3.1 Complex numbers and their geometric interpretation

Complex numbers blend algebra and geometry, giving us a powerful tool for visualizing mathematical concepts. They're not just abstract symbols - they're points on a plane, with real and imaginary parts as coordinates.

This geometric interpretation opens up new ways to think about operations like addition and multiplication. It connects complex numbers to trigonometry, helping us solve problems in physics, engineering, and more.

Complex numbers in the plane

Representing complex numbers geometrically

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• A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit defined as the square root of -1
• The complex plane is a two-dimensional coordinate system where the horizontal axis represents the real part ($a$) and the vertical axis represents the imaginary part ($b$) of a complex number
• The real part of a complex number determines its position along the horizontal axis, while the imaginary part determines its position along the vertical axis
• Any point ($a$, $b$) in the complex plane can be uniquely associated with a complex number $a + bi$, and vice versa

Modulus and argument of complex numbers

• The modulus (or absolute value) of a complex number $a + bi$ is the distance from the origin to the point ($a$, $b$) in the complex plane, calculated using the Pythagorean theorem: $|a + bi| = \sqrt{a^2 + b^2}$
• The argument (or phase) of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point ($a$, $b$), typically expressed in radians or degrees
  • The argument can be calculated using the arctangent function: $\arg(a + bi) = \atan2(b, a)$, where $\atan2$ is the two-argument arctangent function that takes into account the signs of both $a$ and $b$ to determine the quadrant of the angle
• Complex numbers with the same modulus lie on a circle centered at the origin in the complex plane (unit circle for modulus 1), while complex numbers with the same argument lie on a ray emanating from the origin
• The complex conjugate of a complex number $a + bi$ is $a - bi$, which is the reflection of the point ($a$, $b$) across the real axis in the complex plane

Arithmetic operations with complex numbers

Addition and subtraction of complex numbers

• Addition of complex numbers can be visualized as vector addition in the complex plane, where the real and imaginary parts are added separately
  • To add two complex numbers geometrically, represent them as vectors in the complex plane and perform vector addition by placing the tail of one vector at the head of the other, then drawing the resultant vector from the origin to the new head
• Subtraction of complex numbers can be performed by adding the negative of the second complex number, which is equivalent to a 180-degree rotation of the vector representing the second number in the complex plane

Multiplication and division of complex numbers

• Multiplication of complex numbers can be interpreted as a rotation and scaling operation in the complex plane
  • Multiplying a complex number by $i$ results in a 90-degree counterclockwise rotation in the complex plane
  • Multiplying two complex numbers results in a vector with a magnitude equal to the product of the moduli and an angle equal to the sum of the arguments of the two numbers
• Division of complex numbers can be performed by multiplying the numerator and denominator by the complex conjugate of the denominator, which results in a real denominator
  • Geometrically, division can be interpreted as a combination of rotation and scaling operations in the complex plane, where the resulting vector has a magnitude equal to the ratio of the moduli and an angle equal to the difference of the arguments of the two numbers

Geometric properties of complex numbers

Polar form and Euler's formula

• The polar form of a complex number $z = a + bi$ is $z = r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument of the complex number
• Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, establishes a connection between complex numbers and trigonometric functions, allowing for the representation of trigonometric expressions using complex exponentials
  • This formula can be used to derive the addition and subtraction formulas for sine and cosine, as well as to solve trigonometric equations

De Moivre's formula and roots of unity

• De Moivre's formula, $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$, is a powerful tool for calculating powers and roots of complex numbers in trigonometric form
• Complex numbers can be used to find the roots of unity, which are complex numbers that satisfy the equation $z^n = 1$, where $n$ is a positive integer
  • The $n$ roots of unity are evenly spaced around the unit circle in the complex plane, with angles of $\frac{2\pi k}{n}$ for $k = 0, 1, \ldots, n-1$
  • These roots have important applications in various fields, such as signal processing (Fourier analysis) and quantum mechanics (quantum states)

Applications of complex numbers

Geometry and trigonometry

• Complex numbers can be used to represent and solve problems involving two-dimensional vectors, such as force vectors or displacement vectors
• The geometric interpretation of complex numbers can be applied to solve problems in analytic geometry, such as finding the equations of lines, circles, and other curves in the complex plane
  • For example, the equation of a circle with center $c$ and radius $r$ in the complex plane can be written as $|z - c| = r$, where $z$ is a complex variable

Physics and engineering

• Complex numbers are widely used in physics and engineering to model oscillations, waves, and other periodic phenomena
  • In electrical engineering, complex numbers are used to represent impedance, admittance, and other quantities in alternating current (AC) circuits
  • In quantum mechanics, complex numbers are essential for describing the wave function and probability amplitudes of quantum systems
• Fourier transforms, which decompose a signal into its frequency components, rely heavily on complex numbers and their geometric properties
  • The Fourier transform of a real-valued function $f(t)$ is a complex-valued function $F(\omega)$, where the real part represents the amplitude and the imaginary part represents the phase of each frequency component