Math 1.1 - Real Numbers and Algebra Flashcards

Number Types

• Natural Numbers are positive integers starting from 1, such as 1, 2, 3, and so on.
• Whole Numbers include all natural numbers plus zero, represented as 0, 1, 2, 3, etc.
• Integers consist of all whole numbers and their negatives, e.g., -3, -2, -1, 0, 1, 2, 3.
• Rational Numbers are numbers that can be expressed as fractions (m/n) where m and n are integers and n ≠ 0; they can also appear as terminating or repeating decimals.
• Irrational Numbers cannot be expressed as fractions and are represented as non-repeating, non-terminating decimals.
• Real Numbers encompass all rational and irrational numbers and are categorized as negative, zero, or positive.

Notation and Properties

• Exponential Notation involves the format a^n, where 'a' is the base and 'n' represents the exponent.
• The Order of Operations is essential for evaluating expressions correctly and follows PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebraic Properties

• The Commutative Property of Addition states that numbers can be added in any order (a + b = b + a).
• The Commutative Property of Multiplication indicates that the order of multiplication does not affect the product (ab = ba).
• Subtraction and division are not commutative; the order does matter.
• The Associative Property of Multiplication shows that changing the grouping of factors does not alter the product (a(bc) = (ab)c).
• The Associative Property of Addition implies that changing the grouping of addends does not change the sum (a + (b + c) = (a + b) + c).
• The Distributive Property states that a factor times a sum equals the sum of the factor multiplied by each term (a(b + c) = ab + ac).
• The Identity Property of Addition claims that adding zero to any number keeps the number unchanged (a + 0 = a).
• The Identity Property of Multiplication asserts that multiplying any number by one retains the original number (a * 1 = a).
• The Inverse Property of Addition identifies that a number added to its opposite equals zero (a + (-a) = 0).
• The Inverse Property of Multiplication states that every number has a multiplicative inverse (reciprocal), so a * (1/a) = 1.

Algebra Basics

• A Constant is a value that does not change, such as the '5' in the expression x + 5.
• A Variable represents a factor that can change, exemplified by 'x' in the expression x + 5.
• An Algebraic Expression is a combination of constants and variables linked by algebraic operations including addition, subtraction, multiplication, and division.
• An Equation is a mathematical statement that asserts two expressions are equal.
• A Formula is a specific type of equation that expresses a relationship between constant and variable quantities.