Algebra Basics

Study Notes

Algebra

Definition

Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through the use of symbols, equations, and functions.

Key Concepts

Variables and Constants: Letters or symbols that represent unknown values or quantities.
Algebraic Expressions: Combinations of variables, constants, and mathematical operations.
Equations and Inequalities: Statements that express the equality or inequality of two algebraic expressions.
Functions: Relations between variables, often represented as f(x) or g(y).

Algebraic Operations

Addition and Subtraction: Combining like terms in an algebraic expression.
Multiplication and Division: Expanding and simplifying algebraic expressions.
Factoring: Expressing an algebraic expression as a product of simpler expressions.

Calculus

Definition

Calculus is a branch of mathematics that deals with the study of continuous change, particularly in the context of functions and limits.

Key Concepts

Limits: The behavior of a function as the input (or x-value) approaches a specific point.
Derivatives: Measures of how a function changes as its input changes.
Integrals: Measures of the accumulation of a function over a defined interval.

Calculus Operations

Differentiation: Finding the derivative of a function, often using rules such as the power rule and product rule.
Integration: Finding the integral of a function, often using techniques such as substitution and integration by parts.

Applications of Calculus

Optimization: Finding the maximum or minimum of a function, often used in physics, engineering, and economics.
Physics and Engineering: Modeling real-world phenomena, such as motion, force, and energy.
Economics: Modeling economic systems, including supply and demand, and economic growth.

Algebra

Definition

Branch of mathematics that deals with the study of variables and their relationships.

Key Concepts

Variables represent unknown values or quantities.
Algebraic expressions are combinations of variables, constants, and mathematical operations.
Equations and inequalities are statements that express the equality or inequality of two algebraic expressions.
Functions are relations between variables, often represented as f(x) or g(y).

Algebraic Operations

Addition and subtraction involve combining like terms in an algebraic expression.
Multiplication and division involve expanding and simplifying algebraic expressions.
Factoring involves expressing an algebraic expression as a product of simpler expressions.

Calculus

Definition

Branch of mathematics that deals with the study of continuous change, particularly in the context of functions and limits.

Key Concepts

Limits describe the behavior of a function as the input approaches a specific point.
Derivatives measure how a function changes as its input changes.
Integrals measure the accumulation of a function over a defined interval.

Calculus Operations

Differentiation involves finding the derivative of a function using rules such as the power rule and product rule.
Integration involves finding the integral of a function using techniques such as substitution and integration by parts.

Applications of Calculus

Optimization involves finding the maximum or minimum of a function, often used in physics, engineering, and economics.
Calculus is used to model real-world phenomena in physics and engineering, such as motion, force, and energy.
Calculus is used to model economic systems, including supply and demand, and economic growth.

Rates

A rate is a comparison of two quantities, often expressed as a ratio, fraction, or decimal
Rates can be used to describe various phenomena, such as:
Speed (distance per unit of time)
Heart rate (beats per minute)
Interest rate (percentage per year)

Unit Rates

A unit rate is a rate with a denominator of one unit
Unit rates are used to simplify rates and make comparisons easier
Example: 30 miles per hour is a rate, and 30 miles per 1 hour is a unit rate

Order of Operations

Bedmas is a mnemonic device used to remember the order of operations in mathematics
Bedmas stands for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction
The order of operations is as follows:
- Evaluate expressions inside brackets
- Evaluate exponents
- Evaluate division and multiplication from left to right
- Evaluate addition and subtraction from left to right
Example: 2 × 3 + 12 - 8 = 10 (following the order of operations)