Mathematics Fundamentals and Number Systems

Fundamental Concepts

• Mathematics is a science that studies quantities, structures, space, and change.
• It uses symbols and logical reasoning to represent and manipulate these concepts.
• Key branches of mathematics include arithmetic, algebra, geometry, calculus, and statistics.
• Mathematics is used in various fields such as science, engineering, computer science, and finance.

Number Systems

• Natural numbers (counting numbers): 1, 2, 3, ...
• Whole numbers: 0, 1, 2, 3, ... (includes zero)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (positive and negative whole numbers)
• Rational numbers: numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not zero.
• Irrational numbers: numbers that cannot be expressed as a fraction of two integers. Examples include √2 and π.
• Real numbers: the set of all rational and irrational numbers.
• Complex numbers: numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).

Arithmetic Operations

• Addition (+): Combining quantities.
• Subtraction (-): Finding the difference between quantities.
• Multiplication (× or ⋅): Repeated addition.
• Division (÷ or /): Repeated subtraction or finding how many times one quantity goes into another.
• Exponentiation (): Repeated multiplication.
• Factorials (!): The product of all positive integers less than or equal to a given positive integer.

Algebra

• Represents unknown quantities with variables (letters like x, y, z).
• Uses equations and inequalities to solve problems.
• Simplifying expressions, factoring, solving equations, and applying formulas ( quadratic formula, binomial theorem).

Geometry

• Studies shapes, sizes, positions, and properties of figures in space.
• Different types of figures and shapes like points, lines, angles, triangles, polygons, circles, etc.
• Measures and calculations involving shapes and their properties.
• Concepts such as area, perimeter, volume, surface area, and various geometric theorems (Pythagorean theorem, trigonometric ratios).

Calculus

• Deals with continuous change and its properties.
• Two main branches: differential calculus (study of rates of change) and integral calculus (study of accumulated quantities).
• Derivatives and integrals are fundamental tools to find slopes, areas, volumes, motion, etc.

Statistics

• Deals with collection, analysis, interpretation, and presentation of data.
• Statistical methods include measures of central tendency (mean, median, mode), measures of dispersion (variance, standard deviation), and probability calculations.
• Use of statistical graphs, charts, etc.

Logic

• Fundamental in mathematical thinking.
• Uses deductive and inductive reasoning to prove mathematical statements.
• Logical connectives (and, or, not, if-then, etc).