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.

Test your knowledge of the basic concepts of Algebra II with these flashcards. Covering key terms such as closure, integers, and irrational numbers, this quiz will help solidify your understanding of fundamental algebraic principles.