Algebra II Unit 1 - Basic Concepts Flashcards

Fundamental Concepts in Algebra II

• Closure: Property of a set that ensures performing an operation on its members always results in a member of the same set.

• Integers (Z): Include numbers {0, +1, −1, +2, −2,...}.

• Irrational Numbers: Real numbers that cannot be represented as a ratio of two integers, denoted as ǭ.

• Natural Numbers (N): Consist of positive integers {1, 2, 3, 4,...}.

• Rational Numbers (Q): Numbers expressed as a fraction a/b where a and b are integers and b ≠ 0.

• Real Numbers (R): Combination of rational and irrational numbers.

• Whole Numbers: Include non-negative integers {0, 1, 2, 3,...}.

Set Notation and Membership

• Union (∪): Combines two sets to include all unique elements from both.

• Member of (∈): Indicates if an element belongs to a set.

• Not a Member of (∉): Indicates if an element does not belong to a set.

• Subset of (⊂): A set where all its elements are contained within another set.

Properties of Number Sets

• Natural Numbers: Closed under addition and multiplication but not under subtraction or division.

• Real and Rational Numbers: Closed under addition, subtraction, multiplication, and division (excluding division by 0).

Key Mathematical Properties

• Additive Identity Element: 0 acts as the identity since a + 0 = a for any number a.

• Additive Inverse Property: Every number a has an inverse (−a) such that a + (−a) = 0.

• Associative Property of Addition: (a + b) + c = a + (b + c) for any numbers a, b, and c.

• Associative Property of Multiplication: (a⋅b)⋅c = a⋅(b⋅c) for any numbers a, b, and c.

• Commutative Property of Addition: a + b = b + a.

• Commutative Property of Multiplication: a⋅b = b⋅a.

• Distributive Property: a(b + c) = ab + ac for any numbers a, b, and c.

• Multiplicative Identity Element: 1 acts as the identity since a⋅1 = a.

• Multiplicative Inverse Property: For any number a ≠ 0, there exists an inverse (1/a) such that a⋅(1/a) = 1.

Properties of Equality

• Reflexive Property: a = a for any number a.

• Symmetric Property: If a = b, then b = a.

• Transitive Property: If a = b and b = c, then a = c.

• Trichotomy Property: For any numbers, either a < b, a = b, or a > b.

Polynomial Concepts

• Binomial: A polynomial with two terms.

• Monomial: A polynomial with one term.

• Polynomial: A mathematical expression containing constants and variables using addition and multiplication.

• Term: A constant, variable, or the product of a constant and variables.

• Trinomial: A polynomial with three terms.

Equations and Inequalities

• Linear Equation: An equation in which each term is constant or the product of a constant and a single variable.

• Linear Inequality: Similar to linear equations, it involves constants or the product of constants and a variable but indicates a range instead of an exact value.

Additional Concepts

• Substitution Property of Equality: If x = y, y can replace x in an expression.

• Graph: A visual representation showing relationships among numbers.

• Discriminant: The expression b² - 4ac which determines the nature of roots in quadratic equations.

• Quadratic Equation: Form ax² + bx + c = y, where a, b, c are real numbers and a ≠ 0.

• Quadratic Formula: Formula to find the roots of a quadratic equation.