Introduction to Arithmetic

• Math is the science that deals with the logic of shape, quantity and arrangement
• Math is all around, in everything
• Math is a foundational tool for many aspects of life

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics
• It deals with basic operations on numbers
• These basic operations are addition, subtraction, multiplication, and division
• Arithmetic operations are performed according to a specific order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
• Whole numbers are non-negative integers (0, 1, 2,...)
• Integers include all whole numbers as well as their negative counterparts (... -2, -1, 0, 1, 2,...)
• 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; they have non-repeating, non-terminating decimal representations (e.g., π, √2)
• Real numbers encompass all rational and irrational numbers
• Complex numbers have the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1)

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• Variables are symbols (usually letters) that represent unknown or changeable values
• Constants are values that remain fixed
• Expressions are combinations of variables, constants, and operations
• Equations state the equality of two expressions
• Linear equations can be written in the form ax + b = 0, where a and b are constants and x is a variable
• Quadratic equations can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Systems of equations involve two or more equations with the same variables, solved simultaneously to find the values that satisfy all equations
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Factoring is the process of breaking down a polynomial into simpler expressions that, when multiplied together, give the original polynomial

Geometry

• Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Points are fundamental, dimensionless elements in geometry, usually represented by a dot
• A line is a one-dimensional figure that extends infinitely in both directions
• A plane is a two-dimensional flat surface that extends infinitely in all directions
• Line segments are parts of a line with two endpoints
• Rays are parts of a line with one endpoint, extending infinitely in one direction
• Angles are formed by two rays sharing 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 polygons with three sides and three angles
• 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
• 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)
• Quadrilaterals are polygons with four sides and four angles
• Squares have four equal sides and four right angles
• Rectangles have opposite sides that are equal and four right angles
• Parallelograms have opposite sides that are parallel and equal
• Trapezoids have at least one pair of parallel sides
• Circles are sets of points equidistant from a central point
• The radius is the distance from the center to any point on the circle
• The diameter is the distance across the circle through the center (twice the radius)
• The circumference is the distance around the circle (C = 2πr)
• The area of a circle is the space enclosed within the circle (A = πr²)
• Volume is the measure of the amount of space occupied by a three-dimensional object
• Surface area is the total area of the surfaces of a three-dimensional object

Trigonometry

• Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles
• It is particularly useful for right triangles
• Sine (sin) is the ratio of the length of the opposite side to the length of the hypotenuse
• Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse
• Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side
• Reciprocal trigonometric functions; Cosecant (csc) is the reciprocal of sine, Secant (sec) is the reciprocal of cosine, Cotangent (cot) is the reciprocal of tangent
• The unit circle is a circle with a radius of 1, used to define trigonometric functions for all angles
• Trigonometric identities are equations involving trigonometric functions 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

Calculus

• Calculus is the branch of mathematics that deals with continuous change
• Differential calculus concerns rates of change and slopes of curves
• Integral calculus concerns 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
• The derivative of a function measures the instantaneous rate of change of the function
• It represents the slope of the tangent line to the function's graph at a specific point
• The integral of a function calculates the area under the curve of the function's graph
• It is the reverse process of differentiation
• The Fundamental Theorem of Calculus connects differentiation and integration
• It states that the derivative of the integral of a function is the original function, and the integral of the derivative of a function is the original function (up to a constant)
• Applications of calculus are optimization problems (finding maximum or minimum values), related rates problems, and calculating areas, volumes, and arc lengths

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data
• Descriptive statistics involves methods for summarizing and presenting data, such as mean, median, mode, standard deviation, and histograms
• Inferential statistics involves making inferences and generalizations about a population based on a sample of data
• Probability is the measure of the likelihood that an event will occur
• It is expressed as a number between 0 and 1
• A probability distribution describes the likelihood of different outcomes in a random experiment
• Common distributions include the normal distribution, binomial distribution, and Poisson distribution
• Hypothesis testing is a method for testing a claim or hypothesis about a population based on a sample of data
• Regression analysis is a method for modeling the relationship between a dependent variable and one or more independent variables