Complex numbers come alive on the complex plane. This geometric representation lets us visualize these numbers as points or vectors, with the real part on the x-axis and the imaginary part on the y-axis.

The complex plane opens up a world of geometric interpretations. We can see addition as vector addition, multiplication as rotation and scaling, and even grasp concepts like modulus and argument visually. It's a powerful tool for understanding complex numbers.

Plotting complex numbers

The complex plane

• The complex plane represents complex numbers in two dimensions with the real part on the horizontal axis and the imaginary part on the vertical axis
• Each complex number a + bi corresponds to a unique point (a, b) on the complex plane
  • For example, the complex number 2 + 3i is represented by the point (2, 3) on the complex plane
• The real axis contains all real numbers, while the imaginary axis contains all purely imaginary numbers (numbers with a real part of 0)
  • The real number 5 is located at the point (5, 0) on the real axis
  • The imaginary number 4i is located at the point (0, 4) on the imaginary axis
• The complex conjugate a - bi of a complex number a + bi is the reflection of the point (a, b) across the real axis
  • For instance, the complex conjugate of 2 + 3i is 2 - 3i, which is the reflection of (2, 3) across the real axis

Representing complex numbers as vectors

• Complex numbers can be represented as vectors on the complex plane, with the tail at the origin and the head at the point (a, b)
• The vector representation provides a geometric interpretation of complex numbers
  • The vector from the origin to the point (2, 3) represents the complex number 2 + 3i
• Vector representation allows for a visual understanding of complex number operations like addition, subtraction, multiplication, and division

Geometric interpretation of addition and subtraction

Vector addition of complex numbers

• Adding complex numbers is equivalent to vector addition on the complex plane
• To add two complex numbers, place the tail of the second vector at the head of the first vector
  • The resulting vector from the origin to the head of the second vector represents the sum of the two complex numbers
• For example, to add (2 + 3i) and (1 + 2i), place the tail of the vector representing (1 + 2i) at the head of the vector representing (2 + 3i)
  • The resulting vector from the origin to the head of (1 + 2i) represents the sum (3 + 5i)

Subtracting complex numbers

• Subtracting complex numbers is the same as adding the first complex number to the negative of the second complex number
• The negative of a complex number is the vector with the same magnitude but opposite direction on the complex plane
  • The negative of (1 + 2i) is (-1 - 2i), which is the vector pointing in the opposite direction
• To subtract (1 + 2i) from (2 + 3i), add (2 + 3i) and (-1 - 2i) using vector addition
  • The resulting vector represents the difference (1 + i)

Modulus and argument of complex numbers

Modulus (absolute value)

• The modulus |z| of a complex number z = a + bi is the distance from the origin to the point (a, b) on the complex plane
• Calculate the modulus using the formula |z| = √(a^2 + b^2)
  • For example, the modulus of (3 + 4i) is |3 + 4i| = √(3^2 + 4^2) = 5
• The modulus represents the magnitude or length of the vector representing the complex number

Argument (phase)

• The argument arg(z) of a complex number z = a + bi is the angle θ between the positive real axis and the vector representing the complex number
• Calculate the argument using the formula θ = arctan(b/a), typically expressed in radians
  • For example, the argument of (1 + √3i) is arg(1 + √3i) = arctan(√3/1) = π/3 radians or 60 degrees
• The argument lies in the interval (-π, π] or [0, 2π)
• Express a complex number in polar form using its modulus and argument: z = r(cos(θ) + i⋅sin(θ)), where r is the modulus and θ is the argument
  • The complex number (3 + 4i) can be written in polar form as 5(cos(arctan(4/3)) + i⋅sin(arctan(4/3)))

Geometric effects of multiplication vs division

Multiplying complex numbers

• Multiplying complex numbers corresponds to rotation and scaling on the complex plane
• When multiplying z1 and z2, the modulus of the product is the product of the moduli: |z1⋅z2| = |z1|⋅|z2|
• The argument of the product is the sum of the arguments: arg(z1⋅z2) = arg(z1) + arg(z2)
  • Multiplying (2 + 2i) and (1 + i) yields (0 + 4i) because |2 + 2i|⋅|1 + i| = 2√2⋅√2 = 4 and arg(2 + 2i) + arg(1 + i) = π/4 + π/4 = π/2
• Geometrically, multiplying a complex number by another rotates the first number by the argument of the second and scales its modulus by the modulus of the second

Dividing complex numbers

• Dividing complex numbers corresponds to rotation and scaling in the opposite direction on the complex plane
• When dividing z1 by z2, the modulus of the quotient is the quotient of the moduli: |z1/z2| = |z1|/|z2|
• The argument of the quotient is the difference of the arguments: arg(z1/z2) = arg(z1) - arg(z2)
  • Dividing (4 + 4i) by (1 + i) yields (2 + 2i) because |4 + 4i|/|1 + i| = 4√2/√2 = 4 and arg(4 + 4i) - arg(1 + i) = π/4 - π/4 = 0
• Geometrically, dividing a complex number by another rotates the first number by the negative of the argument of the second and scales its modulus by the reciprocal of the modulus of the second