Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Complex Analysis → Complex Numbers ↓


Complex Conjugates


Complex numbers are an essential part of modern mathematics, both theoretical and practical. They extend the concept of one-dimensional real numbers to the two-dimensional complex plane by introducing an imaginary component in the real part. Complex numbers are generally expressed as:

z = a + bi

Here, a is the real part and b is the imaginary part of the complex number z. The term i is an imaginary unit whose defining property is that i2 = -1.

Defining complex conjugates

The complex conjugate of a complex number is a simple but powerful concept. For a given complex number z = a + bi, the complex conjugate is written as:

z̅ = a - bi

Essentially, the complex conjugate is found by changing the sign of the imaginary part while keeping the real part unchanged. In the real number line analogy, this is like reflecting across the real part, thus lying in the same vertical line in the complex plane.

z = a + by z̅ = a - by again I am

Properties of complex conjugates

Complex conjugates have several important properties, which make them particularly useful in complex analysis and linear algebra. Some of these properties are as follows:

1. Conjugate of conjugate

Taking the conjugate of the conjugate gives the original number:

z = (z̅)̅

In words, if you take the conjugate of a complex number twice, you'll get back to the original complex number.

2. Addition and subtraction

The conjugate of the sum (or difference) of two complex numbers is equal to the sum (or difference) of their conjugates:

(z + w)̅ = z̅ + w̅
(z - w)̅ = z̅ - w̅

This property becomes simple when you substitute the definitions of complex conjugates.

3. Multiplication

The conjugate of the product of two complex numbers is equal to the product of their conjugates:

(zw)̅ = z̅w̅

Using the distributive property, you can verify this result by multiplying complex numbers and then taking the conjugate of the result.

4. Partition

The conjugate of the quotient of two complex numbers is equal to the quotient of their conjugates:

(z/w)̅ = z̅/w̅

To see this, both the numerator and denominator must be multiplied by the conjugate of the denominator, leading to simplification.

5. Module

The modulus of a complex number is invariant under conjugation:

|z̅| = |z|

In geometric terms, both z and z̅ are equidistant from the origin in the complex plane because they are reflections of each other on the real axis.

Applications of complex conjugates

The concept of complex conjugates is not just theoretical; it has practical applications in various fields such as engineering, physics, and computer science.

Easily computing division

Complex conjugates can simplify the division of complex numbers. Given two complex numbers:

z = a + bi
w = c + di

The division z/w can be simplified by multiplying the numerator and denominator by the conjugate of the denominator:

(a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)]

Simplifying it by multiplying it out gives the result in the standard form of a complex number. This is a typical solution technique in electrical engineering when calculating impedance in AC circuits.

Roots of polynomials

In polynomial equations with real coefficients, the conjugate root theorem states that if a complex number is a root, then its conjugate is also a root. For example, if z = a + bi is a root, then z̅ = a - bi must also be a root. This is important in areas such as signal processing and control systems.

Complex Fourier Transform

In digital signal processing, the Fourier transform often results in complex numbers. The complex conjugate is often used to calculate real-valued functions and energies. In particular, the symmetry properties of the Fourier coefficients about zero make extensive use of complex conjugates.

Geometrical interpretation

Geometrically, complex conjugation represents the movement of a point in the complex plane across the real axis. If the point (a, b) is represented by the complex number a + bi, then its conjugate a - bi will lie at the point (a, -b). This can be an intuitive way to look at operations with complex conjugates.

Conclusion

Complex conjugates are fundamental to the field of complex numbers. They provide an important tool in mathematical operations with complex numbers and have applications ranging from simplifying mathematical expressions to solving practical problems in engineering and physics. Understanding complex conjugates, their properties, and uses can greatly increase your proficiency in working with complex numbers.


Graduate → 7.1.2


U
username
0%
completed in Graduate

← Prev (7.1.1)
Polar Form
Next (7.1.3) →
Exponential Form

Comments 



Complex Conjugates

https://www.buddymath.com/g/7.1.2?bmal=en