Number Systems Basics

Number Systems

Types of Numbers

• Natural Numbers: 1, 2, 3, ... (positive integers)
• Whole Numbers: 0, 1, 2, 3, ... (non-negative integers)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (all positive and negative whole numbers)
• Rational Numbers: fractions, e.g., 3/4, 22/7, 1/2 (can be expressed as a finite decimal)
• Irrational Numbers: non-repeating, non-terminating decimals, e.g., π, e, √2
• Real Numbers: all rational and irrational numbers

Properties of Numbers

• Even Numbers: divisible by 2, e.g., 2, 4, 6, ...
• Odd Numbers: not divisible by 2, e.g., 1, 3, 5, ...
• Prime Numbers: only divisible by 1 and themselves, e.g., 2, 3, 5, 7, ...
• Composite Numbers: not prime, e.g., 4, 6, 8, ...

Operations with Numbers

• Divisibility Rules:
  • A number is divisible by 2 if its last digit is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if its last digit is 0 or 5.
  • A number is divisible by 9 if the sum of its digits is divisible by 9.
• LCM (Least Common Multiple): the smallest number that is a multiple of two or more numbers.
• HCF (Highest Common Factor): the largest number that divides two or more numbers without leaving a remainder.

Number Systems

Types of Numbers

• Natural Numbers: Start from 1 and go up to infinity, excluding 0.
• Whole Numbers: Include 0 and all positive integers.
• Integers: Encompass all positive and negative whole numbers, including 0.
• Rational Numbers: Can be expressed as a finite decimal, e.g., 3/4, 22/7, 1/2.
• Irrational Numbers: Have non-repeating, non-terminating decimals, e.g., π, e, √2.
• Real Numbers: Combine all rational and irrational numbers.

Properties of Numbers

• Even Numbers: Always divisible by 2, with examples like 2, 4, 6, etc.
• Odd Numbers: Never divisible by 2, with examples like 1, 3, 5, etc.
• Prime Numbers: Only divisible by 1 and themselves, with examples like 2, 3, 5, 7, etc.
• Composite Numbers: Not prime, with examples like 4, 6, 8, etc.

Operations with Numbers

• Divisibility Rules:
  • A number is divisible by 2 if its last digit is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if its last digit is 0 or 5.
  • A number is divisible by 9 if the sum of its digits is divisible by 9.
• LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers.
• HCF (Highest Common Factor): The largest number that divides two or more numbers without leaving a remainder.

Algebra

What is Algebra?

• Algebra deals with variables and their relationships, expressed through symbols, equations, and formulas.
• It's used to solve problems and model real-world situations.

Equations and Inequalities

Linear Equations

• Highest power of the variable(s) is 1.
• Example: 2x + 3 = 7

Quadratic Equations

• Highest power of the variable(s) is 2.
• Example: x^2 + 4x + 4 = 0

Inequalities

• Statements that compare two expressions using >, <, ≥, ≤, or ≠.
• Example: 2x + 3 > 5

Functions

Domain and Range

• Domain: Set of input values for a function.
• Range: Set of output values for a function.

Graphs

• Visual representations of functions on a coordinate plane.

Systems of Equations

Solving Methods

• Substitution Method: Solve by substituting one equation into another.
• Elimination Method: Solve by adding or subtracting equations to eliminate one variable.
• Graphical Method: Solve by finding the point of intersection on a graph.

Quadratic Formula

• x = (-b ± √(b^2 - 4ac)) / 2a: Formula for solving quadratic equations of the form ax^2 + bx + c = 0.

Word Problems

Translation and Modeling

• Translation: Translate word problems into algebraic equations.
• Modeling: Use algebra to model real-world situations, such as optimization, growth, and decay.