1.1 Complex number system

Complex numbers expand the real number system by introducing the imaginary unit i, where i^2 = -1. This extension allows for solutions to equations like x^2 + 1 = 0, previously unsolvable in the real number system.

The complex plane represents these numbers visually, with real and imaginary parts as coordinates. This representation enables geometric interpretations of complex arithmetic and lays the foundation for further study in complex analysis.

Definition of complex numbers

• Complex numbers are an extension of the real number system that includes the imaginary unit $i$, defined as $i^2 = -1$
• A complex number is represented in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit
• Complex numbers allow for the solution of equations that have no real solutions, such as $x^2 + 1 = 0$

Real and imaginary parts

• A complex number $z = a + bi$ consists of two parts: the real part $a$ and the imaginary part $b$
• The real part represents the horizontal component of the complex number on the complex plane
• The imaginary part represents the vertical component of the complex number on the complex plane
• The real and imaginary parts can be extracted from a complex number using the functions $\Re(z) = a$ and $\Im(z) = b$

Imaginary unit i

• The imaginary unit $i$ is defined as the square root of -1, i.e., $i^2 = -1$
• $i$ is a fundamental concept in complex analysis and allows for the extension of the real number system
• Powers of $i$ follow a cyclic pattern: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$
• The imaginary unit enables the representation of complex numbers on a two-dimensional plane

Complex plane representation

• Complex numbers can be represented on a two-dimensional plane called the complex plane or Argand plane

Argand diagram

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• The Argand diagram is a graphical representation of complex numbers on the complex plane
• The horizontal axis represents the real part, and the vertical axis represents the imaginary part
• A complex number $z = a + bi$ is plotted as a point $(a, b)$ on the Argand diagram

Real axis and imaginary axis

• The real axis is the horizontal line on the complex plane, representing real numbers (imaginary part = 0)
• The imaginary axis is the vertical line on the complex plane, representing imaginary numbers (real part = 0)
• The real and imaginary axes intersect at the origin (0, 0), which represents the complex number 0

Algebraic operations on complex numbers

• Complex numbers can be manipulated using algebraic operations similar to those used with real numbers

Addition and subtraction

• To add or subtract complex numbers, add or subtract their real and imaginary parts separately
• $(a + bi) + (c + di) = (a + c) + (b + d)i$
• $(a + bi) - (c + di) = (a - c) + (b - d)i$

Multiplication

• To multiply complex numbers, use the distributive property and the fact that $i^2 = -1$
• $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Division

• To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator
• $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Complex conjugates

• The complex conjugate of $z = a + bi$ is defined as $\bar{z} = a - bi$
• The product of a complex number and its conjugate is always a real number: $(a + bi)(a - bi) = a^2 + b^2$
• Complex conjugates are used in division and other operations to simplify complex expressions

Polar form of complex numbers

• Complex numbers can also be represented in polar form, which expresses the number in terms of its magnitude (modulus) and angle (argument)

Modulus and argument

• The modulus (or absolute value) of a complex number $z = a + bi$ is the distance from the origin to the point $(a, b)$ on the complex plane, given by $|z| = \sqrt{a^2 + b^2}$
• The argument (or phase) of a complex number is the angle between the positive real axis and the line connecting the origin to the point $(a, b)$, given by $\arg(z) = \arctan(\frac{b}{a})$

Euler's formula

• Euler's formula establishes a relationship between complex numbers and trigonometric functions: $e^{i\theta} = \cos\theta + i\sin\theta$
• This formula allows for the conversion between rectangular and polar forms of complex numbers
• Euler's formula is a fundamental result in complex analysis and has numerous applications in mathematics and physics

Trigonometric form

• The trigonometric form of a complex number is $z = r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument
• This form directly relates the rectangular and polar representations of complex numbers
• The trigonometric form is useful for performing operations such as multiplication and division of complex numbers

Exponential form

• The exponential form of a complex number is $z = re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument
• This form is derived from Euler's formula and provides a compact representation of complex numbers
• The exponential form simplifies certain calculations, such as raising complex numbers to powers and finding roots

Properties of complex numbers

• Complex numbers satisfy various algebraic properties that make them a well-behaved number system

Commutative, associative, and distributive laws

• Complex numbers obey the commutative law for addition and multiplication: $a + b = b + a$ and $ab = ba$
• Complex numbers obey the associative law for addition and multiplication: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$
• Complex numbers obey the distributive law: $a(b + c) = ab + ac$

Identity and inverse elements

• The additive identity for complex numbers is 0: $a + 0 = a$ for any complex number $a$
• The multiplicative identity for complex numbers is 1: $a \cdot 1 = a$ for any complex number $a$
• Every complex number $a$ has an additive inverse $-a$ such that $a + (-a) = 0$
• Every non-zero complex number $a$ has a multiplicative inverse $\frac{1}{a}$ such that $a \cdot \frac{1}{a} = 1$

Field properties

• The complex number system forms a field, which is an algebraic structure that satisfies certain properties
• A field is a set with two binary operations (addition and multiplication) that satisfy the commutative, associative, and distributive laws, and has identity and inverse elements
• The field properties ensure that complex numbers behave consistently under algebraic manipulations

Geometric interpretation of complex numbers

• Complex numbers have a rich geometric interpretation that provides insights into their properties and applications

Rotation and scaling in complex plane

• Multiplying a complex number by $e^{i\theta}$ results in a rotation by an angle $\theta$ in the complex plane
• Multiplying a complex number by a real number $r$ results in scaling the modulus by a factor of $r$ without changing the argument
• The combination of rotation and scaling through multiplication allows for geometric transformations in the complex plane

Geometric meaning of addition and multiplication

• Adding two complex numbers corresponds to vector addition in the complex plane, where the resulting complex number is the sum of the component vectors
• Multiplying two complex numbers $z_1 = r_1e^{i\theta_1}$ and $z_2 = r_2e^{i\theta_2}$ results in a complex number with modulus $r_1r_2$ and argument $\theta_1 + \theta_2$, which can be interpreted as a rotation and scaling of $z_1$ by $z_2$
• The geometric interpretation of complex arithmetic provides a visual understanding of the properties and behavior of complex numbers

Applications of complex number system

• The complex number system has numerous applications in various fields of mathematics, science, and engineering

Roots of polynomial equations

• The fundamental theorem of algebra states that every non-constant polynomial equation has at least one complex root
• Complex numbers allow for the solution of polynomial equations that may not have real roots, such as $x^2 + 1 = 0$
• Finding the roots of polynomials is essential in many areas, including algebra, calculus, and numerical analysis

Electrical engineering and impedance

• In electrical engineering, complex numbers are used to represent sinusoidal signals and analyze AC circuits
• Impedance, which is the opposition to the flow of alternating current, is represented as a complex number $Z = R + iX$, where $R$ is resistance and $X$ is reactance
• Complex analysis techniques are used to study the behavior of electrical systems and design circuits

Quantum mechanics and wave functions

• In quantum mechanics, the state of a quantum system is described by a complex-valued wave function $\Psi(x, t)$
• The wave function encodes the probability amplitude of the system, and its squared modulus $|\Psi(x, t)|^2$ represents the probability density
• Complex numbers are essential for describing the superposition and interference of quantum states, which are fundamental concepts in quantum mechanics