Number Sets and Operations

Number Sets

• Natural numbers consist of 1, 2, 3, 4,... Also known as counting numbers.
• Whole numbers include 0, 1, 2, 3, 4,..., encompassing all natural numbers plus zero.
• Integers are all whole numbers, both negative and positive, including zero.

Types of Numbers

• Rational numbers can be expressed as fractions, including integers and finite/recurring decimals.
• Irrational numbers are real numbers that cannot be expressed as a fraction, such as √2 or π.

Properties of Operations

• Commutative property states that the order of numbers does not affect the sum or product (e.g., a + b = b + a).
• Associative property means that the way numbers are grouped does not change their sum or product (e.g., (7 + 4) + 8 = 7 + (4 + 8)).
• Additive identity property indicates adding zero to any number does not change its value (e.g., 4 + 0 = 4).
• Multiplicative identity states that any number multiplied by one remains unchanged (e.g., 4 × 1 = 4).
• Multiplicative property of zero highlights that any number multiplied by zero results in zero (e.g., 4 × 0 = 0).
• Multiplicative property of negative one means any number multiplied by -1 equals its opposite.

Logical Reasoning Properties

• Reflexive property asserts that any number is equal to itself.
• Symmetric property states if a = b, then b = a.
• Transitive property posits if a = b and b = c, then a = c.
• Substitution property allows for replacing one equal value with another in expressions.

Reasoning Techniques

• Deductive reasoning involves forming conclusions based on established facts.
• Counterexample refers to an instance that disproves a statement or claim.
• Open sentences are equations that include a variable, requiring specific values to make them true.
• Inductive reasoning involves making generalizations based on observed patterns.