Basic Concepts in Mathematics

Basic Concepts in Mathematics

• Numbers

  • Natural Numbers: Positive integers (1, 2, 3,...)
  • Whole Numbers: Natural numbers including zero (0, 1, 2,...)
  • Integers: Whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3,...)
  • Rational Numbers: Numbers expressed as a fraction of two integers (e.g., 1/2, -3/4)
  • Irrational Numbers: Non-repeating, non-terminating decimals (e.g., π, √2)
  • Real Numbers: All rational and irrational numbers combined.

• Arithmetic Operations

  • Addition: Combining numbers to get a sum.
  • Subtraction: Finding the difference between numbers.
  • Multiplication: Repeated addition of a number.
  • Division: Splitting a number into equal parts.

• Basic Algebra

  • Variables: Symbols (often x, y) representing numbers.
  • Expressions: Combinations of numbers, variables, and operations (e.g., 2x + 3).
  • Equations: Mathematical statements asserting equality (e.g., 2x + 3 = 7).
  • Solving Equations: Finding the value of variables that make the equation true.

Geometry

• Basic Shapes

  • Triangle: 3 sides, sum of angles = 180°.
  • Square: 4 equal sides, sum of angles = 360°.
  • Rectangle: Opposite sides equal, sum of angles = 360°.
  • Circle: Defined by radius (distance from center) and diameter (distance across).

• Formulas

  • Area of Triangle: ( A = \frac{1}{2} \times base \times height )
  • Area of Rectangle: ( A = length \times width )
  • Area of Circle: ( A = π \times radius^2 )
  • Perimeter of Rectangle: ( P = 2 \times (length + width) )

Measurement

• Units of Measurement
  • Length: meters (m), centimeters (cm), inches (in)
  • Area: square meters (m²), acres, hectares
  • Volume: cubic meters (m³), liters (L), gallons
  • Weight: kilograms (kg), grams (g), pounds (lb)

Basic Statistics

• Measures of Central Tendency

  • Mean: Average value of a set of numbers (sum of all values / number of values).
  • Median: Middle value when data is sorted.
  • Mode: Most frequently occurring value.

• Data Representation

  • Bar Graphs: Visual representation using bars to show frequency.
  • Pie Charts: Circular representation showing proportions.

Functions

• Definition

  • A relationship between two sets that assigns each input exactly one output (e.g., f(x) = x + 2).

• Types of Functions

  • Linear Function: Graph is a straight line, generally in the form ( f(x) = mx + b ).
  • Quadratic Function: Graph is a parabola, generally in the form ( f(x) = ax^2 + bx + c ).

Probability

• Basic Concepts
  • Probability: Measure of the likelihood that an event will occur.
  • Range: Between 0 (impossible) and 1 (certain).
  • Basic Probability Formula: ( P(A) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ outcomes} )

Important Symbols

• Common Mathematical Symbols
  • = : Equals
  • ≠ : Not equal
  • < : Less than

  • : Greater than

  • ≤ : Less than or equal to
  • ≥ : Greater than or equal to
  • ∑ : Summation
  • ∏ : Product

These notes provide a concise summary of fundamental concepts in mathematics, useful for a foundational understanding in the subject.

Numbers

• Natural numbers are positive whole numbers starting from 1 (e.g., 1, 2, 3, ...).
• Whole numbers include zero and all natural numbers (e.g., 0, 1, 2, ...).
• Integers encompass all whole numbers and their negative counterparts (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational numbers are numbers expressible as a fraction of two integers (e.g., 1/2, -3/4).
• Irrational numbers are non-repeating and non-terminating decimals (e.g., π, √2).
• Real numbers encompass all rational and irrational numbers.

Arithmetic Operations

• Addition combines numbers to find their sum.
• Subtraction finds the difference between numbers.
• Multiplication is repeated addition of a number.
• Division splits a number into equal parts.

Basic Algebra

• Variables are symbols, commonly represented by letters like 'x' and 'y', that stand for unknown numbers.
• Expressions combine numbers, variables, and arithmetic operations (e.g., 2x + 3).
• Equations are mathematical statements that express equality (e.g., 2x + 3 = 7).
• Solving equations involves determining the values of variables that make the equation true.

Geometry

• Triangle: A three-sided shape with angles summing to 180°.
• Square: A quadrilateral with four equal sides and angles summing to 360°.
• Rectangle: A quadrilateral with opposite sides equal and angles summing to 360°.
• Circle: Defined by its radius (distance from center to edge) and diameter (distance across the circle).

Formulas

• Area of a triangle:
  • ( A = \frac{1}{2} \times base \times height )
• Area of a rectangle:
  • ( A = length \times width )
• Area of a circle:
  • ( A = π \times radius^2 )
• Perimeter of a rectangle:
  • ( P = 2 \times (length + width) )

Measurement

• Length: Measured in units like meters (m), centimeters (cm), and inches (in).
• Area: Measured in square units like square meters (m²), acres, and hectares.
• Volume: Measured in cubic units like cubic meters (m³), liters (L), and gallons.
• Weight: Measured in units like kilograms (kg), grams (g), and pounds (lb).

Basic Statistics

• Measures of central tendency:
  • Mean: Average value calculated by summing all values and dividing by the total number of values.
  • Median: The middle value in a sorted dataset.
  • Mode: The most frequently occurring value in a dataset.

Data Representation

• Bar graphs: Visual representation using bars to indicate the frequency of data.
• Pie charts: Circular representation showing proportions of data.

Functions

• Definition: A relationship between two sets that assigns a unique output for every input.
• Types of functions:
  • Linear Function: Graph represented by a straight line, typically written as ( f(x) = mx + b ).
  • Quadratic Function: Graph represented by a parabola, typically written as ( f(x) = ax^2 + bx + c ).

Probability

• Probability: Measures the possibility of an event occurring.
• Range: Values range from 0 (impossible) to 1 (certain).
• Basic Formula:
  • ( P(A) = \frac{number\ of\ favorable\ outcomes}{total\ number\ of\ outcomes} )

Important Symbols

• =: Equals
• ≠: Not equal
• <: Less than
• >: Greater than
• ≤: Less than or equal to
• ≥: Greater than or equal to
• ∑: Summation
• ∏: Product