Number Sets and Properties

Overview

This lecture introduces sets of real numbers, their properties, and foundational algebraic concepts, including order of operations and combining like terms.

Sets of Numbers

• A set is a collection of objects; elements are the objects in the set.
• Natural numbers (ℕ): {1, 2, 3, ...}; used for counting.
• Whole numbers (𝕎): {0, 1, 2, 3, ...}; natural numbers plus zero.
• Integers (ℤ): {..., -2, -1, 0, 1, 2, ...}; whole numbers and their negatives.
• Rational numbers (ℚ): numbers expressible as an integer divided by a nonzero integer (e.g., 7/18, -5, 0.333...).
• Irrational numbers (𝕀): numbers not expressible as a ratio of integers; their decimals are non-terminating and non-repeating (e.g., π, -√2).
• Real numbers (ℝ): the set containing all rational and irrational numbers.

Relationships Among Sets

• ℕ ⊆ 𝕎 ⊆ ℤ ⊆ ℚ ⊆ ℝ.
• Irrational numbers are disjoint from rational numbers but both are within real numbers.

Properties of Real Numbers

• Closure: Sums and products of real numbers are real numbers.
• Commutativity: a + b = b + a, a × b = b × a.
• Associativity: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c).
• Distributive property: a × (b + c) = a × b + a × c.
• Identity properties: a + 0 = a (additive identity), a × 1 = a (multiplicative identity).
• Multiplicative property of zero: a × 0 = 0.
• Additive inverse: a + (−a) = 0.
• Multiplicative inverse: a × (1/a) = 1 for a ≠ 0 (also called reciprocal).

The Real Number Line & Absolute Value

• Numbers increase from left to right; negative numbers are left of zero.
• Absolute value |a| is the distance from a to 0; |a| = a if a ≥ 0, |a| = -a if a < 0.
• Distance between x and y: |x − y|.

Exponents (Powers)

• aⁿ = a × a × ... × a (n times), where n is a natural number.
• Base: the number being multiplied; exponent: the number of times it appears.
• Parentheses affect the result: (−3)² = 9 vs. −3² = −9.

Order of Operations

• 1. Grouping symbols (parentheses, brackets, absolute value), innermost first.
• 2. Exponents.
• 3. Multiplication and division (left to right).
• 4. Addition and subtraction (left to right).

Plugging in Values & Simplifying Expressions

• Replace variables with their given values, respect order of operations.
• Combine like terms: only terms with same variables and exponents can be combined.
• Use distributive property to expand products and then combine like terms.

Key Terms & Definitions

• Set — a collection of distinct objects (elements).
• Natural numbers (ℕ) — counting numbers starting at 1.
• Whole numbers (𝕎) — natural numbers plus zero.
• Integers (ℤ) — whole numbers and their negatives.
• Rational numbers (ℚ) — numbers expressible as a ratio of integers.
• Irrational numbers (𝕀) — real numbers not rational, with infinite non-repeating decimals.
• Real numbers (ℝ) — all rational and irrational numbers.
• Absolute value — the nonnegative distance from zero on the real number line.
• Exponent (power) — indicates how many times to multiply the base by itself.
• Like terms — terms with the exact same variable(s) and exponent(s).

Action Items / Next Steps

• Practice identifying and classifying numbers into their correct sets.
• Complete assigned homework on simplifying algebraic expressions and order of operations.