Basics of Arithmetic

• Math is the science and study of quantity, structure, space, and change.
• It uses patterns to formulate new conjectures and establishes truth by rigorous deduction from suitably chosen axioms and definitions.

Arithmetic

• Arithmetic involves the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication, and division.
• Addition is the most basic operation of arithmetic as simple combining of similar objects, for example, three apples plus two apples equals five apples.
• Subtraction is the inverse of addition; if 3 + 2 = 5, then 5 – 2 = 3.
• Multiplication can be thought of as repeated addition; 3 x 4 is the same as 4 + 4 + 4.
• Division is the inverse of multiplication.
• The order of operations is a convention used to standardize how mathematical expressions are evaluated; PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember the order.
• Fractions represent a part of a whole and consist of a numerator (top number) and a denominator (bottom number).
• Decimals are another way of representing fractions, using a base-10 system.

Algebra

• Algebra is a generalization of arithmetic, in which letters and symbols are used to represent numbers, quantities, and relationships.
• Variables are symbols (usually letters) that represent unknown or changing quantities.
• Expressions are combinations of variables, numbers, and operations, like 3x + 2y – 5.
• Equations are statements that two expressions are equal, such as x + 5 = 10.
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations can be written in the form ax + b = 0.
• Quadratic equations can be written in the form ax² + bx + c = 0.
• Systems of equations involve two or more equations with the same variables; the solution is the set of values that satisfy all equations simultaneously.

Geometry

• Geometry deals with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry is based on a set of axioms and deals with shapes like triangles, squares, circles, and their properties.
• A point is a location in space with no dimension.
• A line is a straight path that extends infinitely in both directions.
• A plane is a flat surface that extends infinitely in all directions.
• Angles are formed by two rays sharing a common endpoint (vertex).
• Triangles are three-sided polygons classified by their angles (acute, right, obtuse) and sides (equilateral, isosceles, scalene).
• The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
• Circles are sets of points equidistant from a center point.
• The distance around a circle is called the circumference.
• The distance across a circle through the center is called the diameter.
• Area is the measure of the amount of space inside a two-dimensional shape.
• Volume is the measure of the amount of space inside a three-dimensional object.

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles.
• Sine (sin), cosine (cos), and tangent (tan) are the primary trigonometric functions.
• These functions relate angles of a right triangle to the ratios of its sides.
• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent
• The unit circle is a circle with a radius of 1 used to define trigonometric functions for all real numbers.

Calculus

• Calculus is the study of continuous change.
• Differential calculus deals with rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and the areas under and between curves.
• A limit is the value that a function "approaches" as the input approaches some value.
• A derivative measures the instantaneous rate of change of a function.
• An integral is the reverse operation of a derivative and is used to find the area under a curve.

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Probability is the measure of the likelihood that an event will occur.
• Descriptive statistics involves methods for organizing, summarizing, and presenting data in a meaningful way.
• Inferential statistics involves drawing conclusions and making predictions based on a sample of data.
• Mean is the average of a set of numbers.
• Median is the middle value in a sorted set of numbers.
• Mode is the value that appears most frequently in a set of numbers.
• Range is the difference between the largest and smallest values in a set of numbers.
• Standard deviation measures the spread of data around the mean.
• A probability distribution describes the likelihood of different outcomes in a random experiment.