## 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.