1.3 Geometric representation of complex numbers

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

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• 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