## Introduction to Mathematics Mathematics is the study of quantities, structures, space, and change. It is a fundamental discipline that underpins many aspects of our lives, from the natural sciences to finance, engineering, and technology. This comprehensive set of math notes will provide an overview of the core concepts and techniques in mathematics. ## Number Systems ### Integers Integers are the whole numbers, both positive and negative, including zero. They form the foundation of the number system and are denoted by the symbol ℤ. ### Rational Numbers Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. They are denoted by the symbol ℚ. ### Irrational Numbers Irrational numbers are numbers that cannot be expressed as a ratio of two integers. Examples include π (pi) and √2. Irrational numbers are denoted by the symbol ℝ, which represents the set of all real numbers. ## Algebra ### Variables and Expressions Algebraic expressions involve the use of variables, which represent unknown quantities. These variables are usually represented by letters, such as x, y, or z. ### Equations and Inequalities Equations are mathematical statements that express the equality of two expressions, while inequalities are mathematical statements that express the relationship between two expressions, such as greater than (>), less than (<), or equal to (=). ### Linear Equations Linear equations are equations that can be expressed in the form ax + b = 0, where a and b are constants, and x is the variable. ### Quadratic Equations Quadratic equations are equations that can be expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. ## Geometry ### Shapes and Figures Geometry involves the study of shapes and figures, including points, lines, angles, triangles, quadrilaterals, and other polygons. ### Measurement Geometry also includes the measurement of various properties of shapes and figures, such as length, area, volume, and angle. ### Transformations Geometric transformations, such as translations, reflections, rotations, and dilations, are used to study the properties of shapes and figures. ## Trigonometry ### Trigonometric Functions Trigonometry is the study of the relationships between the sides and angles of triangles. The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). ### Trigonometric Identities Trigonometric identities are equations that are true for all values of the variables involved. They are useful in simplifying and manipulating trigonometric expressions. ### Applications of Trigonometry Trigonometry has many practical applications, including surveying, navigation, engineering, and physics. ## Calculus ### Limits and Continuity Calculus begins with the concept of limits, which are used to study the behavior of functions as they approach a particular value. ### Derivatives Derivatives are used to study the rate of change of a function, and they have many applications in optimization, physics, and other fields. ### Integrals Integrals are used to find the area under a curve, and they are the inverse operation of differentiation. ## Probability and Statistics ### Probability Probability is the study of the likelihood of events occurring. It involves the use of concepts such as sample spaces, events, and conditional probability. ### Statistics Statistics is the study of the collection, organization, analysis, and interpretation of data. It includes concepts such as measures of central tendency, measures of dispersion, and statistical inference. ## Conclusion This comprehensive set of math notes covers the core concepts and techniques in mathematics, from number systems and algebra to geometry, trigonometry, calculus, and probability and statistics. These notes should provide a solid foundation for further study and application of mathematical principles.
