Concepts in Mathematics

Arithmetic: Basic operations; addition, subtraction, multiplication, division.
Algebra: Use of symbols to represent numbers in equations; solving for unknowns.
Geometry: Study of shapes, sizes, and properties of space; includes points, lines, angles, surfaces.
Calculus: Study of change; includes differentiation (rates of change) and integration (area under curves).
Statistics: Collection, analysis, interpretation, presentation, and organization of data.
Probability: Measure of the likelihood that an event will occur.

Pythagorean Theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Fundamental Theorem of Algebra: Any non-constant polynomial equation has at least one complex root.
Mean Value Theorem: A function that is continuous on [a, b] and differentiable on (a, b) has a point c where the derivative equals the average rate of change.

Natural Numbers: Positive integers (1, 2, 3, ...).
Whole Numbers: Non-negative integers (0, 1, 2, ...).
Integers: Whole numbers including negatives (..., -3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers: Fractions or ratios of integers (e.g., 1/2, 3/4).
Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π).
Real Numbers: All rational and irrational numbers.

Function: Relation where each input has exactly one output.
Linear Function: f(x) = mx + b, where m is the slope and b is the y-intercept.
Quadratic Function: f(x) = ax² + bx + c, represented as a parabola.
Exponential Function: f(x) = a * b^x, where b is a constant.

Angle Types: Acute (< 90°), Right (90°), Obtuse (> 90°, < 180°), Straight (180°).
Triangles: Types include equilateral, isosceles, and scalene.
Area Formulas:
- Triangle: A = 1/2 * base * height
- Circle: A = π * r²
- Rectangle: A = length * width
Volume Formulas:
- Cuboid: V = length × width × height
- Cylinder: V = π * r² * height

Understand the Problem: Read carefully and identify what is needed.
Devise a Plan: Choose appropriate strategies (draw a diagram, create an equation).
Carry Out the Plan: Execute the strategies.
Review/Reflect: Check the solution and understand errors if applicable.

Arithmetic is the foundation of mathematics, focused on basic operations like addition, subtraction, multiplication, and division.
Algebra introduces symbols to represent numbers in equations, allowing for solving for unknowns.
Geometry delves into the study of shapes, sizes, and properties of space, encompassing elements like points, lines, angles, and various surfaces.
Calculus analyzes change using concepts like differentiation for rates of change and integration for calculating areas under curves.
Statistics deals with the collection, analysis, interpretation, presentation, and organization of data.
Probability calculates the likelihood of an event occurring.

Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides.
Fundamental Theorem of Algebra asserts that any non-constant polynomial equation possesses at least one complex root.
Mean Value Theorem states that for a function continuous on a closed interval and differentiable on its open counterpart, there exists a point where the derivative equals the average rate of change.

Commutative Property allows changing the order of operands in addition and multiplication without affecting the result.
Associative Property permits grouping of operands in addition and multiplication without altering the outcome.
Distributive Property applies to combining multiplication and addition, where multiplying the sum of numbers by another number is equivalent to multiplying each of the numbers individually by that number and then adding the results.

Natural Numbers consist of positive integers starting from 1 (e.g., 1, 2, 3, ...).
Whole Numbers include zero in addition to natural numbers (e.g., 0, 1, 2, ...).
Integers encompass all whole numbers, including their negatives (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).
Rational Numbers represent all numbers that can be expressed as a fraction or ratio of integers (e.g., 1/2, 3/4).
Irrational Numbers are non-repeating and non-terminating decimals (e.g., √2, π).
Real Numbers include all rational and irrational numbers.

A function assigns exactly one output to each input.
A linear function is represented by the equation f(x) = mx + b, where m is the slope and b is the y-intercept, yielding a straight line when plotted on a graph.
A quadratic function is represented by the equation f(x) = ax² + bx + c, creating a parabola when graphed.
An exponential function is represented by the equation f(x) = a * b^x, where b is a constant.

Angle Types include acute (less than 90 degrees), right (90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), and straight (180 degrees).
Triangles are classified into equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal).
Area Formulas:
- Triangle: A = 1/2 * base * height
- Circle: A = π * r²
- Rectangle: A = length * width
Volume Formulas:
- Cuboid: V = length × width × height
- Cylinder: V = π * r² * height

Mean calculates the average of a set of numbers.
Median represents the middle value when data is sorted in ascending order.
Mode identifies the most frequently occurring value(s) in a dataset.
Range represents the difference between the highest and lowest values in a dataset.

Understand the Problem: Carefully read and identify the required information.
Devise a Plan: Select appropriate strategies, such as drawing a diagram or constructing an equation.
Carry Out the Plan: Implement the chosen strategies.
Review/Reflect: Verify the solution and analyze any errors that may have occurred.