Introduction to Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics may be used purely abstractly, or it may be applied to other disciplines to study a related topic.
• Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and more.
• Some areas of mathematics include arithmetic, algebra, geometry, and calculus.

Arithmetic

• Arithmetic involves the study of numbers, especially the properties of the traditional operations between them, such as addition, subtraction, multiplication, division, exponentiation, and extraction of roots.
• Arithmetic operations are governed by the order of operations.
• The order of operations is a convention that dictates the order in which operations should be performed in mathematical expressions.
• The PEMDAS acronym is a common way to remember the order of operations.
• P stands for Parentheses
• E stands for Exponents
• MD stands for Multiplication and Division
• AS stands for Addition and Subtraction
• In the case of multiple operations of equal precedence, the operations are carried out from left to right.

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic.
• Variables are used to represent numbers, allowing for the formulation of general rules and relationships.
• Algebra allows for the solving of equations.
• Equations are mathematical statements asserting the equality of two expressions.
• Equations are solved by applying the same operation to both sides of the equation, maintaining the equality.
• Common forms of algebraic equations include:
• Linear equations, such as y = mx + b, which represents a line
• Quadratic equations, such as ax² + bx + c = 0, which can be solved using the quadratic formula
• Systems of equations, which involve multiple equations with multiple variables that must be solved simultaneously
• Algebra includes the study of polynomials.
• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Factoring polynomials involves expressing a polynomial as a product of simpler polynomials, which is useful for solving equations and simplifying expressions.

Geometry

• Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry is the study of geometry based on a set of axioms formalized by Euclid around 300 BC.
• Euclidean geometry deals with concepts such as points, lines, angles, and shapes, and it forms the basis for much of classical geometry.
• Concepts from geometry:
• Area of a two-dimensional shape is the measure of the surface it covers, examples include squares, circles, and triangles.
• Volume of a three-dimensional object is the measure of the space it occupies, examples include cubes, spheres, and pyramids.
• The Pythagorean theorem states that in a right-angled 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.
• Trigonometry studies the relationships between the angles and sides of triangles.
• Trigonometry concepts: sine, cosine, and tangent functions, are used to solve problems involving angles and distances.
• Analytic geometry combines algebra and geometry by using coordinate systems to represent geometric shapes and solve geometric problems algebraically.

Calculus

• Calculus is a branch of mathematics that deals with continuous change.
• Differential calculus concerns the study of rates at which quantities change.
• Derivatives are used to find the slope of a curve at a given point and to optimize functions.
• Integral calculus concerns the accumulation of quantities, such as areas under curves and volumes.
• Integrals provide methods for finding the area under a curve, the volume of a solid, and the total change of a quantity.
• The Fundamental Theorem of Calculus connects differentiation and integration.
• The Fundamental Theorem of Calculus states that differentiation and integration are inverse operations, which simplifies many calculations.
• Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a certain value.
• Limits are used to define continuity, derivatives, and integrals.