PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

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Algebra


Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It is a vast and foundational field of mathematics that is based on basic arithmetic operations, such as addition, subtraction, multiplication, and division, but extends them to solve more complex problems. In algebra, we often deal with unknown values and we solve them using various techniques and rules.

Basics of algebra

Algebra introduces the concept of variables, which are symbols (usually letters such as x, y, or z) that represent unknown numbers. For example, in the equation:

x + 5 = 10

We can solve for x and find that x = 5 The idea is to find the value of x that makes the equation true.

Operations in algebra

Just like arithmetic with numbers, variables in algebra can be added, subtracted, multiplied, and divided. Consider these simple operations:

  • a + b = c
  • a - b = c
  • a times b = c
  • a div b = c (where b neq 0)

These operations follow guiding rules such as the commutative property, associative property, and distributive property, which make algebraic manipulation possible.

Equations and expressions

An important part of algebra is understanding the difference between expressions and equations.

An expression is a combination of variables, numbers and operations. For example, 3x + 4 is an expression. It does not represent equality and has no 'solution'.

On the other hand, an equation states that two expressions are equivalent, which means they have the = sign. For example, 3x + 4 = 10 is an equation. We can solve it to find out that x = 2.

Solving linear equations

Linear equations are the simplest type of algebraic equations. They look like ax + b = c. Here's a step-by-step example:

2x + 3 = 7

To solve this equation, follow these steps:

  1. Subtract 3 from both sides to simplify: 
     2x + 3 - 3 = 7 - 3
     2x = 4
  2. Divide both sides by 2 to find the value of x: 
     x = 4 / 2
     x = 2

Therefore, x = 2 is a solution to the equation 2x + 3 = 7.

Illustrating algebra with examples

Imagine a balance scale, where each side must be equal to keep the scale balanced. Solving an algebraic equation is just like keeping the scale balanced. Let's take an example:

X 5 x + 5 = 10

In this example, a variable x and a number 5 both weigh as much as 10 units. Solving for x maintains balance, resulting in x = 5.

Polynomials and factorization

Polynomials are another important topic in algebra. A polynomial is an expression of more than two algebraic terms, specifically a sum of several terms that contain different powers of the same variable(s). The standard form of a polynomial is:

a_n x^n + a_(n-1) x^(n-1) + ... + a_2 x^2 + a_1 x + a_0

where n is a nonnegative integer, and the coefficients a_n, a_(n-1), ..., a_0 are constants.

Factoring polynomials involves expressing the polynomial as a product of its factors. For example, to factor x^2 + 5x + 6:

Find two numbers that when multiplied give 6 (the constant term) and when added give 5 (the coefficient of x term). Thus, we have 2 and 3. Therefore, the expression can be broken down into:

(x + 2)(x + 3)

Quadratic equations

Quadratic equations are algebraic equations of this form:

ax^2 + bx + c = 0

These equations are called quadratic because they include terms up to x^2 (the second power). There are several ways to solve quadratic equations, including:

Factoring method

If a quadratic expression can be factored, it can be solved by setting each factor to zero. For example:

4^2 - 3x - 4 = 0

This can be attributed to the following factors:

(x – 4)(x + 1) = 0

Setting each factor to 0 gives the solutions: x = 4 and x = -1.

Quadratic formula

When a quadratic cannot be easily factored, the quadratic formula can be used:

x = (-b ± √(b² - 4ac)) / (2a)

For example, solving:

2x^2 - 4x - 3 = 0

Use of Quadratic Formula:

x = (4 ± √((4)^2 - 4(2)(-3))) / (2(2))

This gives the solutions: x = 2.5 and x = -0.5.

Algebraic inequalities

Inequalities are similar to equations, but instead of an equals sign (=), they use inequality symbols (>, <, ≥, ≤). Solving them involves the same operations as equations, but those operations affect the direction of the inequality sign.

For example: solve the inequality:

3x + 4 > 10

Subtract 4 from both sides:

3x > 6

Divide both sides by 3:

x > 2

So the solution shows that x must be a number greater than 2. We can express this solution on the number line or with the interval notation, (2,∞).

Algebra in real life

Algebra is widely used in various real-life scenarios, such as calculating distances, predicting profits, and solving logistics problems. A common example of this is determining the cost of goods.

Let's say you are buying x candies at $0.50 per candy and y chocolates at $1.00 per candy, and you want to spend a total of $10. We can represent this with an equation:

0.50x + 1.00y = 10

Let's say you decide to buy 8 chocolates. Substitute y = 8 into the equation:

0.50x + 1.00(8) = 10

Simplify and solve:

0.50x + 8 = 10
0.50x = 2
x = 4

You will buy 4 candies and 8 chocolates to stay within your budget.

Summary

Algebra is a vital tool in understanding mathematical concepts and solving a variety of practical problems. It is based on arithmetic and opens the door to areas of advanced mathematics, enabling individuals to work with unknowns, develop equations, solve problems, and predict outcomes efficiently.

Through equations, inequalities, polynomials, and more, algebra helps us understand patterns and relationships in the world around us.


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