Fundamental Mathematical Concepts PDF

DISCRETE STRUCTURES 1 CS 110  DISCRETE DATA DISCRETE STRUCTURES 1 Definition: Consists of distinct, CS 110 separate values. Characteristics: Countable, Discrete structu...

DISCRETE STRUCTURES 1 CS 110  DISCRETE DATA DISCRETE STRUCTURES 1 Definition: Consists of distinct, CS 110 separate values. Characteristics: Countable, Discrete structures are often whole numbers. mathematical Examples: Number of students frameworks used to in a class, dice rolls, shoe sizes. study distinct, separate CONTINUOUS DATA objects like graphs, sets, Definition: Can take any value and logical statements, within a range. essential for computer Characteristics: Measurable, can science and algorithm include fractions and decimals. design. Examples: Height, temperature, time. FUNDAMENTAL MATHEMATICAL CONCEPTS Arithmetic Algebra Geometry Calculus FUNDAMENTAL MATHEMATICAL Algebra Geometry Calculus CONCEPTS Addition (+) : Combining two Arithmetic numbers to get a sum. Arithmetic is the branch of mathematics dealing with basic Subtraction (−): Finding the operations like addition, subtraction, difference between two multiplication, and division on numbers. numbers. It forms the foundation for more advanced mathematical concepts. BASIC OPERATION PROPERTIES Commutative: Order doesn't Multiplication (×): Repeated matter. addition of a number. Associative: Grouping doesn't Division (÷): Splitting a matter. number into equal parts. Distributive: Distributes one operation over another. FUNDAMENTAL MATHEMATICAL Algebra Geometry Calculus Multiplication CONCEPTS Arithmetic Commutative: Addition a×b=b×a Commutative: Associative: a+b=b+a (a × b) × c = a × (b × c) Associative: Distributive over addition: (a + b) + c = a + (b + c) a × (b + c) = (a × b) + (a × c) Division Subtraction Not Commutative: Not Commutative: a÷b≠b÷a a-b≠b-a Not Associative: Not Associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) (a - b) - c ≠ a - (b - c) FUNDAMENTAL MATHEMATICAL Algebra Geometry Calculus CONCEPTS Arithmetic Properties of Numbers Integers: Whole numbers, both positive and negative, including zero. Rational Numbers: Numbers that can be expressed as a 𝑎 fraction where b ≠ 0. 𝑏 Real Numbers: Includes both rational and irrational numbers (numbers that cannot be expressed as a fraction). FUNDAMENTAL MATHEMATICAL Algebra Geometry Calculus CONCEPTS Arithmetic Number Theory Prime Numbers: Numbers greater than 1 with no divisors other than 1 and itself. Greatest Common Divisor (GCD): Largest number that divides two numbers. Least Common Multiple (LCM): Smallest number that is a multiple of two numbers. FUNDAMENTAL MATHEMATICAL Algebra Geometry Calculus CONCEPTS Arithmetic Arithmetic Application Algorithm Development: Basic arithmetic operations are fundamental in designing and analyzing algorithms. Data Processing: Arithmetic operations are used in tasks like data sorting, searching, and manipulation. Graphics: Arithmetic is used in calculations for rendering images, manipulating pixels, and performing transformations. Cryptography: Arithmetic operations, including modular arithmetic, are crucial for encryption and decryption processes. Simulation and Modeling: Arithmetic is employed in simulations, calculations, and mathematical models to represent real-world systems. FUNDAMENTAL MATHEMATICAL Arithmetic Geometry Calculus CONCEPTS Basic Concepts Algebra Variables: Symbols representing Algebra is a branch of numbers or values. mathematics that deals Expressions: Combinations of with symbols and the rules for manipulating those variables and numbers using symbols to solve equations operations (e.g., 3𝑥+2). and represent relationships. It includes working with Equations: Statements that two expressions, equations, and expressions are equal (e.g., functions. 3𝑥+2=11). FUNDAMENTAL MATHEMATICAL Arithmetic Geometry Calculus CONCEPTS Algebra Operations Addition and Subtraction: Combining like terms (e.g., 2𝑥 + 3𝑥 = 5𝑥). Multiplication: Distributive property (e.g., a (b+c) = ab + ac). Division: Simplifying expressions 6𝑥 (e.g., = 2𝑥). 𝟑 FUNDAMENTAL MATHEMATICAL Arithmetic Geometry Calculus CONCEPTS Algebra Functions A relation between a set of inputs and outputs, typically 𝑓(𝑥). Linear Functions: f(x) = mx + b. Quadratic Functions: f(x) = ax2 + bx + c. Exponential Functions: f(x) = a ⋅ bx. FUNDAMENTAL MATHEMATICAL Arithmetic Geometry Calculus CONCEPTS Algebra Algebra Application Algorithm Design: Formulating and solving problems using algebraic equations. Data Structures: Understanding and implementing data structures like arrays and matrices. Cryptography: Applying algebraic concepts for encoding and decoding information. Computer Graphics: Using linear algebra for transformations and rendering graphics. Machine Learning: Employing algebra for optimization and modeling in data analysis. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Calculus CONCEPTS Basic Concepts Geometry Geometry is the branch of Points: The most fundamental mathematics that studies unit in geometry, having no shapes, sizes, and the properties of space. size, only position. Types of Angles Lines: Infinite sets of points Acute Angle: Less than 90 degrees. extending in both directions Right Angle: Exactly 90 degrees. with no thickness. Obtuse Angle: Greater than 90 Planes: Flat, two-dimensional degrees but less than 180 degrees. surfaces extending infinitely. Straight Angle: Exactly 180 degrees. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Calculus CONCEPTS Geometry Shapes and Properties Triangles Types: Equilateral, Isosceles, Scalene. Properties: Sum of interior angles is 180 degrees. Pythagorean Theorem: In right triangles, 𝑎2 + 𝑏2 = 𝑐2. Quadrilaterals Types: Square, Rectangle, Parallelogram, Trapezoid. Properties: Sum of interior angles is 360 degrees. Circles Terms: Radius, Diameter, Circumference, Arc. Properties: Circumference = 2𝜋𝑟, Area = 𝜋𝑟2. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Calculus CONCEPTS Transformations Geometry Translation: Sliding a shape Coordinate Geometry without rotating or flipping. Points on a Plane: Defined by Rotation: Turning a shape coordinates (𝑥,𝑦). around a fixed point. Distance Formula: Reflection: Flipping a shape d= (𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐 over a line. Midpoint Formula: Scaling: Enlarging or reducing 𝒙𝟏 + 𝒙𝟐 𝒚𝟏 + 𝒚𝟐 ( , ). a shape proportionally. 𝟐 𝟐 FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Calculus CONCEPTS Geometry Geometry Application Computer Graphics: Rendering images, 3D modeling, and transformations. Computer-Aided Design (CAD): Creating and manipulating designs and blueprints. Robotics: Path planning and spatial analysis for robot movement. Geographic Information Systems (GIS): Mapping and spatial data analysis. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Geometry CONCEPTS Calculus Basic Concepts Limits: Describes the value that Calculus is the branch of mathematics focused on a function approaches as the rates of change input approaches a point. (differentiation) and accumulation of quantities Notation: 𝒍𝒊𝒎𝒙 → 𝒂 𝒇 𝒙 = 𝑳 (integration). Continuity: A function is continuous at a point if the limit equals the function's value there. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Geometry CONCEPTS Calculus Differentiation Derivative: Measures the rate at which a function changes as its input changes. 𝒅𝒚 Notation: 𝒇′ 𝒙 or 𝒅𝒙 Basic Rules: 𝑑 Power Rule: 𝑥 𝑛 = 𝑛𝑥 𝑛−1 𝑑𝑥 𝑑 Product Rule: 𝑢𝑣 = 𝑢′ 𝑣 + 𝑢𝑣′ 𝑑𝑥 𝑑 𝑢 𝑢′ 𝑣 −𝑢𝑣 ′ Quotient Rule: = 𝑑𝑥 𝑣 𝑣2 𝑑 Chain Rule: 𝑓 𝑔 𝑥 = 𝑓′ 𝑔 𝑥 ∙ 𝑔′(𝑥) 𝑑𝑥 FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Geometry CONCEPTS Calculus Integration Integral: Measures the accumulation of quantities, such as areas under curves. Notation: ‫𝒙𝒅 𝒙 𝒇 ׬‬ Basic Rules: 𝑥 𝑛+1 Power Rule: ‫׬‬ 𝑥𝑛 𝑑𝑥 = + 𝐶 for 𝑛 ≠ −1 𝑛+1 𝑏 Definite Integral: ‫𝑓 𝑎׬‬ 𝑥 𝑑𝑥 gives the area under 𝑓 𝑥 form 𝑎 to 𝑏. FUNDAMENTAL MATHEMATICAL Arithmetic Algebra Geometry CONCEPTS Calculus Calculus Application Optimization: Finding maximum and minimum values in algorithms and models. Computer Graphics: Rendering techniques, including shading and light modeling. Machine Learning: Training algorithms, including gradient descent for optimizing models. Simulation: Modeling and solving differential equations for dynamic systems.