Integers:

• Definition: Integers are whole numbers, both positive and negative, including zero.
• Notation: Denoted by the symbol ℤ.
• Properties:
  • Closure under addition and subtraction: If a and b are integers, then a + b and a – b are also integers.
  • Closure under multiplication: If a and b are integers, then a * b is also an integer.
  • Associative, commutative, and distributive properties hold true.
• Examples: -3, -2, -1, 0, 1, 2, 3, …
• Operations: Addition, subtraction, multiplication, and division (with certain constraints).

Rational Numbers:

• Definition: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
• Notation: Denoted by the symbol ℚ.
• Properties:
  • Closure under addition, subtraction, multiplication, and division (except division by zero).
  • Associative, commutative, and distributive properties hold true.
• Examples: 1/2, -3/4, 5, -7, 0.25, 3.333…
• Operations: Same as integers, plus division (with constraints).

Irrational Numbers:

• Definition: Irrational numbers are numbers that cannot be expressed as the quotient or fraction of two integers. They have non-repeating, non-terminating decimal representations.
• Properties:
  • They are not expressible as the ratio of two integers.
  • They are non-terminating and non-repeating decimals.
  • Examples include the square root of non-perfect squares (e.g., √2, √3), and transcendental numbers like π and e.
• Examples: √2, √3, π, e, etc.
• Operations: Can be involved in operations with rational and real numbers, but their exact values cannot be expressed in finite terms.

Real Numbers:

• Definition: Real numbers include all rational and irrational numbers. They represent any point on the number line.
• Notation: Denoted by the symbol ℝ.
• Properties:
  • Closure under addition, subtraction, multiplication, and division (except division by zero).
  • Associative, commutative, and distributive properties hold true.
  • Real numbers can be positive, negative, or zero.
• Examples: All rational and irrational numbers.
• Operations: Same as rational numbers, including operations with irrational numbers.

Complex Numbers:

• Definition: Complex numbers are numbers that can be expressed in the form a + bi, where “a” and “b” are real numbers, and “i” is the imaginary unit (√(-1)).
• Notation: Denoted by the symbol ℂ.
• Properties:
  • Closure under addition, subtraction, multiplication, and division.
  • They have a real part and an imaginary part.
  • Complex conjugate: If z = a + bi, then its conjugate is denoted as z* = a – bi.
• Examples: 3 + 4i, -2 – i, 5i, etc.
• Operations: Addition, subtraction, multiplication, division, exponentiation, and roots.