Introduction to 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 is the study of numbers and the basic operations on them.
• Basic operations include addition, subtraction, multiplication, and division.
• It forms the foundation for more advanced mathematical studies.
• Whole numbers are basic counting numbers starting from 0: 0, 1, 2, 3, …
• Integers include all whole numbers and their negatives: …, -3, -2, -1, 0, 1, 2, 3, …
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers cannot be expressed as a fraction; examples include √2 and π.
• Real numbers encompass both rational and irrational numbers.
• Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
• Addition is combining two or more numbers to find their total.
• Subtraction is finding the difference between two numbers.
• Multiplication is repeated addition of a number by itself a specified number of times.
• Division is splitting a number into equal parts.
• Order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed.
• Commutative property: a + b = b + a and a × b = b × a.
• Associative property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Distributive property: a × (b + c) = a × b + a × c.

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in formulas and equations.
• Variables are symbols (usually letters) that represent unknown or changing quantities.
• Constants are values that do not change.
• An expression is a combination of variables, numbers, and operations that represents a mathematical quantity.
• An equation is a statement that two expressions are equal.
• Solve equations by isolating the variable on one side.
• Linear equations can be written in the form ax + b = 0, where a and b are constants and x is the variable.
• Quadratic equations can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring is breaking down a polynomial into simpler terms (factors) that, when multiplied together, give the original polynomial.
• Functions are relationships that assign each input value (x) to a unique output value (y).
• Linear functions have a constant rate of change and can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic functions have a parabolic shape and can be written in the form y = ax² + bx + c.
• Exponential functions have the form y = a^x, where a is a constant base.
• Logarithmic functions are the inverse of exponential functions.
• Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

Geometry

• Geometry deals with the properties and relations of points, lines, surfaces, solids, and higher-dimensional analogs.
• A point is a location in space, usually represented by a dot.
• A line is a straight, one-dimensional figure extending infinitely in both directions.
• A plane is a flat, two-dimensional surface that extends infinitely in all directions.
• Parallel lines are lines in a plane that do not intersect.
• Perpendicular lines are lines that intersect at a right angle (90 degrees).
• An angle is the measure of the rotation between two lines or line segments that share a common endpoint (vertex).
• Acute angles are less than 90 degrees.
• Right angles are exactly 90 degrees.
• Obtuse angles are greater than 90 degrees but less than 180 degrees.
• Straight angles are exactly 180 degrees.
• Triangles are three-sided polygons.
• Equilateral triangles have three equal sides and three equal angles (60 degrees each).
• Isosceles triangles have two equal sides and two equal angles.
• Scalene triangles have no equal sides and no equal angles.
• Right triangles have one right angle.
• Pythagorean theorem: In a right triangle, a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
• Quadrilaterals are four-sided polygons.
• Squares have four equal sides and four right angles.
• Rectangles have opposite sides equal and four right angles.
• Parallelograms have opposite sides parallel and equal.
• Trapezoids have at least one pair of parallel sides.
• A circle is a set of points equidistant from a central point.
• Radius is the distance from the center of the circle to any point on the circle.
• Diameter is the distance across the circle through the center (twice the radius).
• Circumference is the distance around the circle: C = 2πr.
• Area is the amount of space inside a two-dimensional shape.
• Volume is the amount of space inside a three-dimensional shape.

Trigonometry

• Trigonometry is the study of the relationships between the angles and sides of triangles.
• Sine (sin) is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
• Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
• Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
• SOH-CAH-TOA is a mnemonic for remembering the trigonometric ratios.
• The unit circle is a circle with a radius of 1, used to define trigonometric functions for all angles.
• Trigonometric identities are equations that are true for all values of the variables.
• The law of sines relates the lengths of the sides of a triangle to the sines of its angles.
• The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
• Trigonometric functions are used to model periodic phenomena such as waves and oscillations.

Calculus

• Calculus is the study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations
• Differential calculus involves the study of rates at which quantities change.
• Derivatives measure the instantaneous rate of change of a function.
• Integral calculus involves the study of the accumulation of quantities.
• Integrals are used to find areas under curves, volumes, and other quantities.
• Limits are used to describe the behavior of a function as it approaches a particular value.
• Continuity is a property of functions that can be graphed without lifting the pen.
• The fundamental theorem of calculus relates differentiation and integration.
• Sequences are ordered lists of numbers.
• Series are sums of sequences.
• Convergence is the property of a sequence or series approaching a finite limit.
• Divergence is the property of a sequence or series not approaching a finite limit.
• Applications of calculus include optimization, related rates, and modeling physical phenomena.

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics involves summarizing and presenting data.
• Inferential statistics involves making inferences and generalizations about a population based on a sample.
• Probability is the measure of the likelihood that an event will occur.
• A random variable is a variable whose value is a numerical outcome of a random phenomenon.
• A probability distribution is a function that describes the likelihood of obtaining the possible values of a random variable.
• The mean is the average of a set of numbers.
• The median is the middle value in a set of numbers.
• The mode is the value that appears most frequently in a set of numbers.
• Variance measures the spread of data around the mean.
• Standard deviation is the square root of the variance.
• Normal distribution is a bell-shaped probability distribution that is symmetric around the mean.
• Hypothesis testing is a method for making decisions about a population based on sample data.
• Confidence intervals provide a range of values within which a population parameter is likely to fall.
• Regression analysis is a method for modeling the relationship between two or more variables.

Discrete Mathematics

• Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.
• Logic is the study of reasoning.
• Propositional logic deals with statements that are either true or false.
• Predicate logic extends propositional logic to include variables and quantifiers.
• Set theory is the study of sets, which are collections of objects.
• Relations are sets of ordered pairs.
• Functions are special types of relations.
• Combinatorics is the study of counting.
• Permutations are arrangements of objects in a specific order.
• Combinations are selections of objects without regard to order.
• Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
• Trees are a special type of graph.
• Algorithms are step-by-step procedures for solving problems.
• Boolean algebra is a branch of algebra dealing with binary variables and logical operations.