What Is Complex Numbers?
Explore the fascinating world of complex numbers, extending the real number system into new dimensions. This comprehensive discussion covers core concepts, properties, operations, and applications in various scientific and engineering fields.
Sophia Greene
Dec 24, 2024 17 mins read
Complex numbers offer an intriguing extension to the real number system and have many applications across various disciplines of mathematics and science. Complex numbers can be thought of as expressions that contain both real and imaginary parts, typically written as the expression a+bi , where i represents the imaginary unit satisfying \( i^2 = -1 \). Complex numbers were first invented to solve equations that cannot be solved using real numbers alone, such as solving equations like \( x^2 +1=0 \) . Today, complex numbers remain an integral component of modern mathematical theory and applied sciences; their origin dates back to 16th-century Europe and they play an essential role in electrical engineering, fluid dynamics, quantum physics, signal processing as well as theoretical research applications.
Basic Definition of Complex Numbers
Core Concepts of Complex Numbers
Real Numbers and Imaginary Numbers
Real numbers represent quantities along a continuous line, such as integers, fractions and irrational numbers. An imaginary number differs in that when squared it results in negative square roots - this fundamental imaginary unit being denoted as "i", with its square root equalling 1. Complex numbers combine these two types of numbers together to represent quantities with both real and imaginary components.
Basic Components of Complex Numbers and Imaginary Unit (i)
Complex numbers consist of two main parts, the real part and imaginary part. They are usually represented as \( a+bi \) , where a is the real component and bi is imaginary part. An imaginary unit called i , which meets this condition, allows complex numbers to extend the real number system by providing solutions to equations with no real solutions, such as those related to gravity or harmonic oscillators.
Real component a of a complex plane represents points on its horizontal axis; imaginary component bi denotes points on its vertical axis. Together these components offer a comprehensive framework for describing quantities that have both magnitude and direction - applicable across scientific and engineering domains.
The imaginary unit, often denoted as "i," is an essential mathematical concept used to expand the real number system into complex numbers. It is defined by the property \( i^2 = -1 \) which has no real number solution and allows complex numbers such as those expressed using a+bi where both elements a and b are real numbers to exist. Furthermore, this unit makes square root equations much simpler, making it vital for fields like engineering, physics, and complex analysis.
As we all know, 0 is a real number and real numbers belong to complex numbers; therefore it can also be considered a complex number and represented as 0+0i .
Modulus and Absolute Value
The modulus (or absolute value) of a complex number is used as an indication of its magnitude in any direction in the complex plane.For a complex number \( z=a+bi \), the modulus is denoted as \[ \left| z \right| \] and is calculated using the formula:
\[ \left| z \right| = \sqrt{a^2 + b^2} \]
This calculation can be thought of as the Euclidean distance from the point (a,b)which represents z in the complex plane to its origin (0,0).
For example, consider the complex number \( z=3+4i \). The modulus of z is:
\[ \left| z \right| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
This indicates that \( z=3+4i \) to the origin is located five units in the complex plane.
The concept of modulus can be put to good use in various applications, from electrical engineering and complex analysis to electrical engineering and complex analysis. It provides a way to quantify the size of complex numbers as well as playing an essential role in polar representation (where complex numbers are expressed in terms of their modulus and argument (θ)).
Overall, modulus of a complex number is an invaluable concept that helps understand both its geometric and physical significance - thus being an asset in both theoretical and applied mathematics.
Complex Roots and Roots of Unity
Complex roots extend the concept of roots beyond real numbers. A complex root of any number is defined as the solution of an equation in the form \( z^n = k \) with both terms being complex numbers; root of unity are special cases of such complex roots as they satisfy equation \( z^n = 1 \) with all numbers from zero through to n-1 being complex numbers and are evenly distributed around the unit circle in complex plane.
Properties of Complex Numbers
Conjugate of a Complex Number、Complex Conjugate Root Theorem
Conjugating of complex numbers involves changing the sign of their imaginary part while maintaining its real part intact.For a complex number \( z=a+bi \), its conjugate is denoted as \( \overline{z} \) and is given by: \( \overline{z} = a - bi \)
The conjugate of a complex number reflects it across the real axis in the complex plane, providing a useful mathematical operation for solving polynomial equations with real coefficients and simplifying division of complex numbers.For example, if z =3+4i , then its conjugate is \( \overline{z}=3−4i \). Properties of the conjugate include \( | \overline{z}|=z \) and \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \), which are fundamental in complex number theory.
The Complex Conjugate Root Theorem asserts that, for any polynomial with real coefficients and a complex root \( a+bi \), its complex conjugate must also be a root. This ensures that non-real roots of real polynomials always occur in conjugate pairs in order to preserve its real character.
Reciprocal of a Complex Number
The reciprocal of a complex number \( z=a+bi \) is given by \( \frac{1}{z} = \frac{a - bi}{a^2 + b^2} \). This formula employs the conjugate to facilitate the division of complex numbers by standardizing the denominator.
Ordering of Complex Numbers
Complex numbers do not follow an orderly set of rules like real numbers; their structure lies on two dimensions of space rather than three, thus there being no definitive way of defining complex numbers as one after the other in any clear way.
Equality of Complex Numbers
Two complex numbers \( a+bi \) and \( c+di \) are considered equal if their real parts and imaginary parts are equal, i.e. if a=c and b=d respectively.
Others properties
Complex numbers possess many properties beyond those mentioned previously, including "all real numbers are complex numbers, yet not all complex numbers are real. All imaginary numbers are complex numbers but not all imaginary numbers are complex numbers." To accurately differentiate between the complex numbers and real numbers. It is therefore vitally important that one carefully distinguishes them.
Operations on Complex Numbers
Addition of Complex Numbers
Addition of complex numbers is performed by adding their respective real and imaginary parts. Given two complex numbers \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 + b_2 i \), their sum \( z = z_1 + z_2 \) is calculated as: \( z = (a_1 + a_2) + (b_1 + b_2)i \).
Closure Law
The sum of two complex numbers is still a complex number.If \( z_1 \) and \( z_2 \) are complex numbers, then \( z = z_1 + z_2 \) is also a complex number.
Commutative Law
Addition of complex numbers is commutative, meaning that adding two complex numbers together yields equal results: \( z_1 + z_2 = z_2 + z_1 \).
Associative Law
Addition of complex numbers is associative, so \( (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \) for any complex numbers \( z_1, z_2, z_3 \)
Identity Element
The identity element for addition is 0 (which can be written as 0+0i in complex form). For any complex number \( z=a+bi \): \(z+0=z\).
Inverse Element
Addition can be illustrated as the negation of complex numbers. For any complex number z=a+bi , the inverse is \( −z=−a−bi \). Thus: \( z+(−z)=0. \)
Subtraction Operation
Subtracting complex numbers can be accomplished in much the same way as adding them, by first dividing their real and imaginary components before subtracting.For \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 + b_2 i \), the difference \( z = z_1 - z_2 \) is: \(z = (a_1 - a_2) + (b_1 - b_2)i \)
This method ensures a straightforward subtraction operation by adhering to arithmetic rules applicable to real numbers.
Multiplication Operation
Multiplication of complex numbers involves more complexity due to their imaginary units. When multiplying two complex numbers \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 + b_2 i \) respectively, their product \( z = z_1 z_2 \) can be determined using the distributive property:
\[ z = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1)i \]
The key is to remember that i^2 = -1 when combining real and imaginary parts.
Division Operation
Dividing complex numbers requires the use of the conjugate. For \( z_1 = a_1 + b_1 i \) and \(z_2 = a_2 + b_2 i \), the quotient \( z = \frac{z_1}{z_2} \) is obtained by multiplying numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)} = \frac{1}{a_2^2 + b_2^2} \left( (a_1 a_2 + b_1 b_2) + (b_1 a_2 + a_1 b_2)i \right) \]
Identity for Complex Numbers
Complex numbers use zeros for addition, and ones for multiplication; therefore, any complex number such as z=a+bi can be written as: \( z+0=z \) and \( 1=z \), with its original value preserved. Here are six identities commonly encountered when working with complex numbers that frequent users should memorize:
\[ \begin{align*} (z_1 + z_2)^2 &= z_1^2 + z_2^2 + 2z_1z_2 \\ (z_1 + z_2)^2 &= z_1^2 + z_2^2 + 2z_1z_2 \\ z_1^2 - z_2^2 &= (z_1 + z_2)(z_1 - z_2) \\ (z_1 + z_2)^3 &= z_1^3 + 3z_1^2 z_2 + 3z_2^2 z_1 + z_2^3 \\ (z_1 - z_2)^3 &= z_1^3 - 3z_1^2 z_2 + 3z_2^2 z_1 - z_2^3 \\ (z_1 - z_2)^3 &= z_1^3 - 3z_1^2 z_2 + 3z_2^2 z_1 - z_2^3 \\ (z_1 + z_2 + z_3)^2 &= z_1^2 + z_2^2 + z_3^2 + 2z_1z_2 + 2z_1z_3 + 2z_2z_3 \end{align*} \]
Geometric Representation of Complex Numbers
What is the Complex Plane?
The Complex Plane, also referred to as Argand plane, is a two-dimensional plot where complex numbers can be represented graphically. The horizontal axis depicts real part while the vertical one illustrates imaginary part; for a complex number \( z =a+bi \) it would appear as point \( (a,b) \). This geometric representation allows users to visualize complex numbers and their operations more readily thereby improving comprehension and analysis.
Argument of a Complex Number
An argument of a complex number is used to measure the angle between its line of representation in complex lane and positive real axis of that number.It is denoted as \( \arg(z) \) or \( \theta \) and gives tphe direction of the complex number from the origin.
For a complex number z=a+bi, where a is the real part and b is the imaginary part, the argument \( \theta \) is the angle between the positive real axis and the line segment joining the origin and the point \( (a,b) \) in the complex plane.
The argument \( \theta \) can be calculated using the arctangent function:
\[ \theta = \tan^{-1} \left(\frac{b}{a}\right) \]
However, care must be taken to determine the correct quadrant of the angle since the arctangent function only returns values between \( -\frac{\pi}{2} \quad \text{and} \quad \frac{\pi}{2} \).
\[ \begin{cases} \theta = \tan^{-1} \left( \frac{b}{a} \right) & \text{if } a > 0 \text{ and } b \geq 0 \\ \theta = \tan^{-1} \left( \frac{b}{a} \right) + \pi & \text{if } a < 0 \\ \theta = \tan^{-1} \left( \frac{b}{a} \right) + \pi & \text{if } a < 0 \\ \theta = \tan^{-1} \left( \frac{b}{a} \right) + 2\pi & \text{if } a > 0 \text{ and } b < 0 \\ \theta = \frac{\pi}{2} & \text{if } a = 0 \text{ and } b > 0 \\ \theta = -\frac{\pi}{2} & \text{if } a = 0 \text{ and } b < 0 \end{cases} \]
Consider the complex number \( z = 1 + \sqrt{3}i \). To find the argument \( \theta \):
\[ \begin{align*} z &= 1 + \sqrt{3}i \\ |z| &= \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2 \\ \theta &= \tan^{-1} \left( \frac{\sqrt{3}}{1} \right) = \tan^{-1} (\sqrt{3}) = \frac{\pi}{3} \end{align*} \]
Polar Representation of Complex Numbers
Polar representation expresses complex numbers by their modulus and argument. Instead of writing complex numbers like "z" as an equation like so:
\( z = r (\cos \theta + i \sin \theta) \)，where \( r =∣z∣ \) is the modulus and \( \theta = \arg(z) \) is the argument. For example, \( z=1+i \) can be represented as \( \sqrt{2} \left( \cos 4\pi + i \sin 4\pi \right) \)
Application of Rotating Point Coordinates
Rotating points on the complex plane with complex numbers is straightforward due to their polar representation. A complex number such as \( z = e^{i\varphi} \). can be written as two polar forms: one representing its modulus (or magnitude) while the other represents its argument (or angle).
To rotate a complex number z by an angle \( \varphi \), multiply z by \( e^{i\varphi} \). This operation results in a new complex number \( z' = z \cdot e^{i\varphi} = r e^{i\theta} \cdot e^{i\varphi} = r e^{i(\theta + \varphi)} \)
The modulus remains constant at r, while the argument becomes \( (\theta + \varphi) \)，effectively rotating a point by ph radians in the complex plane.
Consider the complex number \( z=1+i \), which has a modulus \( r = \sqrt{2} \) and an argument \( \theta = 4\pi \). To rotate z by \( 2\pi \) radians:
\[ z' = \sqrt{2} e^{i4\pi} \cdot e^{i2\pi} = \sqrt{2} e^{i(4\pi + 2\pi)} = \sqrt{2} e^{i6\pi} \]
Converting back to rectangular form:
\[ z' = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) = \sqrt{2} \left( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -1 + i \]
Supplementary Knowledge of Complex Numbers
Euler's Formula
Euler's formula provides an incredible bridge between complex analysis and trigonometry, stated as \( :e^{i\theta} = \cos \theta + i \sin \theta \)
This elegant equation serves as the basis for the polar representation of complex numbers. If angle \( \theta = \pi \) is chosen as the variable value for this formula, \( e^{i\pi} + 1 = 0 \) appears, unifying fundamental constants from across mathematical fields into one identity value.
Euler's formula asserts that complex exponentiation with an imaginary exponent can be expressed using trigonometric functions, making it an invaluable tool for simplifying complex number expressions and has far-reaching applications in mathematics and engineering.
Signal Processing : Euler's formula can be used to represent sinusoidal signals as complex exponentials, simplifying analysis and manipulation in the frequency domain.Electrical Engineering : Euler's formula plays an essential role in the analysis of AC circuits, where voltage and current can be represented as phasors using Euler's formula. Quantum Mechanics : Euler's formula serves to describe wave functions and quantum states that often take the form of complex exponentials.
Control Systems : It assists in the design and analysis of control systems by simplifying representations of oscillatory behavior and feedback loops.
Rotations in the Complex Plane : Euler's formula can be used to perform rotations and transformations on complex planes, making it an indispensable tool in fields like computer graphics and robotics.
De Moivre's Theorem
De Moivre's theorem extends the concept of powers and roots to complex numbers. It states that for any integer \( n \) and a complex number \( z = re^{i\theta} \), the following holds:
\[ z^n = r^n e^{in\theta} = r^n (\cos n\theta + i\sin n\theta). \]
It simplifies computations involving powers and roots of complex numbers, making it essential for solving polynomial equations and analyzing periodic functions.
Furthermore, this theorem has applications in signal processing, electrical engineering, quantum mechanics to describe waveforms and oscillations as well as deriving trigonometric identities and simplifying multiple-angle formulas in mathematics.
Gaussian Integers
Gaussian integers are complex numbers with both real and imaginary components expressed as integers, typically expressed in the form a+bi where both components are integers. These numbers form an array-like grid on the complex plane and form part of the set of complex numbers; their unique factorization properties allow them to have many applications within number theory, such as solving polynomial equations with integer solutions or studying unique factorization properties within all complex numbers.
Mandelbrot Set
The Mandelbrot set is a collection of complex numbers that produces a fractal group when iterated through a particular quadratic equation \( z_{n+1} = z_n^2 + c \). This subset of the complex plane features an intricate boundary that exhibits self-similarity at various scales; thus making it ideal for study within chaos theory and fractal geometry.
Applications of Complex Numbers in Physics
Circuit Analysis
Complex numbers are widely used in electrical engineering to model impedance. Impedance represents resistance and reactance in AC circuits.The impedance \( Z \) is given as \( Z =R+iX \), where \( R \) is the resistance, and \( X \) the reactance, facilitating the analysis of circuit behavior.
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion in which an object's restoring force directly corresponds with its displacement from equilibrium, acting in the opposite direction from that displacement. This type of periodic movement is distinguished by its sinusoidal patterns both in displacement and velocity/acceleration over time.
In SHM, an object oscillates back and forth about an equilibrium position. The motion takes place in such a way that the object's acceleration a(t) is proportional to its displacement x (t) by a constant factor \( -\omega^2 \), where \( \omega \) is the angular frequency: \( a(t) = -\omega^2 x(t) \)
Given this, the displacement \( x(t) \) as a function of time t can be described by:
\( x(t) = A \cos(\omega t + \varphi) \)，where A is the amplitude of motion,\( \omega \) is the angular frequency, and \( \varphi \) is the phase constant.
SHM can be seen in small oscillations of a pendulum, where its restoring force is gravity acting along its arc; when mass attached to a spring is dislodged from equilibrium and released it also experiences SHM; furthermore vibrations within molecules often approximate simple harmonic motion which provides key insights into thermodynamics and quantum mechanics.
Quantum Mechanics
The Schrodinger equation, an essential building block of quantum mechanics, uses complex wave functions \( \psi(x, t) \) to describe the quantum state of particles. These wave functions represent probabilities while also showing complex behaviors influenced by potential fields - underscoring how fundamental complex numbers are in quantum systems.
Signal Processing
Signal processing entails analyzing, altering, and synthesizing signals to extract useful information or optimize transmission and storage. Signal manipulation employs mathematical techniques for manipulating audio, video, sensor data or any other form. As an example, in audio processing signal processing techniques are employed to enhance sound quality in music and speech, making them clearer and more enjoyable. Image processing techniques use digital cameras and smartphones to improve and compress pictures for easier storage, leading to higher-quality photographs with reduced storage requirements and bandwidth usage. Meanwhile, signal processing ensures efficient data transmission over networks while decreasing errors and bandwidth usage. Biomedical engineers use signal processing to analyze ECG or MRI signals for medical diagnostics, aiding physicians in patient care. Furthermore, radar and sonar systems employ signal processing to detect and interpret signals for navigation and military use - increasing accuracy and reliability while improving accuracy and reliabilty.
Complex Function
Complex functions are mathematical functions that convert complex numbers to complex numbers. More specifically, if z is a complex number such that its value equals \( x +yi \) where x and y are real numbers then its complex function \( f(z) \) can be expressed in terms of this formula: \( F(z) = u(x,y) + iv(x,y) \), where both functions of two real variables (u and v respectively) have real values.
Complex functions extend the concept of functions to the complex domain, offering more intricate structures and behaviors than real functions. They inherit many fundamental properties associated with complex numbers - including addition, multiplication and representation in the complex plane - making complex functions ideal for operations and transformations in various mathematical or applied contexts.
Differentiability is one of the hallmarks of complex functions. If a function \( f(z) \) meets the Cauchy-Riemann equations - two partial differential equations designed to ensure its holomorphicity at every point within its domain - then it is considered differentiable in the complex plane.
Complex functions with differentiable solutions everywhere within their domain are central to complex analysis, offering powerful properties like being represented as power series.
Related Examples
Calculations Involving the Imaginary Part "i" and Magnitude Calculation
To calculate the magnitude and imaginary parts of a complex number \( z=3+4i \), the modulus is:
\[ |z| = \sqrt{3^2 + 4^2} = 5 \]
The imaginary part is directly 4i, making these operations essential to understanding complex number properties and their geometric equivalents in the complex plane.
Addition, Subtraction, Multiplication, and Division Calculations
Addition
Problem: Add the complex numbers \( z_1 = 3 + 4i \quad \text{and} \quad z_2 = 1 + 2i \)
Solution:
Step 1: Identify the real and imaginary parts of each complex number.
\[ z_1 = 3 + 4i \quad \text{and} \quad z_2 = 1 + 2i \]
Step 2: Add the real parts. \( 3+1=4 \)
Step 3: Add the imaginary parts. \( 4i+2i=6i \)
Step 4: Combine the results. \( z_1 + z_2 = 4 + 6i \)
Answer: \( 4+6i \)
Subtraction
Problem: Subtract the complex number \( z_2 = 1 + 2i \quad \text{from} \quad z_1 = 3 + 4i \)
Solution:
Step 1: Identify the real and imaginary parts of each complex number.
\[ z_1 = 3 + 4i \quad \text{and} \quad z_2 = 1 + 2i \]
Step 2: Subtract the real parts. \( 3−1=2 \)
Step 3: Subtract the imaginary parts. \( 4i−2i=2i \)
Step 4: Combine the results. \( z_1 - z_2 = 2 + 2i \)
Answer: \( 2+2i \)
Multiplication
Problem: Multiply the complex numbers \( z_1 = 2 + 3i \quad \text{and} \quad z_2 = 1 + 4i \)
Solution:
Step 1: Use the distributive property (FOIL method) to expand \( (2+3i )(1+4i ) \).
\[ z_1 \cdot z_2 = 2 \cdot 1 + 2 \cdot 4i + 3i \cdot 1 + 3i \cdot 4i \]
Step 2: Perform the multiplication for each term.
\[2 \cdot 1 = 2\]
\[2 \cdot 4i = 8i\]
\[3i \cdot 1 = 3i\]
\[3i \cdot 4i = 12i^2\]
Step 3: Recall that \( i^2 = -1 \) and combine like terms. \( 2+8i+3i+12(−1)=2+11i−12 \)
Step 4: Simplify the expression. \( 2−12+11i=−10+11i \)
Answer: \( −10+11i \)
Division
Problem: Divide the complex number
\[z_1 = 2 + 3i\]
by
\[z_2 = 1 + 4i\]
Solution:
Step 1: Write the division as a fraction.
\[\frac{z_1}{z_2} = \frac{2 + 3i}{1 + 4i}\]
Step 2: Multiply the numerator and the denominator by the conjugate of the denominator.
\[\frac{2 + 3i}{1 + 4i} = \frac{(2 + 3i)(1 - 4i)}{(1 + 4i)(1 - 4i)}\]
Step 3: Use the distributive property to simplify. Numerator:
\[(2 + 3i)(1 - 4i) = 2 \cdot 1 - 2 \cdot 4i + 3i \cdot 1 - 3i \cdot 4i = 2 - 8i + 3i - 12i^2\]
\[2 - 8i + 3i + 12 = 14 - 5i\]
Denominator:
\[(1 + 4i)(1 - 4i) = 1 \cdot 1 - (4i)^2 = 1 - 16i^2 = 1 + 16 = 17\]
Step 4: Simplify the fraction.
\[\frac{14 - 5i}{17} = \frac{14}{17} - \frac{5i}{17}\]
Answer:
\[\frac{14}{17} - \frac{5i}{17}\]