Math: Arithmetic and Algebra

Math is the science and study of quantity, structure, space, and change.

• Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

Arithmetic

• Arithmetic is the study of numbers and the basic operations on them.
• Basic arithmetic operations include addition, subtraction, multiplication, and division.
• Addition is the process of finding the total of two or more numbers.
• Subtraction is the process of finding the difference between two numbers.
• Multiplication is the process of finding the product of two or more numbers.
• Division is the process of splitting a number into equal parts.
• The order of operations is a convention used to standardize the sequence in which operations are performed in a mathematical expression: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities.
• Variables are symbols that represent unknown or changing quantities.
• Equations are mathematical statements that show the equality between two expressions.
• Solving an equation involves finding the value(s) of the variable(s) that make the equation true.
• Common algebraic techniques include combining like terms, factoring, and using the distributive property.

Geometry

• Geometry is the study of shapes, sizes, positions, and properties of space.
• Basic geometric shapes include points, lines, angles, triangles, squares, circles, and cubes.
• Area is the measure of the two-dimensional space inside a shape.
• Volume is the measure of the three-dimensional space inside a solid object.
• Perimeter is the distance around a two-dimensional shape.
• Key geometric concepts include congruence, similarity, and symmetry.
• 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 ($a^2 + b^2 = c^2$).

Trigonometry

• Trigonometry is the study of the relationships between the sides and angles of triangles.
• The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan).
• These functions relate the angles of a right triangle to the ratios of its sides.
• Trigonometry is used in various fields, including navigation, engineering, and physics.

Calculus

• Calculus is the study of continuous change.
• It is divided into two main branches: differential calculus and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and areas under curves.
• Key concepts in calculus include limits, derivatives, and integrals.

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Descriptive statistics involves summarizing and presenting data in a meaningful way.
• Inferential statistics involves making inferences and generalizations about a population based on a sample of data.
• Common statistical measures include mean, median, mode, standard deviation, and variance.

Number Theory

• Number theory is a branch of mathematics that studies the properties and relationships of numbers, especially integers.
• Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves.
• The fundamental theorem of arithmetic states that every integer greater than 1 can be uniquely represented as a product of prime numbers.
• Concepts like divisibility, congruences, and Diophantine equations are central to number theory.

Set Theory

• Set theory is a branch of mathematics that studies sets, which are collections of objects.
• Sets can be finite or infinite.
• Elements are the objects that belong to a set.
• Basic set operations include union, intersection, and complement.
• Venn diagrams are used to visually represent sets and their relationships.

Mathematical Logic

• Mathematical logic is the study of formal systems of reasoning.
• It deals with the principles of valid inference and proof.
• Propositional logic and predicate logic are two main branches of mathematical logic.
• Logic is used in computer science, philosophy, and mathematics to formalize arguments and prove theorems.