Algebra Overview

Study Notes

Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through symbols, equations, and functions.
There are several branches of algebra, including:
- Elementary algebra: deals with solving linear equations and inequalities, graphing lines and circles, and quadratic equations.
- Intermediate algebra: covers systems of equations, functions, and graphing.
- College algebra: includes advanced topics like polynomial and rational functions, sequences, and series.
- Abstract algebra: focuses on advanced structures like groups, rings, and fields.

Variables and Expressions: a variable is a symbol that represents a value, and an expression is a combination of variables, numbers, and operations.
Equations and Inequalities: an equation is a statement that two expressions are equal, while an inequality is a statement that two expressions are not equal.
Functions: a function is a relation between a set of inputs (domain) and a set of possible outputs (range).

Addition and Multiplication: algebraic operations that combine expressions and variables.
Distributive Property: a property that allows us to multiply a single value to multiple terms inside a parenthesis.
Commutative Property: a property that states that the order of numbers or variables does not change the result of an operation.
Associative Property: a property that states that the order in which we perform operations does not change the result.

Linear Equations: equations in which the highest power of the variable is 1.
Quadratic Equations: equations in which the highest power of the variable is 2.
Systems of Equations: a set of two or more equations that must be solved simultaneously.
Inequality Solving: methods for solving inequalities, including graphing and algebraic methods.

Graphing Lines: methods for graphing linear equations, including slope-intercept form and standard form.
Graphing Circles: methods for graphing circles, including center-radius form and standard form.
Graphing Quadratic Functions: methods for graphing quadratic functions, including vertex form and standard form.

Science and Engineering: algebra is used to model and solve problems in physics, engineering, computer science, and other fields.
Data Analysis: algebra is used to analyze and interpret data in statistics, economics, and social sciences.
Cryptography: algebra is used to develop secure encryption methods and codes.

Algebra studies variables and relationships through symbols, equations, and functions.
The branches of algebra are: elementary algebra, intermediate algebra, college algebra, and abstract algebra.

A variable is a symbol representing a value, and an expression is a combination of variables, numbers, and operations.
An equation is a statement that two expressions are equal, while an inequality is a statement that two expressions are not equal.
A function is a relation between a set of inputs (domain) and a set of possible outputs (range).

Algebraic operations include addition and multiplication of expressions and variables.
The Distributive Property allows multiplying a single value to multiple terms inside a parenthesis.
The Commutative Property states that the order of numbers or variables does not change the result of an operation.
The Associative Property states that the order in which we perform operations does not change the result.

Linear equations have a highest power of 1, quadratic equations have a highest power of 2.
Systems of equations involve solving two or more equations simultaneously.
Inequality solving methods include graphing and algebraic methods.

Algebra is used in science and engineering to model and solve problems.
Algebra is used in data analysis to analyze and interpret data in statistics, economics, and social sciences.
Algebra is used in cryptography to develop secure encryption methods and codes.