Branches of Mathematics

Branches of Mathematics

• Arithmetic: Study of numbers and their operations (e.g., addition, subtraction, multiplication, division)
• Algebra: Study of variables and their relationships, often expressed through the use of symbols, equations, and functions
• Geometry: Study of shapes, sizes, and positions of objects
• Calculus: Study of change and motion, including limits, derivatives, and integrals
• Trigonometry: Study of triangles and the relationships between their sides and angles
• Statistics: Study of the collection, analysis, and interpretation of data
• Topology: Study of the properties of shapes and spaces that are preserved under continuous transformations

Key Concepts

• Numbers:
  • Natural numbers: 1, 2, 3, ...
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: fractions (e.g., 3/4, 22/7)
  • Irrational numbers: cannot be expressed as a finite decimal or fraction (e.g., π, e)
  • Real numbers: includes all rational and irrational numbers
• Operations:
  • Binary operations: take two inputs (e.g., 2 + 3, 4 × 5)
  • Unary operations: take one input (e.g., -x, 1/x)
• Equations and Inequalities:
  • Linear equations: degree of 1 (e.g., 2x + 3 = 5)
  • Quadratic equations: degree of 2 (e.g., x^2 + 4x + 4 = 0)
  • Inequalities: statements that compare two expressions using <, >, ≤, or ≥

Theorems and Axioms

• Famous theorems:
  • Pythagorean theorem: a^2 + b^2 = c^2 (right-angled triangles)
  • Fermat's last theorem: no integer solutions for a^n + b^n = c^n (n > 2)
• Axioms: fundamental assumptions that form the basis of mathematics
  • Peano axioms: define the natural numbers
  • Euclid's axioms: define geometry

Branches of Mathematics

• Arithmetic studies numbers and their operations, such as addition, subtraction, multiplication, and division.
• Algebra involves variables and their relationships, often expressed through symbols, equations, and functions.
• Geometry examines shapes, sizes, and positions of objects.
• Calculus explores change and motion, including limits, derivatives, and integrals.
• Trigonometry focuses on triangles and the relationships between their sides and angles.
• Statistics deals with the collection, analysis, and interpretation of data.
• Topology studies the properties of shapes and spaces that are preserved under continuous transformations.

Key Concepts

Numbers

• Natural numbers are 1, 2, 3, and so on.
• Integers are ..., -3, -2, -1, 0, 1, 2, 3, and so on.
• Rational numbers are fractions, such as 3/4 and 22/7.
• Irrational numbers cannot be expressed as a finite decimal or fraction, for example π and e.
• Real numbers include all rational and irrational numbers.

Operations

• Binary operations take two inputs, such as 2 + 3 and 4 × 5.
• Unary operations take one input, such as -x and 1/x.

Equations and Inequalities

• Linear equations have a degree of 1, such as 2x + 3 = 5.
• Quadratic equations have a degree of 2, such as x^2 + 4x + 4 = 0.
• Inequalities are statements that compare two expressions using <, ≤, or ≥.

Theorems and Axioms

Famous Theorems

• The Pythagorean theorem states that a^2 + b^2 = c^2 for right-angled triangles.
• Fermat's last theorem states that there are no integer solutions for a^n + b^n = c^n when n is greater than 2.

Axioms

• Peano axioms define the natural numbers.
• Euclid's axioms define geometry.