Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Complex Analysis ↓


Complex Numbers


Understanding complex numbers in mathematics may seem a bit difficult at first glance. But a simple approach and visual aids make this interesting topic easier to understand.

What are complex numbers?

Complex numbers are numbers that include the square root of negative one, i, which is called the imaginary unit. A complex number is expressed as:

a + bi

In this expression, a and b are real numbers, while i is the imaginary unit, which satisfies:

i² = -1

Here, a is the real part of the complex number, and b is the imaginary part. For example, in the complex number 3 + 4i, 3 is the real part and 4 is the imaginary part.

Visualization of complex numbers

Complex numbers can be represented visually on the complex plane. The complex plane is similar to the Cartesian coordinate system, but has a real axis and an imaginary axis. Here's how complex numbers are plotted:

RealImaginary(3, 4i)

The horizontal axis (x-axis) represents the real part, while the vertical axis (y-axis) represents the imaginary part of the complex number. The point (3, 4i) corresponds to the complex number 3 + 4i. The distance from the origin to this point is called the modulus of the complex number.

Operations with complex numbers

Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.

Addition and subtraction

Addition and subtraction of complex numbers is simple. You add or subtract the real parts and imaginary parts separately. Here are some examples:

(3 + 4i) + (1 + 2i) = (3 + 1) + (4 + 2)i = 4 + 6i
(5 + 3i) - (2 + 1i) = (5 - 2) + (3 - 1)i = 3 + 2i

Multiplication

Multiplying complex numbers involves using the distributive property and remembering that i² = -1. For example:

(3 + 4i) * (2 + i) = 3*2 + 3*i + 4i*2 + 4i*i
= 6 + 3i + 8i + 4(-1)
= 6 + 11i - 4
= 2 + 11i

Division

To divide, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a + bi is a - bi.

(3 + 4i) / (1 + 2i)

Multiply the numerator and denominator by the conjugate of the denominator:

(3 + 4i) * (1 - 2i) / ((1 + 2i) * (1 - 2i))

Results obtained from calculations:

= (3*1 - 3*2i + 4i*1 - 4i*2i) / (1*1 - (2i)^2)
= (3 - 6i + 4i + 8) / (1 + 4)
= (11 - 2i) / 5
= 11/5 - (2/5)i

Modulus and argument of a complex number

Every complex number has a length called the modulus or absolute value. This is the distance from the origin to a point on the complex plane. The modulus of a + bi is given by:

|a + bi| = √(a² + b²)

The modulus for the complex number 3 + 4i is:

|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5

The argument of a complex number, often denoted as arg(z), is the angle it makes with the positive real axis. It can be calculated using the arctangent function:

arg(a + bi) = arctan(b/a)

The argument is usually measured in radians.

Complex conjugates

The complex conjugate of a complex number a + bi is a - bi. It is found by changing the sign of the imaginary part. Complex conjugates have special properties, for example:

(a + bi) * (a - bi) = a² + b²

Polar form of complex numbers

Complex numbers can also be represented in polar form, which is particularly useful for multiplication and division. The polar form is expressed as:

r(cos θ + i sin θ)

where r is the modulus, and θ is the argument of the complex number. This is often written using Euler's formula as follows:

re^(iθ)

Example calculation in polar form

Let's consider an example with the complex number 3 + 4i. First, we find the modulus and argument:

  • Modulus: r = √(3² + 4²) = 5
  • Reason: θ = arctan(4/3)

The polar form is:

5(cos(arctan(4/3)) + i sin(arctan(4/3)))

Using Euler's form:

5e^(i*arctan(4/3))

Why are complex numbers important?

Complex numbers are widely used in mathematics, engineering, and physics. They provide solutions to polynomial equations that do not have real number solutions and are essential in understanding oscillations, waves, and much more.

Conclusion

Complex numbers extend real numbers and open up new possibilities in mathematics by allowing us to work with negative square roots. By understanding their basic operations and properties, such as modulus, argument, and polar form, you will have a strong foundation for further applications in science and engineering.

This was a simple introduction to complex numbers, focusing on their basic properties and operations. As you further your studies, these concepts will be important in understanding more advanced topics in mathematics and related fields.


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