Introduction to Mathematics: Arithmetic and Algebra

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be used as a tool for quantifying, measuring and describing the world

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, combines two numbers into a single number, the sum.
• Subtraction is the inverse of addition; it finds the difference between two numbers.
• Multiplication combines two numbers to produce a third number, the product. It can be seen as repeated addition.
• Division is the inverse of multiplication; it finds how many times one number is contained in another.

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic, using variables (letters) to represent numbers or quantities.
• Algebraic expressions consist of numbers, variables, and arithmetic operations.
• Equations state the equality between two expressions.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear equations have the form ax + b = 0, where a and b are constants and x is the variable.
• Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
• Systems of equations involve two or more equations with the same variables.

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 postulates, including the parallel postulate.
• Key concepts include points, lines, planes, angles, and shapes.
• Triangles are fundamental shapes, classified by their sides (equilateral, isosceles, scalene) and angles (acute, obtuse, right).
• The Pythagorean theorem relates the sides of a right triangle: a^2 + b^2 = c^2, where c is the hypotenuse.
• Circles are sets of points equidistant from a center. Key concepts include radius, diameter, circumference, and area.
• Solid geometry extends plane geometry to three dimensions, dealing with shapes like cubes, spheres, cylinders, and cones.

Trigonometry

• Trigonometry studies the relationships between the angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) relate angles to ratios of sides in right triangles.
• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.
• Trigonometric identities are equations that are true for all values of the variables.
• The unit circle is a circle with radius 1, used to define trigonometric functions for all angles.

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 areas under and between curves.
• Limits describe the behavior of a function as its input approaches a specific value.
• Derivatives measure the instantaneous rate of change of a function.
• Integrals calculate the area under a curve.
• The Fundamental Theorem of Calculus connects differentiation and integration.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics summarize and describe the main features of a dataset.
• Inferential statistics use data to make inferences and generalizations about a population.
• Key concepts include mean, median, mode, variance, standard deviation, and probability.
• Probability measures the likelihood of an event occurring.
• Distributions describe the possible values and probabilities of a variable.
• The normal distribution is a common distribution with a bell-shaped curve.

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous.
• Logic is the study of reasoning and argumentation.
• Set theory deals with collections of objects, called sets.
• Relations and functions describe how elements of sets are related to each other.
• Graph theory studies the properties of graphs, which are mathematical structures used to model pairwise relations between objects.
• Combinatorics involves counting and arranging objects.

Number Theory

• Number theory studies the properties and relationships of numbers, especially integers.
• Prime numbers are integers greater than 1 that are divisible only by 1 and themselves.
• Divisibility rules provide shortcuts for determining whether a number is divisible by another number.
• Modular arithmetic deals with remainders after division.
• Congruences are equations that state that two numbers have the same remainder when divided by a given modulus.

Mathematical Proofs

• A mathematical proof is a logical argument that establishes the truth of a statement.
• Direct proofs start with known facts and use logical steps to arrive at the desired conclusion.
• Indirect proofs (proof by contradiction) assume the negation of the statement and show that it leads to a contradiction.
• Mathematical induction is a method for proving statements that hold for all natural numbers.
• Proofs often involve definitions, axioms, theorems, and logical inference rules.

Coordinate Geometry

• Coordinate geometry uses a coordinate system to represent and analyze geometric shapes.
• The Cartesian coordinate system uses two perpendicular axes (x-axis and y-axis) to define the position of points in a plane.
• Equations of lines can be written in various forms such as slope-intercept form (y = mx + b) and point-slope form.
• The distance formula calculates the distance between two points in the coordinate plane.
• The midpoint formula finds the coordinates of the midpoint of a line segment.
• Conic sections (circles, ellipses, parabolas, hyperbolas) can be described by equations in the coordinate plane.