Number Systems and Operations

Integers

• A set of whole numbers, including positive, negative, and zero
• Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Properties:
  • Closure: The result of adding, subtracting, multiplying, or dividing integers is always an integer.
  • Commutativity: The order of integers in an operation does not change the result.
  • Associativity: The order in which integers are grouped in an operation does not change the result.

Fractions

• A way to represent part of a whole
• Consists of a numerator (top number) and a denominator (bottom number)
• Examples: 1/2, 3/4, 2/3
• Properties:
  • Equivalent fractions: Fractions that have the same value but different numerators and denominators.
  • Simplification: Fractions can be simplified by dividing both numerator and denominator by their greatest common divisor.

Decimals

• A way to represent fractions in a decimal form
• Examples: 0.5, 0.25, 0.333...
• Properties:
  • Terminating decimals: Decimals that end in a finite number of digits (e.g., 0.5)
  • Non-terminating decimals: Decimals that do not end in a finite number of digits (e.g., 0.333...)

Rational and Irrational Numbers

• Rational numbers:
  • Can be expressed as the ratio of two integers (fractions)
  • Examples: 1/2, 3/4, 22/7
• Irrational numbers:
  • Cannot be expressed as the ratio of two integers
  • Examples: π, e, √2

Non-Terminating and Non-Recurring Decimals

• Non-terminating decimals:
  • Decimals that do not end in a finite number of digits
  • Examples: 0.333..., 0.14159...
• Non-recurring decimals:
  • Decimals that do not have a repeating pattern
  • Examples: π, e, √2

Non-Terminating and Recurring Decimals

• Non-terminating decimals:
  • Decimals that do not end in a finite number of digits
  • Examples: 0.333..., 0.14159...
• Recurring decimals:
  • Decimals that have a repeating pattern
  • Examples: 0.12341234..., 0.45674567...