Algebra Basics

Algebra

• Variables and Expressions:
  • A variable is a symbol that represents a value that can change.
  • An algebraic expression is a combination of variables, constants, and mathematical operations.
• Equations and Inequalities:
  • An equation is a statement that two expressions are equal.
  • An inequality is a statement that one expression is greater than, less than, or equal to another.
  • Solving equations and inequalities involves finding the value(s) of the variable(s) that make the statement true.
• Linear Equations:
  • A linear equation is an equation in which the highest power of the variable(s) is 1.
  • Linear equations can be solved using addition, subtraction, multiplication, and division.
• Quadratic Equations:
  • A quadratic equation is an equation in which the highest power of the variable(s) is 2.
  • Quadratic equations can be solved using factoring, quadratic formula, or completing the square.

Combinatorics

• Permutations:
  • A permutation is an arrangement of objects in a specific order.
  • The number of permutations of n objects is n! (n factorial).
• Combinations:
  • A combination is a selection of objects without regard to order.
  • The number of combinations of n objects taken k at a time is C(n, k) = n! / (k!(n-k)!).
• Pascal's Triangle:
  • A triangular array of numbers in which each number is the sum of the two numbers directly above it.
  • Used to calculate binomial coefficients and combinations.

Probability

• Basic Concepts:
  • An experiment is an action or situation that can produce a set of outcomes.
  • An outcome is a specific result of an experiment.
  • The sample space is the set of all possible outcomes.
• Events:
  • An event is a set of one or more outcomes of an experiment.
  • The probability of an event is a number between 0 and 1 that represents the likelihood of the event occurring.
• Independent Events:
  • Events are independent if the occurrence of one event does not affect the probability of another event.
  • The probability of independent events is multiplied to find the probability of both events occurring.

AM-GM Inequality

• Statement:
  • The arithmetic mean (AM) of a set of non-negative numbers is greater than or equal to the geometric mean (GM) of the same set of numbers.
• Formula:
  • AM = (a1 + a2 + ... + an) / n
  • GM = √(a1 * a2 * ... * an)
• Applications:
  • Used to prove various inequalities in mathematics, such as the Cauchy-Schwarz Inequality.
  • Has applications in optimization problems, geometry, and other areas of mathematics.

Algebra

• A variable is a symbol representing a value that can change.
• An algebraic expression is a combination of variables, constants, and mathematical operations.

Equations and Inequalities

• An equation is a statement that two expressions are equal, and solving it involves finding the value(s) of the variable(s) that make the statement true.
• An inequality is a statement that one expression is greater than, less than, or equal to another, and solving it involves finding the value(s) of the variable(s) that make the statement true.

Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• Linear equations can be solved using addition, subtraction, multiplication, and division.

Quadratic Equations

• A quadratic equation is an equation in which the highest power of the variable(s) is 2.
• Quadratic equations can be solved using factoring, quadratic formula, or completing the square.

Combinatorics

Permutations

• A permutation is an arrangement of objects in a specific order.
• The number of permutations of n objects is n! (n factorial).

Combinations

• A combination is a selection of objects without regard to order.
• The number of combinations of n objects taken k at a time is C(n, k) = n!/ (k!(n-k)!).

Pascal's Triangle

• A triangular array of numbers in which each number is the sum of the two numbers directly above it.
• Used to calculate binomial coefficients and combinations.

Probability

Basic Concepts

• An experiment is an action or situation that can produce a set of outcomes.
• An outcome is a specific result of an experiment.
• The sample space is the set of all possible outcomes.

Events

• An event is a set of one or more outcomes of an experiment.
• The probability of an event is a number between 0 and 1 that represents the likelihood of the event occurring.

Independent Events

• Events are independent if the occurrence of one event does not affect the probability of another event.
• The probability of independent events is multiplied to find the probability of both events occurring.

AM-GM Inequality

Statement

• The arithmetic mean (AM) of a set of non-negative numbers is greater than or equal to the geometric mean (GM) of the same set of numbers.

Formula

• AM = (a1 + a2 +...+ an) / n
• GM = √(a1 * a2 ... an)

Applications

• Used to prove various inequalities in mathematics, such as the Cauchy-Schwarz Inequality.
• Has applications in optimization problems, geometry, and other areas of mathematics.

Learn the fundamentals of algebra, including variables and expressions, equations and inequalities. Understand how to solve for variables and grasp the basics of algebraic operations.