Key Concepts in Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division).
Algebra: Use of symbols and letters to represent numbers and quantities.
Geometry: Study of shapes, sizes, and properties of space.
Trigonometry: Relations between angles and sides of triangles.
Calculus: Study of change and motion; includes derivatives and integrals.
Statistics: Collection, analysis, interpretation, presentation of data.
Probability: Study of uncertainty and likelihood of events.

Variables: Symbols representing unknown values (e.g., x, y).
Expressions: Combinations of variables and numbers (e.g., 2x + 3).
Equations: Mathematical statements expressing equality (e.g., 2x + 3 = 7).
Functions: Relations between sets, assigning each input to a single output (e.g., f(x) = x^2).

Points, Lines, and Planes: Basic building blocks.
Angles: Measured in degrees; types include acute, right, obtuse.
Shapes:
- Triangles: Three sides, types based on angles (acute, right, obtuse).
- Quadrilaterals: Four-sided figures (e.g., squares, rectangles).
- Circles: Defined by a center and radius.

Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse).
Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a for solving quadratic equations.
Fundamental Theorem of Calculus: Connects differentiation and integration.

Limits: Understanding the behavior of functions as they approach a certain point.
Derivatives: Measure of how a function changes as its input changes.
Integrals: Measure of the area under a curve; used to find total quantities.

Mean, Median, Mode: Measures of central tendency.
Standard Deviation: Measure of data dispersion.
Probability Rules:
- Sum of probabilities in a sample space equals 1.
- Independent events: P(A and B) = P(A) * P(B).

Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Properties:
- Commutative (a + b = b + a)
- Associative ((a + b) + c = a + (b + c))
- Distributive (a(b + c) = ab + ac).

These concepts form the foundation of mathematics and are essential for further study in more advanced topics.

Arithmetic focuses on the basic operations of addition, subtraction, multiplication, and division.
Algebra utilizes symbols and letters to represent numerical quantities and solve for unknown values.
Geometry investigates the properties, shapes, and sizes of spatial objects (like triangles and quadrilaterals).
Trigonometry explores the relationships between angles and the sides of triangles.
Calculus examines change and motion, using concepts like derivatives and integrals.
Statistics involves collecting, analyzing, interpreting, and presenting data.
Probability deals with the uncertainty and likelihood of events.

Variables are symbols that represent unknown numerical values.
Mathematical expressions combine variables and numbers.
Equations express equality between mathematical statements.
Functions are relationships between sets, where each input corresponds to a unique output.

Points, lines, and planes form the fundamental elements of geometry.
Angles are measured in degrees with classifications like acute, right, and obtuse.
Shapes are categorized by their sides and angles:
- Triangles: Have three sides and different types based on their angles.
- Quadrilaterals: Have four sides, with examples including squares and rectangles.
- Circles: Defined by a radius and central point.

Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (longest side) equals the sum of squares of the other two sides (a² + b² = c²).
Quadratic Formula: Solves for the roots (solutions) of quadratic equations in the form ax² + bx + c = 0.
Fundamental Theorem of Calculus: Establishes a fundamental connection between differentiation and integration.

Mean, Median, and Mode: Describe the central tendency of a dataset.
Standard Deviation: Measures the spread or dispersion of data points.
Probability Rules:
- The total probability of all outcomes in a sample space equals 1.
- For independent events, the probability of both events occurring is the product of their individual probabilities.

Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) ensures consistent calculation order.
Properties:
- Commutative Property: The order of operations doesn't affect the outcome of addition or multiplication (a + b = b + a).
- Associative Property: The grouping of numbers in addition or multiplication doesn't alter the result ((a + b) + c = a + (b + c)).
- Distributive Property: Multiplication can be distributed over addition or subtraction (a(b + c) = ab + ac).