About Lesson
-
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.