k is a constant > |a - b|) is an ellipse with foci at 'a' and 'b'
• |z - a| - |z - b| = k (where k is a constant < |a - b|) is a hyperbola with foci at 'a' and 'b'